前言

算法导论专题的最后一部分，也是数据结构中最难最综合的内容。图论的算法基于离散数学的结论，给出了具体的算法实现过程，属于是计算机解决数学问题的一大典型。

并且在编写图论算法中，大量使用了相对高级的数据结构例如不相交集合以及斐波那契堆等，一个良好的数据结构或许决定着算法的运算速度，这个概念在之前是无论如何也不会想到的。也难怪，数据结构的算法中并不会强调关于图论的实现，属于是超出能力范畴了…

邻接表与邻接矩阵

邻接表数据结构

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// 边表结点
typedef struct EdgeNode 
{
    int adjvex;                   // 邻接点域,存储该顶点对应的下标
    EdgeType weight;              // 用于存储权值,对于非网图可以不需要
    struct EdgeNode *next;        // 链域,指向下一个邻接点
} EdgeNode;

// 顶点表结点
typedef struct VextexNode
{
    int data;                       // 顶点域,存储顶点信息
    EdgeNode *next;                 // 边表头指针
    int color;                      // 用于DFS和BFS的颜色标记（白色0：未发现，灰色1：已发现，黑色2：已完成）
    int d;                          // DFS和BFS的发现时间
    int f;                          // DFS的完成时间
    struct VextexNode* predecessor; // BFS和DFS的前驱节点
    EdgeType key;                   // 连接MST最小权重边的权重 或 最短路径权重估计上界
} VextexNode;

// 图邻接表表示法
typedef struct GraphAdjList
{
    VextexNode* adjList;
    int numNodes;       // 图中当前顶点数
    bool weighted;      // 是否为加权图
} GraphAdjList;

邻接矩阵数据结构

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// 图邻接矩阵表示法
typedef struct GraphAdjMatrix
{
    int numNodes;           // 图中当前顶点数
    int* data;              // 储存顶点信息
    EdgeType** adjarr;      // 图的邻接矩阵
    bool weighted;          // 是否为加权图
}GraphAdjMatrix;

相关功能

无向图邻接表边的数量

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// 输出无向图邻接表边的数量
int GraphAdjList_Edge_Count(GraphAdjList* G)
{
    int count_edge = 0;
    EdgeNode* node;
    for (int i = 0; i < G->numNodes; i++)
    {
        node = (&G->adjList[i])->next;
        while (node != NULL)
        {
            count_edge++;
            node = node->next;
        }
    }
    return count_edge;
}

创建图邻接表

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// 创建图邻接表
GraphAdjList* Create_GraphAdjList(int numVertices, bool weighted)
{
    GraphAdjList* G = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    G->numNodes = numVertices;
    G->weighted = weighted;
    G->adjList = (VextexNode*)malloc(sizeof(VextexNode) * numVertices);

    for (int i = 0; i < numVertices; i++)
    {
        G->adjList[i].data = i;
        G->adjList[i].next = NULL;
    }
    return G;
}

图邻接表添加边

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// 图邻接表添加边
void AddEdge_GraphAdjList(GraphAdjList* G, int pre_data, int after_data, EdgeType weight)
{
    EdgeNode* newEdge = (EdgeNode*)malloc(sizeof(EdgeNode));
    newEdge->adjvex = after_data;
    if (G->weighted == 1)
    {
        newEdge->weight = weight;
    }
    else
    {
        newEdge->weight = 1;
    }
    newEdge->next = G->adjList[pre_data].next;
    G->adjList[pre_data].next = newEdge;
}


图邻接表修改边

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// 图邻接表修改边
void ChangeEdge_GraphAdjList(GraphAdjList* G, int pre_data, int after_data, EdgeType change_weight)
{
    EdgeNode* edge;
    edge = G->adjList[pre_data].next;
    while (edge != NULL && edge->adjvex != after_data)
    {
        edge = edge->next;
    }
    if (edge != NULL)
    {
        edge->weight += change_weight;
    }
    else
    {
        AddEdge_GraphAdjList(G, pre_data, after_data, change_weight);   // 没有找到边，添加边
    }
}

输出图邻接表

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// 输出图邻接表
void Print_GraphAdjList(GraphAdjList* graphadjlist)
{
    printf("邻接表:\n");
    if (graphadjlist->weighted == 1)
    {
        printf("加权图\n");
    }
    else
    {
        printf("非加权图\n");
    }
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        printf("节点: %d 边: ", i);
        EdgeNode* p;
        p = graphadjlist->adjList[i].next;
        while (p != NULL)
        {
            printf("(%d, %d)  ", i, p->adjvex);
            if (graphadjlist->weighted == 1)
            {
                printf("[%d]   ", p->weight);
            }
            p = p->next;
        }
        printf("\n");
    }
}

删除图邻接表

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// 删除图邻接表
void Delete_GraphAdjList(GraphAdjList* graphadjlist)
{
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        EdgeNode* p;
        p = graphadjlist->adjList[i].next;
        while (p != NULL)
        {
            graphadjlist->adjList[i].next = p->next;
            free(p);
            p = graphadjlist->adjList[i].next;
        }
    }
    free(graphadjlist->adjList);
    free(graphadjlist);
}

输出图邻接矩阵

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// 输出图邻接矩阵
void Print_GraphAdjMatrix(GraphAdjMatrix* graphadjmatrix)
{
    printf("邻接矩阵:\n");
    if (graphadjmatrix->weighted == 1)
    {
        printf("加权图\n");
    }
    else
    {
        printf("非加权图\n");
    }
    for (int i = 0; i < graphadjmatrix->numNodes; i++)
    {
        for (int j = 0; j < graphadjmatrix->numNodes; j++)
        {
            printf("%d	", graphadjmatrix->adjarr[i][j]);
        }
        printf("\n");
    }
}

删除图邻接表

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// 删除图邻接表
void Delete_GraphAdjMatrix(GraphAdjMatrix* graphadjmatrix)
{
    for (int i = 0; i < graphadjmatrix->numNodes; i++)
	{
		free(graphadjmatrix->adjarr[i]);
	}
	free(graphadjmatrix->adjarr);
    free(graphadjmatrix->data);
    free(graphadjmatrix);
}

邻接矩阵转化图的邻接表

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// 邻接矩阵转化图的邻接表
GraphAdjList* GraphMatrix_Transfor_GraphAdjList(GraphAdjMatrix* graphadjmatrix)
{
    GraphAdjList* graphadjlist;
    graphadjlist = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    graphadjlist->numNodes = graphadjmatrix->numNodes;
    graphadjlist->weighted = graphadjmatrix->weighted;
    graphadjlist->adjList = (VextexNode*)malloc(sizeof(VextexNode) * graphadjlist->numNodes);

    for (int i = 0; i < graphadjmatrix->numNodes; i++)
    {
        graphadjlist->adjList->data = graphadjmatrix->data[i];
        graphadjlist->adjList->next = NULL;
        EdgeNode* last;
        last = NULL;
        for (int j = 0; j < graphadjmatrix->numNodes; j++)
        {
            if (graphadjmatrix->adjarr[i][j] != 0)
            {
                EdgeNode* edge;
                edge = (EdgeNode*)malloc(sizeof(EdgeNode));
                edge->adjvex = j;
                edge->weight = graphadjmatrix->adjarr[i][j];
                edge->next = NULL;
                if (last == NULL)
                {
                    graphadjlist->adjList[i].next = edge;
                }
                else
                {
                    last->next = edge;
                }
                last = edge;
            }
        }
    }
    
    return graphadjlist;
}

邻接表转化邻接矩阵

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// 邻接表转化邻接矩阵
GraphAdjMatrix* GraphAdjList_Transfor_GraphMatrix(GraphAdjList* graphadjlist)
{
    GraphAdjMatrix* graphadjmatrix;
    graphadjmatrix = (GraphAdjMatrix*)malloc(sizeof(GraphAdjMatrix));
    graphadjmatrix->numNodes = graphadjlist->numNodes;
    graphadjmatrix->weighted = graphadjlist->weighted;
    graphadjmatrix->adjarr = (EdgeType **)calloc(graphadjmatrix->numNodes, sizeof(EdgeType *));
    for (int i = 0; i < graphadjmatrix->numNodes; i++)
	{
		graphadjmatrix->adjarr[i] = (EdgeType *)calloc(graphadjmatrix->numNodes, sizeof(EdgeType));
	}

    for (int i = 0; i < graphadjmatrix->numNodes; i++)
    {
        EdgeNode* node = graphadjlist->adjList[i].next;
        while (node != NULL)
        {
            graphadjmatrix->adjarr[i][node->adjvex] = node->weight;
            node = node->next;
        }
    }
    
    return graphadjmatrix;
}

广度优先遍历/深度优先遍历

链队列数据结构及功能

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//链队列结构类型
typedef struct QueueNode
{
	VextexNode* node;
	QueueNode* next;
}QueueNode;

typedef struct LinkQueue
{
	QueueNode* front;//队头指针
	QueueNode* rear;//队尾指针
}LinkQueue;

LinkQueue* InitLinkQueue()
{
	LinkQueue* q;
	q = (LinkQueue*)malloc(sizeof(LinkQueue));
	q->front = (QueueNode*)malloc(sizeof(QueueNode));
	if (q->front == NULL)
	{
		printf("开辟空间失败\n");
		exit(0);
	}
	q->rear = q->front;
	q->front->next = NULL;
	q->front->node = NULL;
	return q;
}

//空队	1-非空	0-空
bool EmptyLinkQueue(LinkQueue* q)
{
	if (q->front == q->rear)
	{
		return 0;
	}
	else
	{
		return 1;
	}
}

//入队
void EnLinkQueue(LinkQueue* q, VextexNode* e)
{
	QueueNode* p;
	p = (QueueNode*)malloc(sizeof(QueueNode));
	if (p == NULL)
	{
		printf("开辟空间失败\n");
		exit(0);
	}
	p->node = e;
	p->next = NULL;
	q->rear->next = p;
	q->rear = p;
}

//出队
VextexNode* DeLinkQueue(LinkQueue* q)
{
	if (EmptyLinkQueue(q) == 0)
	{
		printf("队空\n");
        return NULL;
	}
	else
	{
		VextexNode* node;
		QueueNode* queuenode;
		queuenode = q->front->next;
		node = queuenode->node;
		q->front->next = queuenode->next;
		if (queuenode == q->rear)
		{
			q->rear = q->front;
		}
		free(queuenode);
		return node;
	}
}

广度优先遍历

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// 广度优先遍历
void BFS_GraphAdjList(GraphAdjList* graphadjlist, VextexNode* s)
{
    // 先标记
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        if (i != s->data)
        {
            graphadjlist->adjList[i].color = 0;
            graphadjlist->adjList[i].predecessor = NULL;
            graphadjlist->adjList[i].d = -1;
        }
    }
    
    // 入队s
    s->color = 1;
    s->d = 0;
    s->predecessor = NULL;
    LinkQueue* Q;
    Q = InitLinkQueue();
    EnLinkQueue(Q, s);

    while (EmptyLinkQueue(Q) == 1)
    {
        VextexNode* u;
        u = DeLinkQueue(Q);
        EdgeNode* p;
        p = u->next;
        while (p != NULL)
        {
            VextexNode* v;
            v = &(graphadjlist->adjList[p->adjvex]);
            // 发现为标记节点
            if(v->color == 0)
            {
                v->color = 1;
                v->d = u->d + 1;
                v->predecessor = u;
                EnLinkQueue(Q, v);
            }
            p = p->next;
        }
        u->color = 2;   //标黑
    }
}

打印广度优先树

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// 打印广度优先树
void Print_BFTree(GraphAdjList* graphadjlist, VextexNode* s)
{
    printf("广度优先遍历树:\n");
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        printf("节点: %d 距离 %d 最短路径为 %d  : %d ", i, s->data, graphadjlist->adjList[i].d, i);
        VextexNode* p;
        p = graphadjlist->adjList[i].predecessor;
        while (p != NULL)
        {
            printf("<-- %d  ", p->data);
            p = p->predecessor;
        }
        printf("\n");
    }
}

深度优先遍历

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// 深度优先访问
void DFS_Visit(GraphAdjList* graphadjlist, VextexNode* u, int* time)
{
    // u被发现，时间戳+1
    (*time) ++;
    u->d = *time;
    u->color = 1;
    
    EdgeNode* p;
    p = u->next;
    while (p != NULL)
    {
        VextexNode* v;
        v = &(graphadjlist->adjList[p->adjvex]);
        if (v->color == 0)
        {
            v->predecessor = u;
            DFS_Visit(graphadjlist, v, time);
        }
        p = p->next;
    }
    
    //全部扫描完成
    u->color = 2;
    (*time)++;
    u->f = *time;

}

// 深度优先遍历
void DFS_GraphAdjList(GraphAdjList* graphadjlist)
{
    // 先标记
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        graphadjlist->adjList[i].color = 0;
        graphadjlist->adjList[i].predecessor = NULL;
    }
    int time;
    time = 0;   // 时间戳

    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        if (graphadjlist->adjList[i].color == 0)
        {
            DFS_Visit(graphadjlist, &(graphadjlist->adjList[i]), &time);
        }
    }
}

打印深度优先树

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// 打印深度优先树
void Print_DFTree(GraphAdjList* graphadjlist)
{
    printf("深度优先遍历树:\n");
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        printf("节点: %d 发现时间为: %d 终止时间为: %d ,深度优先树路径为: %d ", i, graphadjlist->adjList[i].d, graphadjlist->adjList[i].f, i);
        VextexNode* p;
        p = graphadjlist->adjList[i].predecessor;
        if (graphadjlist->weighted == true)
        {
            printf(",路径权重为: %d  ", graphadjlist->adjList[i].key);
        }

        printf("%d  ", graphadjlist->adjList[i].data);
        while (p != NULL)
        {
            printf("<-- %d   ", p->data);
            p = p->predecessor;
        }
        printf("\n");
    }
}

最小生成树

不相交集合数据结构及功能

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// 不相交集合森林表示
typedef struct Set_Tree_Node
{
    VextexNode* x;
    struct Set_Tree_Node* p;
    int rank;
}Set_Tree_Node;

// 建立新的树
Set_Tree_Node* Make_Set_Tree(VextexNode* x)
{
    Set_Tree_Node* t;
    t = (Set_Tree_Node*)malloc(sizeof(Set_Tree_Node));
    t->rank = 0;
    t->x = x;
    t->p = t;

    return t;
}

// 路径压缩
Set_Tree_Node* Find_Set_Tree(Set_Tree_Node* t)
{
    if (t->p != t)
    {
        t->p = Find_Set_Tree(t->p);
    }
    return t->p;
}

// 按秩合并 
void Union_Set_Tree(Set_Tree_Node* x, Set_Tree_Node* y)
{
    Set_Tree_Node* t1, *t2;
    t1 = Find_Set_Tree(x);
    t2 = Find_Set_Tree(y);

    if (t1 != t2)
    {
        if (t1->rank > t2->rank)
        {
            t2->p = t1;
        }
        else
        {
            t1->p = t2;
            if (t1->rank == t2->rank)
            {
                t2->rank ++;
            }
        }
    }
}

边数据结构

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// 边表示
typedef struct Edge
{
    int pre_data;
    int afte_data;
    EdgeType weight;
}Edge;

// 图邻接表转化边链表
Edge** nonDirected_GraphAdjList_Transfor_EdgeList(GraphAdjList* G)
{
    Edge** edgelist;
    
    edgelist = (Edge**)malloc(sizeof(Edge*) * GraphAdjList_Edge_Count(G)/2);
    int j = 0;
    for (int i = 0; i < G->numNodes; i++)
    {
        EdgeNode* node;
        node = (&G->adjList[i])->next;
        while (node != NULL)
        {
            if (node->adjvex > i)
            {
                edgelist[j] = (Edge*)malloc(sizeof(Edge));
                edgelist[j]->pre_data = i;
                edgelist[j]->afte_data = node->adjvex;
                edgelist[j]->weight = node->weight;
                j++;
            }
            node = node->next;
        }
    }

    return edgelist;
}

随机快速排序

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//(随机)快速排序
void Swap_Array_ij(Edge** a, int num, int i, int j)
{
	int t1, t2;
	t1 = a[i]->pre_data;
	a[i]->pre_data = a[j]->pre_data;
	a[j]->pre_data = t1;

	t2 = a[i]->afte_data;
	a[i]->afte_data = a[j]->afte_data;
	a[j]->afte_data = t2;

    EdgeType t;
	t = a[i]->weight;
	a[i]->weight = a[j]->weight;
	a[j]->weight = t;
}

int Random_Partition(Edge** a, int p, int r)
{
	//随机主元
	int random_num = rand() % (r - p + 1) + p;
	EdgeType x = a[random_num]->weight;
	Swap_Array_ij(a, r - p + 1, random_num, r);

	int i = p - 1;
	for (int j = p; j < r; j++)
	{
		if (a[j]->weight <= x)
		{
			i++;
			//交换i与j
			Swap_Array_ij(a, r - p + 1, i, j);
		}
	}
	//交换i+1与r
	Swap_Array_ij(a, r - p + 1, i + 1, r);

	return i + 1;
}

void Random_Quick_Sort(Edge** a, int p, int r)
{
	if (p < r)
	{
		int q;
		q = Random_Partition(a, p, r);
		Random_Quick_Sort(a, p, q - 1);
		Random_Quick_Sort(a, q + 1, r);
	}
}

Kruskal算法

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// Kruskal算法
GraphAdjList* MST_Kruskal(GraphAdjList* G)
{
    GraphAdjList* A;
    A = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    A->numNodes = G->numNodes;
    A->weighted = G->weighted;
    A->adjList = (VextexNode*)malloc(sizeof(VextexNode) * G->numNodes);
    for (int i = 0; i < A->numNodes; i++)
    {
        A->adjList[i].data = G->adjList[i].data; // 设置顶点数据
        A->adjList[i].next = NULL; // 初始化为空
    }

    Set_Tree_Node** set_list;
    set_list = (Set_Tree_Node**)malloc(sizeof(Set_Tree_Node*) * A->numNodes);

    for (int i = 0; i < A->numNodes; i++)
    {
        set_list[i] = Make_Set_Tree(&G->adjList[i]);
    }
    
    Edge** edgelist;
    edgelist = nonDirected_GraphAdjList_Transfor_EdgeList(G);
    int edge_count = GraphAdjList_Edge_Count(G)/2;
    Random_Quick_Sort(edgelist, 0, edge_count - 1);

    int finish_count = 0;
    for (int i = 0; i < edge_count; i++)
    {
        if ((Find_Set_Tree(set_list[edgelist[i]->pre_data]) != Find_Set_Tree(set_list[edgelist[i]->afte_data])) && finish_count < A->numNodes - 1)
        {
            // 将边添加入A中
            AddEdge_GraphAdjList(A, edgelist[i]->pre_data, edgelist[i]->afte_data, edgelist[i]->weight);
            AddEdge_GraphAdjList(A, edgelist[i]->afte_data, edgelist[i]->pre_data, edgelist[i]->weight);

            // 合并
            Union_Set_Tree(set_list[edgelist[i]->pre_data], set_list[edgelist[i]->afte_data]);
            finish_count++;
        }
    }

    // 删除边链表
    for (int i = 0; i < edge_count; i++)
    {
        free(edgelist[i]);
    }
    free(edgelist);
    free(set_list);
    return A;
}

斐波那契堆数据结构及功能

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// 斐波那契堆节点
typedef struct Fib_Heap_Node
{
    VextexNode* x;
    Fib_Heap_Node* p;       // 父节点
    Fib_Heap_Node* child;   // 某一个孩子节点
    Fib_Heap_Node* left;    // 左兄弟
    Fib_Heap_Node* right;   // 右兄弟
    int degree;             // 孩子数目（度）
    bool mark;              // 标记
}Fib_Heap_Node;

// 斐波那契堆
typedef struct Fib_Heap
{
    int size;
    Fib_Heap_Node* min_node;
}Fib_Heap;

// 创建斐波那契堆
Fib_Heap* Create_Fib_Heap()
{
    Fib_Heap* H;
    H = (Fib_Heap*)malloc(sizeof(Fib_Heap));
    H->min_node = NULL;
    H->size = 0;
    return H;
}

// 递归删除斐波那契堆
void Free_Fib_Heap_Node(Fib_Heap_Node* node)
{
    if (node != NULL)
    {
        Fib_Heap_Node* start = node;
        do {
            Fib_Heap_Node* child = node->child;
            Fib_Heap_Node* next = node->right;

            // 递归释放子节点
            Free_Fib_Heap_Node(child);

            // 释放当前节点
            free(node);

            node = next;
        } while (node != start);
    }
}

void Free_Fib_Heap(Fib_Heap* H)
{
    if (H != NULL) {
        // 释放所有根节点
        Free_Fib_Heap_Node(H->min_node);

        // 释放堆结构本身
        free(H);
    }
}

// 斐波那契堆树根的个数
int Fib_Heap_Root_Count(Fib_Heap* H)
{
    if (H->min_node == NULL)
    {
        return 0;
    }
    else
    {
        int count = 1;
        Fib_Heap_Node* p;
        p = H->min_node->right;
        while (p != H->min_node)
        {
            count++;
            p = p->right;
        }
        return count;
    }
}

// 插入根节点   插在最小节点与根节点右边节点之间
void Fib_Heap_Insert_Root(Fib_Heap* H, Fib_Heap_Node* node)
{  
    // 只有一个根
    if (H->min_node->right == H->min_node)
    {
        H->min_node->right = node;
        H->min_node->left = node;
        node->left = H->min_node;
        node->right = H->min_node;
    }
    else
    {
        node->p = NULL;
        Fib_Heap_Node* p;
        p = H->min_node->right;

        H->min_node->right = node;
        p->left = node;
        node->left = H->min_node;
        node->right = p;
    }
    node->p = NULL;
}

// 插入节点
Fib_Heap_Node* Fib_Heap_Insert(Fib_Heap* H, VextexNode* x)
{
    Fib_Heap_Node* node;
    node = (Fib_Heap_Node*)malloc(sizeof(Fib_Heap_Node));
    node->x = x;
    node->child = NULL;
    node->degree = 0;
    node->mark = 0;

    // 堆空
    if (H->min_node == NULL)
    {
        H->min_node = node;
        node->p = NULL;
        node->left = node;
        node->right = node;
    }
    else
    {
        // 插入节点为根
        Fib_Heap_Insert_Root(H, node);

        if (x->key < H->min_node->x->key)
        {
            H->min_node = node; // 改变min指针
        }
    }
    H->size ++;

    return node;
}

// 合并堆
Fib_Heap* Fib_Heap_Union(Fib_Heap* H1, Fib_Heap* H2)
{
    Fib_Heap* H;
    H = Create_Fib_Heap();
    H->min_node = H1->min_node;
    H->size = H1->size + H2->size;
    // 根合并
    if (H1->min_node == NULL)
    {
        // 跳过
    }
    else if (H->min_node->right == H->min_node)   // H1只有一棵树
    {
        if (H2->min_node->right == H2->min_node)   // H2只有一棵树
        {
            H->min_node->right = H2->min_node;
            H->min_node->left = H2->min_node;
            H2->min_node->right = H->min_node;
            H2->min_node->left = H->min_node;
        }
        else
        {
            Fib_Heap_Insert_Root(H2, H->min_node);
        }
    }
    else    // H1有多棵树
    {
        if (H2->min_node->right == H2->min_node)   // H2只有一棵树
        {
            Fib_Heap_Insert_Root(H, H2->min_node);
        }
        else
        {
            Fib_Heap_Node* p,* q;
            p = H->min_node->right;
            q = H2->min_node->left;

            H->min_node->right = H2->min_node;
            H2->min_node->left = H->min_node;
            q->right = p;
            p->left = q;
        }
    }

    if ((H1->min_node == NULL) || (H2->min_node && H2->min_node->x->key < H1->min_node->x->key))
    {
        H->min_node = H2->min_node;
    }
    free(H1);
    free(H2);
    return H;
}

// 链接根节点   把y接为x的儿子
void Fib_Heap_Link(Fib_Heap* H, Fib_Heap_Node* y, Fib_Heap_Node* x)
{
    // 把y从根节点中移除
    y->left->right = y->right;
    y->right->left = y->left;

    // 把y填入x的孩子中
    y->p = x;
    Fib_Heap_Node* p;
    p = x->child;
    if (p == NULL)  // x原来没孩子
    {
        x->child = y;
        y->left = y;
        y->right = y;
    }
    else
    {
        p->right->left = y;
        y->right = p->right;
        p->right = y;
        y->left = p;
    }
    x->degree ++;

    y->mark = 0;
}

// 合并根链表
void Fib_Heap_Consolidate(Fib_Heap* H)
{
    Fib_Heap_Node** A;
    A = (Fib_Heap_Node**)malloc(sizeof(Fib_Heap_Node*) * H->size);
    for (int i = 0; i < H->size; i++)
    {
        A[i] = NULL;
    }
    Fib_Heap_Node** root_list;
    int count;
    count = Fib_Heap_Root_Count(H);
    Fib_Heap_Node* p;
    p = H->min_node;
    Fib_Heap_Node* x, *y;
    int d;
    for (int i = 0; i < count; i++)
    {
        x = p;
        p = p->right;
        d = x->degree;  // 当前节点的度数
        while (A[d] != NULL)
        {
            y = A[d];
            // 比谁小，然后交换
            if (x->x->key > y->x->key)
            {
                Fib_Heap_Node* t;
                t = x;
                x = y;
                y = t;
            }
            Fib_Heap_Link(H, y, x);
            A[d] = NULL;    //清空
            d++;    //存入下一个
        }
        A[d] = x;
    }

    // 根据 A 重建 H
    H->min_node = NULL;
    for (int i = 0; i < H->size; i++)
    {
        if (A[i] != NULL)
        {
            if (H->min_node == NULL)
            {
                H->min_node = A[i];
                H->min_node->right = H->min_node;
                H->min_node->left = H->min_node;
            }
            else
            {
                Fib_Heap_Insert_Root(H, A[i]);
                if (A[i]->x->key < H->min_node->x->key)
                {
                    H->min_node = A[i];
                }
            }
        }
    }
    
    free(A);
}

// 提取最小节点
VextexNode* Fib_Heap_Extract_Min(Fib_Heap* H)
{
    Fib_Heap_Node* z;
    z = H->min_node;
    if (z != NULL)
    {
        if (z->child != NULL)
        {
            // 令z的孩子都为根
            Fib_Heap_Node* child, *next;
            child = z->child;
            for (int i = 0; i < z->degree; i++)
            {
                next = child->right;
                Fib_Heap_Insert_Root(H, child);
                child = next;
            }
        }

        VextexNode* element;
        element = z->x;

        // z为唯一节点
        if (H->size == 1)
        {
            H->min_node = NULL;
        }
        else
        {
            // 从斐波那契堆移除z
            z->left->right = z->right;
            z->right->left = z->left;

            H->min_node = z->right;
            Fib_Heap_Consolidate(H);
        }
        H->size--;

        free(z);
        return element;
    }
    else
    {
        printf("空堆");
        return NULL;
    }
}

// 切断 切断x与父节点y，使x为根节点
void Fib_Heap_Cut(Fib_Heap* H, Fib_Heap_Node* x, Fib_Heap_Node* y)
{
    // x为y的孩子指针
    if (y->child == x)
    {
        // y只有一个孩子
        if (x->right == x)
        {
            y->child = NULL;
            Fib_Heap_Insert_Root(H, x);
        }
        else
        {
            y->child = x->right;
            x->right->left = x->left;
            x->left->right = x->right;
            Fib_Heap_Insert_Root(H, x);
        }
    }
    else
    {
        x->right->left = x->left;
        x->left->right = x->right;
        Fib_Heap_Insert_Root(H, x);
    }
    x->mark = 0;
}

// 级联切断
void Fib_Heap_Cascading_Cut(Fib_Heap* H, Fib_Heap_Node* y)
{
    Fib_Heap_Node* z;
    z = y->p;
    // y非根节点
    if (z != NULL)
    {
        if (y->mark == 0)
        {
            y->mark = 1;
        }
        else
        {
            Fib_Heap_Cut(H, y, z);
            Fib_Heap_Cascading_Cut(H, z);
        }
    }
}

// 关键字减值
void Fib_Heap_Decrease_Node(Fib_Heap* H, Fib_Heap_Node* x, EdgeType k)
{
    if (k > x->x->key)
    {
        printf("不能让值变大\n");
    }
    else
    {
        x->x->key = k;
        Fib_Heap_Node *y;
        y = x->p;

        if (y != NULL && x->x->key < y->x->key)
        {
            Fib_Heap_Cut(H, x, y);
            Fib_Heap_Cascading_Cut(H, y);
        }
        if (x->x->key < H->min_node->x->key)
        {
            H->min_node = x;
        }
    }
}

Prim算法

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// Prim算法
GraphAdjList* MST_Prim(GraphAdjList* G, VextexNode* r)
{
    GraphAdjList* A;
    A = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    A->numNodes = G->numNodes;
    A->weighted = G->weighted;
    A->adjList = (VextexNode*)malloc(sizeof(VextexNode) * G->numNodes);

    EdgeNode* node;
    node = G->adjList[r->data].next;
    if (node == NULL)
    {
        printf("图不连通\n");
        Delete_GraphAdjList(A);
        return NULL;
    }
    else
    {
        bool *vextexlist;   // 标记已经发现的顶点
        vextexlist = (bool*)calloc(G->numNodes, sizeof(bool));

        Fib_Heap_Node** heap_node_list;
        heap_node_list = (Fib_Heap_Node**)malloc(sizeof(Fib_Heap_Node*) * G->numNodes);
        for (int i = 0; i < G->numNodes; i++)
        {
            heap_node_list[i] = NULL;
            (&G->adjList[i])->key = INT_MAX;
        }

        Fib_Heap* H;
        H = Create_Fib_Heap();
        r->key = 0;
        heap_node_list[r->data] = Fib_Heap_Insert(H, r);
        
        while (H->min_node != NULL)
        {
            VextexNode* u;
            u = Fib_Heap_Extract_Min(H);
            vextexlist[u->data] = 1;

            EdgeNode* edge;
            edge = u->next;
            while (edge != NULL)
            {
                if (vextexlist[edge->adjvex] != 1 && edge->weight <= (&G->adjList[edge->adjvex])->key)
                {
                    if (heap_node_list[edge->adjvex] == NULL)
                    {
                        (&G->adjList[edge->adjvex])->key = edge->weight;
                        heap_node_list[edge->adjvex] = Fib_Heap_Insert(H, &G->adjList[edge->adjvex]);
                    }
                    else
                    {
                        Fib_Heap_Decrease_Node(H, heap_node_list[edge->adjvex], edge->weight);
                    }
                }
                edge = edge->next;
            }

            if (u != r)
            {
                edge = u->next;
                while (edge != NULL)
                {
                    if (vextexlist[edge->adjvex] == 1 && edge->weight == u->key) 
                    {
                        break;
                    }
                    edge = edge->next;
                }
                // 将边添加入A中
                AddEdge_GraphAdjList(A, u->data, edge->adjvex, edge->weight);
                AddEdge_GraphAdjList(A, edge->adjvex, u->data, edge->weight);
            }
        }
        Free_Fib_Heap(H);
        free(vextexlist);
        free(heap_node_list);

        return A;
    }
}

单源最短路径

松弛边

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// 松弛边 u -> v
void Relax_Edge(VextexNode* u, VextexNode* v, EdgeType weight)
{
    if (u->key != INT_MAX && v->key > u->key + weight)
    {
        v->key = u->key + weight;
        v->predecessor = u;
    }
}

Bellman_Ford算法

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// Bellman_Ford算法
bool Bellman_Ford(GraphAdjList* G, VextexNode* s)
{
    // 初始化
    for (int i = 0; i < G->numNodes; i++)
    {
        G->adjList[i].predecessor = NULL;
        G->adjList[i].key = INT_MAX;
    }
    s->key = 0;

    for (int k = 0; k < G->numNodes - 1; k++)
    {
        for (int i = 0; i < G->numNodes; i++)
        {
            EdgeNode* edge = G->adjList[i].next;
            while (edge != NULL)
            {
                Relax_Edge(&G->adjList[i], &G->adjList[edge->adjvex], edge->weight);
                edge = edge->next;
            }
        }
    }

    // 检查是否有负环路
    for (int i = 0; i < G->numNodes; i++)
    {
        EdgeNode* edge;
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            if(G->adjList[edge->adjvex].key > G->adjList[i].key + edge->weight)
            {
                return false;
            }
            edge = edge->next;
        }
    }
    return true;
}

Dijkstra算法

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// Dijkstra算法
void Dijkstra(GraphAdjList* G, VextexNode* s)
{
    // 初始化
    for (int i = 0; i < G->numNodes; i++)
    {
        G->adjList[i].predecessor = NULL;
        G->adjList[i].key = INT_MAX;
    }
    s->key = 0;

    Fib_Heap_Node** heap_node_list;
    heap_node_list = (Fib_Heap_Node**)malloc(sizeof(Fib_Heap_Node*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        heap_node_list[i] = NULL;
    }

    // 与s连接的点集合
    Fib_Heap* H;
    H = Create_Fib_Heap();
    heap_node_list[s->data] = Fib_Heap_Insert(H, s);

    while (H->min_node != NULL)
    {
        VextexNode* u;
        u = Fib_Heap_Extract_Min(H);

        EdgeNode* edge;
        edge = u->next;
        while (edge != NULL)
        {
            if (heap_node_list[edge->adjvex] != NULL)
            {
                Relax_Edge(u, &G->adjList[edge->adjvex], edge->weight);
                if ((&G->adjList[edge->adjvex])->predecessor == u)
                {
                    Fib_Heap_Decrease_Node(H, heap_node_list[edge->adjvex], u->key + edge->weight);
                }
            }
            else
            {
                G->adjList[edge->adjvex].key = u->key + edge->weight;
                G->adjList[edge->adjvex].predecessor = u;
                heap_node_list[edge->adjvex] = Fib_Heap_Insert(H, &G->adjList[edge->adjvex]);
            }
            edge = edge->next;
        }
    }
    Free_Fib_Heap(H);
    free(heap_node_list);
}

节点对最短路径

最短路径数据结构及功能

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typedef struct Graph_Shortest_Paths
{
    int numNodes;   // 节点个数
    EdgeType** D;   // 路径权重
    int** Pai;        // 前驱图
}Graph_Shortest_Paths;

// 删除节点对最短路径
void Delete_Graph_Shortest_Paths(Graph_Shortest_Paths* P)
{
    for (int i = 0; i < P->numNodes; i++)
    {
        free(P->D[i]);
        free(P->Pai[i]);
    }
    
    free(P);
}

// 输出节点对最短路径
void Print_Graph_Shortest_Paths(Graph_Shortest_Paths* P)
{
    printf("节点对最短路径权重\n");
    for (int i = 0; i < P->numNodes; i++)
    {
        for (int j = 0; j < P->numNodes; j++)
        {
            printf("%d  ", P->D[i][j]);
        }
        printf("\n");
    }
    
    printf("节点对最短路径前驱\n");
    for (int i = 0; i < P->numNodes; i++)
    {
        for (int j = 0; j < P->numNodes; j++)
        {
            printf("%d  ", P->Pai[i][j]);
        }
        printf("\n");
    }
}

// 初始化节点对最短路径
Graph_Shortest_Paths* Init__Graph_Shortest_Paths(EdgeType** W, int numNodes)
{
    Graph_Shortest_Paths* PP;
    PP = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    PP->numNodes = numNodes;

    EdgeType** L;
    int** Pai;
    L = (EdgeType**)malloc(sizeof(EdgeType*) * numNodes);
    Pai = (int**)malloc(sizeof(int*) * numNodes);
    for (int i = 0; i < numNodes; i++)
    {
        L[i] = (EdgeType*)malloc(sizeof(EdgeType) * numNodes);
        Pai[i] = (int*)malloc(sizeof(int) * numNodes);
        for (int j = 0; j < numNodes; j++)
        {
            L[i][j] = W[i][j];
            if (i == j || W[i][j] == INT_MAX)
            {
                Pai[i][j] = -1;
            }
            else
            {
                Pai[i][j] = i;
            }
        }
    }
    
    PP->D = L;
    PP->Pai = Pai;
    return PP;
}

自底向上最短路径权重算法

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// 自底向上最短路径权重算法
Graph_Shortest_Paths* Extend_Shortest_Paths(EdgeType** W, Graph_Shortest_Paths* P)
{
    EdgeType** LL;
    int** PPai;
    LL = (EdgeType**)malloc(sizeof(EdgeType*) * P->numNodes);
    PPai = (int**)malloc(sizeof(int*) * P->numNodes);
    for (int i = 0; i < P->numNodes; i++)
    {
        LL[i] = (EdgeType*)malloc(sizeof(EdgeType) * P->numNodes);
        PPai[i] = (int*)malloc(sizeof(int) * P->numNodes);
        for (int j = 0; j < P->numNodes; j++)
        {
            PPai[i][j] = P->Pai[i][j];
        }
    }
        
    Graph_Shortest_Paths* PP;
    PP = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    PP->numNodes = P->numNodes;
    PP->D = LL;
    PP->Pai = PPai;
    
    for (int i = 0; i < P->numNodes; i++)
    {
        for (int j = 0; j < P->numNodes; j++)
        {
            if (i == j)
            {
                LL[i][j] = 0;
            }
            else
            {
                LL[i][j] = INT_MAX;
                for (int k = 0; k < P->numNodes; k++)
                {
                    if (i != j && k != j && P->D[i][k] != INT_MAX && W[k][j] != INT_MAX && P->D[i][k] + W[k][j] < LL[i][j])
                    {
                        LL[i][j] = P->D[i][k] + W[k][j];
                        // 快速算法
                        if (P->D == W)
                        {
                            PPai[i][j] = P->Pai[k][j];
                        }
                        else
                        {
                            PPai[i][j] = k;
                        }
                    }
                }
            }
            
        }
    }

    return PP;
}

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// 慢速算法
Graph_Shortest_Paths* Slow_All_Pairs_Shortest_Paths(GraphAdjList* G)
{
    //构造边权重矩阵
    EdgeType** W;
    W = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        W[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        for (int j = 0; j < G->numNodes; j++)
        {
            if (i == j)
            {
                W[i][j] = 0;
            }
            else
            {
                W[i][j] = INT_MAX;
            }
        }
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            W[i][edge->adjvex] = edge->weight;
            edge = edge->next;
        }
    }

    Graph_Shortest_Paths* P, *Pm, *t;
    P = Init__Graph_Shortest_Paths(W, G->numNodes);

    for (int m = 1; m < G->numNodes - 1; m++)
    {
        Pm = Extend_Shortest_Paths(W, P);
        t = P;
        P = Pm;
        Delete_Graph_Shortest_Paths(t);
    }

    for (int i = 0; i < G->numNodes; i++)
    {
        free(W[i]);
    }
    free(W);

    return P;
}

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// 快速算法
Graph_Shortest_Paths* Faster_All_Pairs_Shortest_Paths(GraphAdjList* G)
{
    //构造边权重矩阵
    EdgeType** W;
    W = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        W[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        for (int j = 0; j < G->numNodes; j++)
        {
            if (i == j)
            {
                W[i][j] = 0;
            }
            else
            {
                W[i][j] = INT_MAX;
            }
        }
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            W[i][edge->adjvex] = edge->weight;
            edge = edge->next;
        }
    }

    Graph_Shortest_Paths* P, *Pm, *t;
    P = Init__Graph_Shortest_Paths(W, G->numNodes);

    int m = 1;
    while (m < G->numNodes)
    {
        Pm = Extend_Shortest_Paths(P->D, P);
        t = P;
        P = Pm;
        Delete_Graph_Shortest_Paths(t);
        m = 2 * m;
    }
    
    for (int i = 0; i < G->numNodes; i++)
    {
        free(W[i]);
    }
    free(W);

    return P;
}

Floyd_Warshall算法

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// Floyd_Warshall算法
Graph_Shortest_Paths* Floyd_Warshall(GraphAdjList* G)
{
    //构造边权重矩阵
    EdgeType** D, **Dn;
    int** Pai, ** PPai;
    D = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    Dn = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    Pai = (int**)malloc(sizeof(int*) * G->numNodes);
    PPai = (int**)malloc(sizeof(int*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        D[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        Dn[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        Pai[i] = (int*)malloc(sizeof(int) * G->numNodes);
        PPai[i] = (int*)malloc(sizeof(int) * G->numNodes);
        for (int j = 0; j < G->numNodes; j++)
        {
            if (i == j)
            {
                D[i][j] = 0;
            }
            else
            {
                D[i][j] = INT_MAX;
            }
            PPai[i][j] = -1;
        }
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            D[i][edge->adjvex] = edge->weight;
            PPai[i][edge->adjvex] = i;
            edge = edge->next;
        }
    }

    Graph_Shortest_Paths* PP;
    PP = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    PP->numNodes = G->numNodes;
    PP->D = D;
    PP->Pai = Pai;

    for (int k = 0; k < G->numNodes; k++)
    {
        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                if (D[i][k] != INT_MAX && D[k][j] != INT_MAX && D[i][k] + D[k][j] < D[i][j])
                {
                    D[i][j] = D[i][k] + D[k][j];
                    Pai[i][j] = PPai[k][j];
                }
                else
                {
                    Pai[i][j] = PPai[i][j];
                }
            }
        }

        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                PPai[i][j] = Pai[i][j];
            }
        }
    }
    
    for (int i = 0; i < G->numNodes; i++)
    {
        free(PPai[i]);
    }
    free(PPai);
    
    return PP;
}

传递闭包

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// 传递闭包
bool** Graph_Transitive_Closure(GraphAdjList* G)
{
    // 初始化T
    bool** T, ** TT;
    T = (bool**)malloc(sizeof(bool*) * G->numNodes);
    TT = (bool**)malloc(sizeof(bool*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        T[i] = (bool*)calloc(G->numNodes, sizeof(bool));
        TT[i] = (bool*)calloc(G->numNodes, sizeof(bool));
        T[i][i] = TT[i][i] = true;
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            T[i][edge->adjvex] = TT[i][edge->adjvex] = true;
            edge = edge->next;
        }
    }

    for (int k = 0; k < G->numNodes; k++)
    {
        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                if (T[i][j] == 0 && (TT[i][k] == true && TT[k][j] == true))
                {
                    T[i][j] = true;
                }
            }
        }
        
        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                TT[i][j] = T[i][j];
            }
        }
    }
    
    for (int i = 0; i < G->numNodes; i++)
    {
        free(TT[i]);
    }
    free(TT);

    return T;
}

Johnson算法

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// Johnson算法
Graph_Shortest_Paths* Graph_Johnson(GraphAdjList* G)
{
    // 增加点
    GraphAdjList* GG;
    GG = Create_GraphAdjList(G->numNodes + 1, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(GG, i, p->adjvex, p->weight);
            p = p->next;
        }
        AddEdge_GraphAdjList(GG, GG->numNodes - 1, i, 0);
    }
    
    Graph_Shortest_Paths* P;
    P = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    P->numNodes = G->numNodes;
    P->D = NULL;
    P->Pai = NULL;

    if (Bellman_Ford(GG, &GG->adjList[GG->numNodes - 1]) == false)
    {
        printf("有负环路\n");
    }
    else
    {
        EdgeType* h;
        h = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        for (int i = 0; i < G->numNodes; i++)
        {
            h[i] = GG->adjList[i].key;
        }
        // 更新边
        GraphAdjList* Gh;
        Gh = Create_GraphAdjList(G->numNodes, true);
        EdgeNode* p;
        for (int i = 0; i < Gh->numNodes; i++)
        {
            p = G->adjList[i].next;
            while (p != NULL)
            {
                AddEdge_GraphAdjList(Gh, i, p->adjvex, p->weight + h[i] - h[p->adjvex]);
                p = p->next;
            }
        }

        P->D = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
        P->Pai = (int**)malloc(sizeof(int*) * G->numNodes);
        for (int i = 0; i < G->numNodes; i++)
        {
            P->D[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
            P->Pai[i] = (int*)malloc(sizeof(int) * G->numNodes);
            for (int j = 0; j < G->numNodes; j++)
            {
                P->D[i][j] = INT_MAX;
            }
            P->Pai[i][i] = -1;
        }

        for (int i = 0; i < Gh->numNodes; i++)
        {
            Dijkstra(Gh, &Gh->adjList[i]);
            for (int j = 0; j < Gh->numNodes; j++)
            {
                P->D[i][j] = Gh->adjList[j].key - h[i] + h[j];
                if (i != j)
                {
                    P->Pai[i][j] = Gh->adjList[j].predecessor->data;
                }
            }
        }
        
        free(h);
        Delete_GraphAdjList(Gh);
    }
    return P;
}

最大流

Ford_Fulkerson

深度优先遍历寻找增广路径

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// 深度优先遍历寻找增广路径
bool DFS_FindAugmentingPath(GraphAdjList* G, int u, int t, bool* visited, int* path)
{
    if (u == t) 
    {
        return true; // 如果到达汇点，返回成功
    }
    visited[u] = 1; // 标记当前节点为已访问

    EdgeNode* edge;
    edge = G->adjList[u].next;
    while (edge != NULL)
    {
        int v = edge->adjvex;
        if (!visited[v] && edge->weight > 0)    // 如果未访问且残存容量大于0
        { 
            path[v] = u; // 记录路径
            if (DFS_FindAugmentingPath(G, v, t, visited, path))
            {
                return true; // 如果找到一条路径，返回成功
            }
        }
        edge = edge->next;
    }
    return false; // 未找到路径，返回失败
}

增广路径的残存容量

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// 增广路径的残存容量
EdgeType Augmenting_Path_Capacity(GraphAdjList* G, int* path, int s, int t)
{
    int path_flow = INT_MAX;
    for (int v = t; v != s; v = path[v])
    {
        // 前驱节点
        int u = path[v];
        // 查找边(u, v)的残存容量
        EdgeNode* edge = G->adjList[u].next;
        while (edge != NULL)
        {
            if (edge->adjvex == v)  //找到节点v
            {
                if (path_flow > edge->weight)
                {
                    path_flow = edge->weight;
                }
                break;
            }
            edge = edge->next;
        }
    }
    return path_flow;
}

Ford_Fulkerson算法

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// Ford_Fulkerson算法
GraphAdjList* Ford_Fulkerson(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    bool* visit_list;
    visit_list = (bool*)malloc(sizeof(bool) * G->numNodes);
    int* pre_path;
    pre_path = (int*)malloc(sizeof(int) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        visit_list[i] = 0;
        pre_path[i] = -1;
    }
    
    EdgeType path_capacity;
    while (DFS_FindAugmentingPath(Ef, s, t, visit_list, pre_path))
    {
        path_capacity = Augmenting_Path_Capacity(Ef, pre_path, s, t);
        // 更新流网络与残差网络
        for (int v = t; pre_path[v] != -1; v = pre_path[v]) // 遍历path路径上的节点
        {
            int u = pre_path[v];
            // 更新流
            EdgeNode* edge;
            edge = G->adjList[u].next;
            while (edge != NULL)
            {
                if (edge->adjvex == v)
                {
                    ChangeEdge_GraphAdjList(Flow, u, v, path_capacity);
                    break;
                }
                edge = edge->next;
            }
            if (edge == NULL)
            {
                ChangeEdge_GraphAdjList(Flow, v, u, -path_capacity);
            }
            
            // 更新残差网络
            ChangeEdge_GraphAdjList(Ef, u, v, -path_capacity);
            ChangeEdge_GraphAdjList(Ef, v, u, path_capacity);
        }
        for (int i = 0; i < G->numNodes; i++)
        {
            visit_list[i] = 0;
            pre_path[i] = -1;
        }
    }
    
    free(visit_list);
    free(pre_path);
    Delete_GraphAdjList(Ef); // 删除残差网络

    return Flow;
}

Edmonds_Karp

广度优先遍历寻找增广路径

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// 广度优先遍历寻找增广路径
bool BFS_FindAugmentingPath(GraphAdjList* G, int s, int t, int* path)
{
    LinkQueue* q = InitLinkQueue(); // 初始化队列
    bool visited[G->numNodes]; // 访问标记数组
    for (int i = 0; i < G->numNodes; i++)
    {
        visited[i] = false;
        path[i] = -1; // 初始化父节点数组
    }

    visited[s] = true; // 标记源点为已访问
    EnLinkQueue(q, &G->adjList[s]); // 源点入队

    while (EmptyLinkQueue(q))   // 队列非空
    {
        VextexNode* uNode = DeLinkQueue(q); // 出队
        int u = uNode->data; // 获取当前顶点编号

        EdgeNode* e = uNode->next; // 获取邻接边
        while (e != NULL)
        {
            int v = e->adjvex; // 邻接顶点编号
            if (!visited[v] && e->weight > 0)   // 如果未访问且边权重（容量）大于0
            {
                visited[v] = true;
                path[v] = u; // 记录路径
                EnLinkQueue(q, &G->adjList[v]); // 邻接顶点入队
                if (v == t) // 如果到达汇点
                { 
                    free(q->front); // 清理队列
                    free(q);
                    return true; // 找到增广路径
                }
            }
            e = e->next; // 移动到下一条边
        }
    }

    free(q->front); // 清理队列
    free(q);
    return false; // 未找到增广路径
}

Edmonds_Karp算法

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// Edmonds_Karp算法
GraphAdjList* Edmonds_Karp(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    int* pre_path;
    pre_path = (int*)malloc(sizeof(int) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        pre_path[i] = -1;
    }
    
    EdgeType path_capacity;
    while (BFS_FindAugmentingPath(Ef, s, t, pre_path))
    {
        path_capacity = Augmenting_Path_Capacity(Ef, pre_path, s, t);
        // 更新流网络与残差网络
        for (int v = t; pre_path[v] != -1; v = pre_path[v]) // 遍历path路径上的节点
        {
            int u = pre_path[v];
            // 更新流
            EdgeNode* edge;
            edge = G->adjList[u].next;
            while (edge != NULL)
            {
                if (edge->adjvex == v)
                {
                    ChangeEdge_GraphAdjList(Flow, u, v, path_capacity);
                    break;
                }
                edge = edge->next;
            }
            if (edge == NULL)
            {
                ChangeEdge_GraphAdjList(Flow, v, u, -path_capacity);
            }
            
            // 更新残差网络
            ChangeEdge_GraphAdjList(Ef, u, v, -path_capacity);
            ChangeEdge_GraphAdjList(Ef, v, u, path_capacity);
        }
        for (int i = 0; i < G->numNodes; i++)
        {
            pre_path[i] = -1;
        }
    }
    
    free(pre_path);
    Delete_GraphAdjList(Ef); // 删除残差网络

    return Flow;
}

重贴标签算法

溢出节点链表

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// 溢出节点链表
typedef struct OverflowingNode{
    int vertexIndex;         // 存储顶点索引
    OverflowingNode* next;   // 指向下一个活跃节点
}OverflowingNode;

// 创建空头节点
OverflowingNode* Init_OverflowingList()
{
    OverflowingNode* head = (OverflowingNode*)malloc(sizeof(OverflowingNode));
    head->vertexIndex = -1; // 空头节点不存储有效的顶点索引
    head->next = NULL;
    return head;
}

// 判断空链表
bool Empty_OverflowingList(OverflowingNode* head)
{
    if (head->next == NULL)
    {
        return true;
    }
    else
    {
        return false;
    }
}

// 头插法添加节点
void Add_OverflowingNode(OverflowingNode* head, int vertexIndex)
{
    OverflowingNode* newNode;
    newNode = (OverflowingNode*)malloc(sizeof(OverflowingNode));
    newNode->vertexIndex = vertexIndex;
    // 插入到头节点之后
    newNode->next = head->next;
    head->next = newNode;
}

// 移除溢出节点
void Remove_OverflowingNode(OverflowingNode* head, int vertexIndex)
{
    OverflowingNode* prev = head;
    OverflowingNode* current = head->next;
    while (current != NULL)
    {
        if (current->vertexIndex == vertexIndex)
        {
            prev->next = current->next;
            free(current);
            break;
        }
        prev = current;
        current = current->next;
    }
}

// 删除溢出节点链表
void Delete_OverflowingList(OverflowingNode* head)
{
    OverflowingNode* current = head;
    while (current != NULL)
    {
        OverflowingNode* temp = current;
        current = current->next;
        free(temp);
    }
}

Push算法

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// Push算法
void Push(GraphAdjList* G, GraphAdjList* Ef, GraphAdjList* Flow, int u, int v)
{
    EdgeNode* edge;
    edge = Ef->adjList[u].next;
    while (edge != NULL)
    {
        if (edge->adjvex == v)
        {
            break;
        }
        edge = edge->next;
    }
    
    EdgeType delta;
    delta = (edge->weight < Ef->adjList[u].key) ? edge->weight : Ef->adjList[u].key;

    edge = G->adjList[u].next;
    while (edge != NULL)
    {
        if (edge->adjvex == v)
        {
            ChangeEdge_GraphAdjList(Flow, u, v, delta);
            ChangeEdge_GraphAdjList(Ef, u, v, -delta);
            ChangeEdge_GraphAdjList(Ef, v, u, delta);
            break;
        }
        edge = edge->next;
    }
    if (edge == NULL)
    {
        ChangeEdge_GraphAdjList(Flow, v, u, -delta);
        ChangeEdge_GraphAdjList(Ef, u, v, -delta);
        ChangeEdge_GraphAdjList(Ef, v, u, delta);
    }

    Ef->adjList[u].key -= delta;
    Ef->adjList[v].key += delta;
}

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// 判断能否使用Push算法
bool exists_Push(GraphAdjList* Ef, int u, int v)
{
    EdgeNode* edge;
    edge = Ef->adjList[u].next;
    while (edge != NULL)
    {
        if (edge->adjvex == v)
        {
            break;
        }
        edge = edge->next;
    }

    if (edge != NULL && edge->weight > 0)
    {
        if (Ef->adjList[u].f == Ef->adjList[v].f + 1)
        {
            return true;
        }
        else
        {
            return false;
        }
    }
    else
    {
        return false;
    }
}

Relabel算法

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// Relabel算法
void Relabel(GraphAdjList* Ef, int u)
{
    int min_height = INT_MAX; // 初始化为最大整数，以确保能找到最小的高度
    for (EdgeNode* edge = Ef->adjList[u].next; edge != NULL; edge = edge->next) // 只考虑残余容量大于0的边
    {
        if (edge->weight > 0)
        {
            int neighbor_height = Ef->adjList[edge->adjvex].f;
            if (neighbor_height < min_height)
            {
                min_height = neighbor_height;
            }
        }
    }
    // 只有当找到至少一个有效的邻接节点时，才更新高度
    if (min_height != INT_MAX)
    {
        Ef->adjList[u].f = min_height + 1;
    }
}

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// 判断能否使用Relabel算法
bool exists_Relabel(GraphAdjList* Ef, int u)
{
    for (EdgeNode* edge = Ef->adjList[u].next; edge != NULL; edge = edge->next)
    {
        if (Ef->adjList[edge->adjvex].f <= Ef->adjList[u].f && edge->weight > 0)
        {
            return false;
        }
    }
    return true;
}

Push_Relabel算法

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// Push_Relabel算法
GraphAdjList* Generic_Push_Relabel(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    // 创建残差网络
    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    // 初始化
    for (int i = 0; i < Ef->numNodes; i++)
    {
        Ef->adjList[i].f = 0;
        Ef->adjList[i].key = 0;
    }
    Ef->adjList[s].f = Ef->numNodes;

    // 溢出节点链表
    OverflowingNode* overflowinglist;
    overflowinglist = Init_OverflowingList();

    // 初始化预流
    EdgeNode* edge;
    edge = Ef->adjList[s].next;
    for (EdgeNode* edge = Ef->adjList[s].next; edge != NULL; edge = edge->next)
    {
        EdgeType flow = edge->weight;
        AddEdge_GraphAdjList(Flow, s, edge->adjvex, flow);
        ChangeEdge_GraphAdjList(Ef, s, edge->adjvex, -flow);
        ChangeEdge_GraphAdjList(Ef, edge->adjvex, s, flow);

        Ef->adjList[edge->adjvex].key = flow;
        Ef->adjList[s].key -= flow;
        Add_OverflowingNode(overflowinglist, edge->adjvex);
    }

    while (!Empty_OverflowingList(overflowinglist))
    {
        OverflowingNode* currentNode;
        currentNode = overflowinglist->next;
        while (currentNode != NULL)
        {
            int u = currentNode->vertexIndex;

            // 尝试对所有邻接节点v进行Push操作
            bool pushed = false;
            for (EdgeNode* edge = Ef->adjList[u].next; edge != NULL; edge = edge->next)
            {
                int v = edge->adjvex;
                if (exists_Push(Ef, u, v))
                {
                    Push(G, Ef, Flow, u, v);
                    pushed = true;

                    // 如果v变成了新的活跃节点，添加到活跃节点列表
                    if (Ef->adjList[v].key > 0 && v != s && v != t)
                    {
                        Add_OverflowingNode(overflowinglist, v);
                    }
                    break; // 只要成功进行一次Push，就跳出循环
                }
            }

            // 如果没有成功进行Push操作，则尝试Relabel
            if (!pushed)
            {
                Relabel(Ef, u);
            }

            // 移动到下一个活跃节点
            currentNode = currentNode->next;

            // 检查u是否仍然是活跃节点
            if (Ef->adjList[u].key <= 0)
            {
                Remove_OverflowingNode(overflowinglist, u); // 移除不再活跃的节点u
            }
        }
    }

    Delete_GraphAdjList(Ef);
    free(overflowinglist);
    return Flow;
}

释放溢出节点

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// 释放溢出节点
void Discharge(GraphAdjList* G, GraphAdjList* Ef, GraphAdjList* Flow, int u)
{
    EdgeNode* currentNode;
    currentNode = Ef->adjList[u].next;
    while (Ef->adjList[u].key > 0)
    {
        if (currentNode == NULL)    // 重贴标签
        {
            Ef->adjList[u].f += 1;
            currentNode = Ef->adjList[u].next;
        }
        else if (Ef->adjList[u].f == Ef->adjList[currentNode->adjvex].f + 1 && currentNode->weight > 0)   // push许可边
        {
            Push(G, Ef, Flow, u, currentNode->adjvex);
        }
        else
        {
            currentNode = currentNode->next;
        }
    }
}

前置重贴标签算法

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// 前置重贴标签算法
GraphAdjList* Relabel_To_Front(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    // 创建残差网络
    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    // 初始化
    for (int i = 0; i < Ef->numNodes; i++)
    {
        Ef->adjList[i].f = 0;
        Ef->adjList[i].key = 0;
    }
    Ef->adjList[s].f = Ef->numNodes;
    
    // 溢出节点链表
    OverflowingNode* OverflowingList;
    OverflowingList = Init_OverflowingList();
    for (int i = 0; i < G->numNodes; i++)
    {
        if (i != s && i != t)
        {
            Add_OverflowingNode(OverflowingList, i);
        }
    }
    
    // 初始化预流
    EdgeNode* edge;
    edge = Ef->adjList[s].next;
    for (EdgeNode* edge = Ef->adjList[s].next; edge != NULL; edge = edge->next)
    {
        EdgeType flow = edge->weight;
        AddEdge_GraphAdjList(Flow, s, edge->adjvex, flow);
        ChangeEdge_GraphAdjList(Ef, s, edge->adjvex, -flow);
        ChangeEdge_GraphAdjList(Ef, edge->adjvex, s, flow);

        Ef->adjList[edge->adjvex].key = flow;
        Ef->adjList[s].key -= flow;
    }
    
    OverflowingNode* u,* pre;
    u = OverflowingList->next;
    pre = OverflowingList;
    int old_height;
    while (u != NULL)
    {
        if (Ef->adjList[u->vertexIndex].key > 0)
        {
            old_height = Ef->adjList[u->vertexIndex].f;
            Discharge(G, Ef, Flow, u->vertexIndex);
            int old_index = u->vertexIndex;

            if (Ef->adjList[u->vertexIndex].f > old_height)
            {
                // 移动u到表头
                pre->next = u->next;
                u->next = OverflowingList->next;
                OverflowingList->next = u;

                pre = u;
                u = u->next;
            }
        }
        else
        {
            u = u->next;
            pre = pre->next;
        }
    }
    
    Delete_GraphAdjList(Ef);
    Delete_OverflowingList(OverflowingList);

    return Flow;
}

总代码

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#include <stdio.h>
#include <stdlib.h>
#include "time.h"
#include <limits.h>
#include <math.h>

typedef int EdgeType;    // 边上的权值类型应由用户定义

// 边表结点
typedef struct EdgeNode 
{
    int adjvex;                   // 邻接点域,存储该顶点对应的下标
    EdgeType weight;              // 用于存储权值,对于非网图可以不需要
    struct EdgeNode *next;        // 链域,指向下一个邻接点
} EdgeNode;

// 顶点表结点
typedef struct VextexNode
{
    int data;                       // 顶点域,存储顶点信息
    EdgeNode *next;                 // 边表头指针
    int color;                      // 用于DFS和BFS的颜色标记（白色0：未发现，灰色1：已发现，黑色2：已完成）
    int d;                          // DFS和BFS的发现时间
    int f;                          // DFS的完成时间
    struct VextexNode* predecessor; // BFS和DFS的前驱节点
    EdgeType key;                   // 连接MST最小权重边的权重 或 最短路径权重估计上界
} VextexNode;

// 图邻接表表示法
typedef struct GraphAdjList
{
    VextexNode* adjList;
    int numNodes;       // 图中当前顶点数
    bool weighted;      // 是否为加权图
} GraphAdjList;

// 图邻接矩阵表示法
typedef struct GraphAdjMatrix
{
    int numNodes;           // 图中当前顶点数
    int* data;              // 储存顶点信息
    EdgeType** adjarr;      // 图的邻接矩阵
    bool weighted;          // 是否为加权图
}GraphAdjMatrix;

// 输出无向图邻接表边的数量
int GraphAdjList_Edge_Count(GraphAdjList* G)
{
    int count_edge = 0;
    EdgeNode* node;
    for (int i = 0; i < G->numNodes; i++)
    {
        node = (&G->adjList[i])->next;
        while (node != NULL)
        {
            count_edge++;
            node = node->next;
        }
    }
    return count_edge;
}

// 创建图邻接表
GraphAdjList* Create_GraphAdjList(int numVertices, bool weighted)
{
    GraphAdjList* G = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    G->numNodes = numVertices;
    G->weighted = weighted;
    G->adjList = (VextexNode*)malloc(sizeof(VextexNode) * numVertices);

    for (int i = 0; i < numVertices; i++)
    {
        G->adjList[i].data = i;
        G->adjList[i].next = NULL;
    }
    return G;
}

// 图邻接表添加边
void AddEdge_GraphAdjList(GraphAdjList* G, int pre_data, int after_data, EdgeType weight)
{
    EdgeNode* newEdge = (EdgeNode*)malloc(sizeof(EdgeNode));
    newEdge->adjvex = after_data;
    if (G->weighted == 1)
    {
        newEdge->weight = weight;
    }
    else
    {
        newEdge->weight = 1;
    }
    newEdge->next = G->adjList[pre_data].next;
    G->adjList[pre_data].next = newEdge;
}

// 图邻接表修改边
void ChangeEdge_GraphAdjList(GraphAdjList* G, int pre_data, int after_data, EdgeType change_weight)
{
    EdgeNode* edge;
    edge = G->adjList[pre_data].next;
    while (edge != NULL && edge->adjvex != after_data)
    {
        edge = edge->next;
    }
    if (edge != NULL)
    {
        edge->weight += change_weight;
    }
    else
    {
        AddEdge_GraphAdjList(G, pre_data, after_data, change_weight);   // 没有找到边，添加边
    }
}

// 输出图邻接表
void Print_GraphAdjList(GraphAdjList* graphadjlist)
{
    printf("邻接表:\n");
    if (graphadjlist->weighted == 1)
    {
        printf("加权图\n");
    }
    else
    {
        printf("非加权图\n");
    }
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        printf("节点: %d 边: ", i);
        EdgeNode* p;
        p = graphadjlist->adjList[i].next;
        while (p != NULL)
        {
            printf("(%d, %d)  ", i, p->adjvex);
            if (graphadjlist->weighted == 1)
            {
                printf("[%d]   ", p->weight);
            }
            p = p->next;
        }
        printf("\n");
    }
}

// 删除图邻接表
void Delete_GraphAdjList(GraphAdjList* graphadjlist)
{
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        EdgeNode* p;
        p = graphadjlist->adjList[i].next;
        while (p != NULL)
        {
            graphadjlist->adjList[i].next = p->next;
            free(p);
            p = graphadjlist->adjList[i].next;
        }
    }
    free(graphadjlist->adjList);
    free(graphadjlist);
}

// 输出图邻接矩阵
void Print_GraphAdjMatrix(GraphAdjMatrix* graphadjmatrix)
{
    printf("邻接矩阵:\n");
    if (graphadjmatrix->weighted == 1)
    {
        printf("加权图\n");
    }
    else
    {
        printf("非加权图\n");
    }
    for (int i = 0; i < graphadjmatrix->numNodes; i++)
    {
        for (int j = 0; j < graphadjmatrix->numNodes; j++)
        {
            printf("%d	", graphadjmatrix->adjarr[i][j]);
        }
        printf("\n");
    }
}

// 删除图邻接表
void Delete_GraphAdjMatrix(GraphAdjMatrix* graphadjmatrix)
{
    for (int i = 0; i < graphadjmatrix->numNodes; i++)
	{
		free(graphadjmatrix->adjarr[i]);
	}
	free(graphadjmatrix->adjarr);
    free(graphadjmatrix->data);
    free(graphadjmatrix);
}

// 邻接矩阵转化图的邻接表
GraphAdjList* GraphMatrix_Transfor_GraphAdjList(GraphAdjMatrix* graphadjmatrix)
{
    GraphAdjList* graphadjlist;
    graphadjlist = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    graphadjlist->numNodes = graphadjmatrix->numNodes;
    graphadjlist->weighted = graphadjmatrix->weighted;
    graphadjlist->adjList = (VextexNode*)malloc(sizeof(VextexNode) * graphadjlist->numNodes);

    for (int i = 0; i < graphadjmatrix->numNodes; i++)
    {
        graphadjlist->adjList->data = graphadjmatrix->data[i];
        graphadjlist->adjList->next = NULL;
        EdgeNode* last;
        last = NULL;
        for (int j = 0; j < graphadjmatrix->numNodes; j++)
        {
            if (graphadjmatrix->adjarr[i][j] != 0)
            {
                EdgeNode* edge;
                edge = (EdgeNode*)malloc(sizeof(EdgeNode));
                edge->adjvex = j;
                edge->weight = graphadjmatrix->adjarr[i][j];
                edge->next = NULL;
                if (last == NULL)
                {
                    graphadjlist->adjList[i].next = edge;
                }
                else
                {
                    last->next = edge;
                }
                last = edge;
            }
        }
    }
    
    return graphadjlist;
}

// 邻接表转化邻接矩阵
GraphAdjMatrix* GraphAdjList_Transfor_GraphMatrix(GraphAdjList* graphadjlist)
{
    GraphAdjMatrix* graphadjmatrix;
    graphadjmatrix = (GraphAdjMatrix*)malloc(sizeof(GraphAdjMatrix));
    graphadjmatrix->numNodes = graphadjlist->numNodes;
    graphadjmatrix->weighted = graphadjlist->weighted;
    graphadjmatrix->adjarr = (EdgeType **)calloc(graphadjmatrix->numNodes, sizeof(EdgeType *));
    for (int i = 0; i < graphadjmatrix->numNodes; i++)
	{
		graphadjmatrix->adjarr[i] = (EdgeType *)calloc(graphadjmatrix->numNodes, sizeof(EdgeType));
	}

    for (int i = 0; i < graphadjmatrix->numNodes; i++)
    {
        EdgeNode* node = graphadjlist->adjList[i].next;
        while (node != NULL)
        {
            graphadjmatrix->adjarr[i][node->adjvex] = node->weight;
            node = node->next;
        }
    }
    
    return graphadjmatrix;
}

//链队列结构类型
typedef struct QueueNode
{
	VextexNode* node;
	QueueNode* next;
}QueueNode;

typedef struct LinkQueue
{
	QueueNode* front;//队头指针
	QueueNode* rear;//队尾指针
}LinkQueue;

LinkQueue* InitLinkQueue()
{
	LinkQueue* q;
	q = (LinkQueue*)malloc(sizeof(LinkQueue));
	q->front = (QueueNode*)malloc(sizeof(QueueNode));
	if (q->front == NULL)
	{
		printf("开辟空间失败\n");
		exit(0);
	}
	q->rear = q->front;
	q->front->next = NULL;
	q->front->node = NULL;
	return q;
}

//空队	1-非空	0-空
bool EmptyLinkQueue(LinkQueue* q)
{
	if (q->front == q->rear)
	{
		return 0;
	}
	else
	{
		return 1;
	}
}

//入队
void EnLinkQueue(LinkQueue* q, VextexNode* e)
{
	QueueNode* p;
	p = (QueueNode*)malloc(sizeof(QueueNode));
	if (p == NULL)
	{
		printf("开辟空间失败\n");
		exit(0);
	}
	p->node = e;
	p->next = NULL;
	q->rear->next = p;
	q->rear = p;
}

//出队
VextexNode* DeLinkQueue(LinkQueue* q)
{
	if (EmptyLinkQueue(q) == 0)
	{
		printf("队空\n");
        return NULL;
	}
	else
	{
		VextexNode* node;
		QueueNode* queuenode;
		queuenode = q->front->next;
		node = queuenode->node;
		q->front->next = queuenode->next;
		if (queuenode == q->rear)
		{
			q->rear = q->front;
		}
		free(queuenode);
		return node;
	}
}

// 广度优先遍历
void BFS_GraphAdjList(GraphAdjList* graphadjlist, VextexNode* s)
{
    // 先标记
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        if (i != s->data)
        {
            graphadjlist->adjList[i].color = 0;
            graphadjlist->adjList[i].predecessor = NULL;
            graphadjlist->adjList[i].d = -1;
        }
    }
    
    // 入队s
    s->color = 1;
    s->d = 0;
    s->predecessor = NULL;
    LinkQueue* Q;
    Q = InitLinkQueue();
    EnLinkQueue(Q, s);

    while (EmptyLinkQueue(Q) == 1)
    {
        VextexNode* u;
        u = DeLinkQueue(Q);
        EdgeNode* p;
        p = u->next;
        while (p != NULL)
        {
            VextexNode* v;
            v = &(graphadjlist->adjList[p->adjvex]);
            // 发现为标记节点
            if(v->color == 0)
            {
                v->color = 1;
                v->d = u->d + 1;
                v->predecessor = u;
                EnLinkQueue(Q, v);
            }
            p = p->next;
        }
        u->color = 2;   //标黑
    }
}

// 打印广度优先树
void Print_BFTree(GraphAdjList* graphadjlist, VextexNode* s)
{
    printf("广度优先遍历树:\n");
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        printf("节点: %d 距离 %d 最短路径为 %d  : %d ", i, s->data, graphadjlist->adjList[i].d, i);
        VextexNode* p;
        p = graphadjlist->adjList[i].predecessor;
        while (p != NULL)
        {
            printf("<-- %d  ", p->data);
            p = p->predecessor;
        }
        printf("\n");
    }
}

// 深度优先访问
void DFS_Visit(GraphAdjList* graphadjlist, VextexNode* u, int* time)
{
    // u被发现，时间戳+1
    (*time) ++;
    u->d = *time;
    u->color = 1;
    
    EdgeNode* p;
    p = u->next;
    while (p != NULL)
    {
        VextexNode* v;
        v = &(graphadjlist->adjList[p->adjvex]);
        if (v->color == 0)
        {
            v->predecessor = u;
            DFS_Visit(graphadjlist, v, time);
        }
        p = p->next;
    }
    
    //全部扫描完成
    u->color = 2;
    (*time)++;
    u->f = *time;

}

// 深度优先遍历
void DFS_GraphAdjList(GraphAdjList* graphadjlist)
{
    // 先标记
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        graphadjlist->adjList[i].color = 0;
        graphadjlist->adjList[i].predecessor = NULL;
    }
    int time;
    time = 0;   // 时间戳

    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        if (graphadjlist->adjList[i].color == 0)
        {
            DFS_Visit(graphadjlist, &(graphadjlist->adjList[i]), &time);
        }
    }
}

// 打印深度优先树
void Print_DFTree(GraphAdjList* graphadjlist)
{
    printf("深度优先遍历树:\n");
    for (int i = 0; i < graphadjlist->numNodes; i++)
    {
        printf("节点: %d 发现时间为: %d 终止时间为: %d ,深度优先树路径为: %d ", i, graphadjlist->adjList[i].d, graphadjlist->adjList[i].f, i);
        VextexNode* p;
        p = graphadjlist->adjList[i].predecessor;
        if (graphadjlist->weighted == true)
        {
            printf(",路径权重为: %d  ", graphadjlist->adjList[i].key);
        }

        printf("%d  ", graphadjlist->adjList[i].data);
        while (p != NULL)
        {
            printf("<-- %d   ", p->data);
            p = p->predecessor;
        }
        printf("\n");
    }
}

// 最小生成树
// 不相交集合森林表示
typedef struct Set_Tree_Node
{
    VextexNode* x;
    struct Set_Tree_Node* p;
    int rank;
}Set_Tree_Node;

// 建立新的树
Set_Tree_Node* Make_Set_Tree(VextexNode* x)
{
    Set_Tree_Node* t;
    t = (Set_Tree_Node*)malloc(sizeof(Set_Tree_Node));
    t->rank = 0;
    t->x = x;
    t->p = t;

    return t;
}

// 路径压缩
Set_Tree_Node* Find_Set_Tree(Set_Tree_Node* t)
{
    if (t->p != t)
    {
        t->p = Find_Set_Tree(t->p);
    }
    return t->p;
}

// 按秩合并 
void Union_Set_Tree(Set_Tree_Node* x, Set_Tree_Node* y)
{
    Set_Tree_Node* t1, *t2;
    t1 = Find_Set_Tree(x);
    t2 = Find_Set_Tree(y);

    if (t1 != t2)
    {
        if (t1->rank > t2->rank)
        {
            t2->p = t1;
        }
        else
        {
            t1->p = t2;
            if (t1->rank == t2->rank)
            {
                t2->rank ++;
            }
        }
    }
}

// 边表示
typedef struct Edge
{
    int pre_data;
    int afte_data;
    EdgeType weight;
}Edge;

// 图邻接表转化边链表
Edge** nonDirected_GraphAdjList_Transfor_EdgeList(GraphAdjList* G)
{
    Edge** edgelist;
    
    edgelist = (Edge**)malloc(sizeof(Edge*) * GraphAdjList_Edge_Count(G)/2);
    int j = 0;
    for (int i = 0; i < G->numNodes; i++)
    {
        EdgeNode* node;
        node = (&G->adjList[i])->next;
        while (node != NULL)
        {
            if (node->adjvex > i)
            {
                edgelist[j] = (Edge*)malloc(sizeof(Edge));
                edgelist[j]->pre_data = i;
                edgelist[j]->afte_data = node->adjvex;
                edgelist[j]->weight = node->weight;
                j++;
            }
            node = node->next;
        }
    }

    return edgelist;
}

//(随机)快速排序
void Swap_Array_ij(Edge** a, int num, int i, int j)
{
	int t1, t2;
	t1 = a[i]->pre_data;
	a[i]->pre_data = a[j]->pre_data;
	a[j]->pre_data = t1;

	t2 = a[i]->afte_data;
	a[i]->afte_data = a[j]->afte_data;
	a[j]->afte_data = t2;

    EdgeType t;
	t = a[i]->weight;
	a[i]->weight = a[j]->weight;
	a[j]->weight = t;
}

int Random_Partition(Edge** a, int p, int r)
{
	//随机主元
	int random_num = rand() % (r - p + 1) + p;
	EdgeType x = a[random_num]->weight;
	Swap_Array_ij(a, r - p + 1, random_num, r);

	int i = p - 1;
	for (int j = p; j < r; j++)
	{
		if (a[j]->weight <= x)
		{
			i++;
			//交换i与j
			Swap_Array_ij(a, r - p + 1, i, j);
		}
	}
	//交换i+1与r
	Swap_Array_ij(a, r - p + 1, i + 1, r);

	return i + 1;
}

void Random_Quick_Sort(Edge** a, int p, int r)
{
	if (p < r)
	{
		int q;
		q = Random_Partition(a, p, r);
		Random_Quick_Sort(a, p, q - 1);
		Random_Quick_Sort(a, q + 1, r);
	}
}

// Kruskal算法
GraphAdjList* MST_Kruskal(GraphAdjList* G)
{
    GraphAdjList* A;
    A = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    A->numNodes = G->numNodes;
    A->weighted = G->weighted;
    A->adjList = (VextexNode*)malloc(sizeof(VextexNode) * G->numNodes);
    for (int i = 0; i < A->numNodes; i++)
    {
        A->adjList[i].data = G->adjList[i].data; // 设置顶点数据
        A->adjList[i].next = NULL; // 初始化为空
    }

    Set_Tree_Node** set_list;
    set_list = (Set_Tree_Node**)malloc(sizeof(Set_Tree_Node*) * A->numNodes);

    for (int i = 0; i < A->numNodes; i++)
    {
        set_list[i] = Make_Set_Tree(&G->adjList[i]);
    }
    
    Edge** edgelist;
    edgelist = nonDirected_GraphAdjList_Transfor_EdgeList(G);
    int edge_count = GraphAdjList_Edge_Count(G)/2;
    Random_Quick_Sort(edgelist, 0, edge_count - 1);

    int finish_count = 0;
    for (int i = 0; i < edge_count; i++)
    {
        if ((Find_Set_Tree(set_list[edgelist[i]->pre_data]) != Find_Set_Tree(set_list[edgelist[i]->afte_data])) && finish_count < A->numNodes - 1)
        {
            // 将边添加入A中
            AddEdge_GraphAdjList(A, edgelist[i]->pre_data, edgelist[i]->afte_data, edgelist[i]->weight);
            AddEdge_GraphAdjList(A, edgelist[i]->afte_data, edgelist[i]->pre_data, edgelist[i]->weight);

            // 合并
            Union_Set_Tree(set_list[edgelist[i]->pre_data], set_list[edgelist[i]->afte_data]);
            finish_count++;
        }
    }

    // 删除边链表
    for (int i = 0; i < edge_count; i++)
    {
        free(edgelist[i]);
    }
    free(edgelist);
    free(set_list);
    return A;
}

// 斐波那契堆节点
typedef struct Fib_Heap_Node
{
    VextexNode* x;
    Fib_Heap_Node* p;       // 父节点
    Fib_Heap_Node* child;   // 某一个孩子节点
    Fib_Heap_Node* left;    // 左兄弟
    Fib_Heap_Node* right;   // 右兄弟
    int degree;             // 孩子数目（度）
    bool mark;              // 标记
}Fib_Heap_Node;

// 斐波那契堆
typedef struct Fib_Heap
{
    int size;
    Fib_Heap_Node* min_node;
}Fib_Heap;

// 创建斐波那契堆
Fib_Heap* Create_Fib_Heap()
{
    Fib_Heap* H;
    H = (Fib_Heap*)malloc(sizeof(Fib_Heap));
    H->min_node = NULL;
    H->size = 0;
    return H;
}

// 递归删除斐波那契堆
void Free_Fib_Heap_Node(Fib_Heap_Node* node)
{
    if (node != NULL)
    {
        Fib_Heap_Node* start = node;
        do {
            Fib_Heap_Node* child = node->child;
            Fib_Heap_Node* next = node->right;

            // 递归释放子节点
            Free_Fib_Heap_Node(child);

            // 释放当前节点
            free(node);

            node = next;
        } while (node != start);
    }
}

void Free_Fib_Heap(Fib_Heap* H)
{
    if (H != NULL) {
        // 释放所有根节点
        Free_Fib_Heap_Node(H->min_node);

        // 释放堆结构本身
        free(H);
    }
}

// 斐波那契堆树根的个数
int Fib_Heap_Root_Count(Fib_Heap* H)
{
    if (H->min_node == NULL)
    {
        return 0;
    }
    else
    {
        int count = 1;
        Fib_Heap_Node* p;
        p = H->min_node->right;
        while (p != H->min_node)
        {
            count++;
            p = p->right;
        }
        return count;
    }
}

// 插入根节点   插在最小节点与根节点右边节点之间
void Fib_Heap_Insert_Root(Fib_Heap* H, Fib_Heap_Node* node)
{  
    // 只有一个根
    if (H->min_node->right == H->min_node)
    {
        H->min_node->right = node;
        H->min_node->left = node;
        node->left = H->min_node;
        node->right = H->min_node;
    }
    else
    {
        node->p = NULL;
        Fib_Heap_Node* p;
        p = H->min_node->right;

        H->min_node->right = node;
        p->left = node;
        node->left = H->min_node;
        node->right = p;
    }
    node->p = NULL;
}

// 插入节点
Fib_Heap_Node* Fib_Heap_Insert(Fib_Heap* H, VextexNode* x)
{
    Fib_Heap_Node* node;
    node = (Fib_Heap_Node*)malloc(sizeof(Fib_Heap_Node));
    node->x = x;
    node->child = NULL;
    node->degree = 0;
    node->mark = 0;

    // 堆空
    if (H->min_node == NULL)
    {
        H->min_node = node;
        node->p = NULL;
        node->left = node;
        node->right = node;
    }
    else
    {
        // 插入节点为根
        Fib_Heap_Insert_Root(H, node);

        if (x->key < H->min_node->x->key)
        {
            H->min_node = node; // 改变min指针
        }
    }
    H->size ++;

    return node;
}

// 合并堆
Fib_Heap* Fib_Heap_Union(Fib_Heap* H1, Fib_Heap* H2)
{
    Fib_Heap* H;
    H = Create_Fib_Heap();
    H->min_node = H1->min_node;
    H->size = H1->size + H2->size;
    // 根合并
    if (H1->min_node == NULL)
    {
        // 跳过
    }
    else if (H->min_node->right == H->min_node)   // H1只有一棵树
    {
        if (H2->min_node->right == H2->min_node)   // H2只有一棵树
        {
            H->min_node->right = H2->min_node;
            H->min_node->left = H2->min_node;
            H2->min_node->right = H->min_node;
            H2->min_node->left = H->min_node;
        }
        else
        {
            Fib_Heap_Insert_Root(H2, H->min_node);
        }
    }
    else    // H1有多棵树
    {
        if (H2->min_node->right == H2->min_node)   // H2只有一棵树
        {
            Fib_Heap_Insert_Root(H, H2->min_node);
        }
        else
        {
            Fib_Heap_Node* p,* q;
            p = H->min_node->right;
            q = H2->min_node->left;

            H->min_node->right = H2->min_node;
            H2->min_node->left = H->min_node;
            q->right = p;
            p->left = q;
        }
    }

    if ((H1->min_node == NULL) || (H2->min_node && H2->min_node->x->key < H1->min_node->x->key))
    {
        H->min_node = H2->min_node;
    }
    free(H1);
    free(H2);
    return H;
}

// 链接根节点   把y接为x的儿子
void Fib_Heap_Link(Fib_Heap* H, Fib_Heap_Node* y, Fib_Heap_Node* x)
{
    // 把y从根节点中移除
    y->left->right = y->right;
    y->right->left = y->left;

    // 把y填入x的孩子中
    y->p = x;
    Fib_Heap_Node* p;
    p = x->child;
    if (p == NULL)  // x原来没孩子
    {
        x->child = y;
        y->left = y;
        y->right = y;
    }
    else
    {
        p->right->left = y;
        y->right = p->right;
        p->right = y;
        y->left = p;
    }
    x->degree ++;

    y->mark = 0;
}

// 合并根链表
void Fib_Heap_Consolidate(Fib_Heap* H)
{
    Fib_Heap_Node** A;
    A = (Fib_Heap_Node**)malloc(sizeof(Fib_Heap_Node*) * H->size);
    for (int i = 0; i < H->size; i++)
    {
        A[i] = NULL;
    }
    Fib_Heap_Node** root_list;
    int count;
    count = Fib_Heap_Root_Count(H);
    Fib_Heap_Node* p;
    p = H->min_node;
    Fib_Heap_Node* x, *y;
    int d;
    for (int i = 0; i < count; i++)
    {
        x = p;
        p = p->right;
        d = x->degree;  // 当前节点的度数
        while (A[d] != NULL)
        {
            y = A[d];
            // 比谁小，然后交换
            if (x->x->key > y->x->key)
            {
                Fib_Heap_Node* t;
                t = x;
                x = y;
                y = t;
            }
            Fib_Heap_Link(H, y, x);
            A[d] = NULL;    //清空
            d++;    //存入下一个
        }
        A[d] = x;
    }

    // 根据 A 重建 H
    H->min_node = NULL;
    for (int i = 0; i < H->size; i++)
    {
        if (A[i] != NULL)
        {
            if (H->min_node == NULL)
            {
                H->min_node = A[i];
                H->min_node->right = H->min_node;
                H->min_node->left = H->min_node;
            }
            else
            {
                Fib_Heap_Insert_Root(H, A[i]);
                if (A[i]->x->key < H->min_node->x->key)
                {
                    H->min_node = A[i];
                }
            }
        }
    }
    
    free(A);
}

// 提取最小节点
VextexNode* Fib_Heap_Extract_Min(Fib_Heap* H)
{
    Fib_Heap_Node* z;
    z = H->min_node;
    if (z != NULL)
    {
        if (z->child != NULL)
        {
            // 令z的孩子都为根
            Fib_Heap_Node* child, *next;
            child = z->child;
            for (int i = 0; i < z->degree; i++)
            {
                next = child->right;
                Fib_Heap_Insert_Root(H, child);
                child = next;
            }
        }

        VextexNode* element;
        element = z->x;

        // z为唯一节点
        if (H->size == 1)
        {
            H->min_node = NULL;
        }
        else
        {
            // 从斐波那契堆移除z
            z->left->right = z->right;
            z->right->left = z->left;

            H->min_node = z->right;
            Fib_Heap_Consolidate(H);
        }
        H->size--;

        free(z);
        return element;
    }
    else
    {
        printf("空堆");
        return NULL;
    }
}

// 切断 切断x与父节点y，使x为根节点
void Fib_Heap_Cut(Fib_Heap* H, Fib_Heap_Node* x, Fib_Heap_Node* y)
{
    // x为y的孩子指针
    if (y->child == x)
    {
        // y只有一个孩子
        if (x->right == x)
        {
            y->child = NULL;
            Fib_Heap_Insert_Root(H, x);
        }
        else
        {
            y->child = x->right;
            x->right->left = x->left;
            x->left->right = x->right;
            Fib_Heap_Insert_Root(H, x);
        }
    }
    else
    {
        x->right->left = x->left;
        x->left->right = x->right;
        Fib_Heap_Insert_Root(H, x);
    }
    x->mark = 0;
}

// 级联切断
void Fib_Heap_Cascading_Cut(Fib_Heap* H, Fib_Heap_Node* y)
{
    Fib_Heap_Node* z;
    z = y->p;
    // y非根节点
    if (z != NULL)
    {
        if (y->mark == 0)
        {
            y->mark = 1;
        }
        else
        {
            Fib_Heap_Cut(H, y, z);
            Fib_Heap_Cascading_Cut(H, z);
        }
    }
}

// 关键字减值
void Fib_Heap_Decrease_Node(Fib_Heap* H, Fib_Heap_Node* x, EdgeType k)
{
    if (k > x->x->key)
    {
        printf("不能让值变大\n");
    }
    else
    {
        x->x->key = k;
        Fib_Heap_Node *y;
        y = x->p;

        if (y != NULL && x->x->key < y->x->key)
        {
            Fib_Heap_Cut(H, x, y);
            Fib_Heap_Cascading_Cut(H, y);
        }
        if (x->x->key < H->min_node->x->key)
        {
            H->min_node = x;
        }
    }
}

// Prim算法
GraphAdjList* MST_Prim(GraphAdjList* G, VextexNode* r)
{
    GraphAdjList* A;
    A = (GraphAdjList*)malloc(sizeof(GraphAdjList));
    A->numNodes = G->numNodes;
    A->weighted = G->weighted;
    A->adjList = (VextexNode*)malloc(sizeof(VextexNode) * G->numNodes);

    EdgeNode* node;
    node = G->adjList[r->data].next;
    if (node == NULL)
    {
        printf("图不连通\n");
        Delete_GraphAdjList(A);
        return NULL;
    }
    else
    {
        bool *vextexlist;   // 标记已经发现的顶点
        vextexlist = (bool*)calloc(G->numNodes, sizeof(bool));

        Fib_Heap_Node** heap_node_list;
        heap_node_list = (Fib_Heap_Node**)malloc(sizeof(Fib_Heap_Node*) * G->numNodes);
        for (int i = 0; i < G->numNodes; i++)
        {
            heap_node_list[i] = NULL;
            (&G->adjList[i])->key = INT_MAX;
        }

        Fib_Heap* H;
        H = Create_Fib_Heap();
        r->key = 0;
        heap_node_list[r->data] = Fib_Heap_Insert(H, r);
        
        while (H->min_node != NULL)
        {
            VextexNode* u;
            u = Fib_Heap_Extract_Min(H);
            vextexlist[u->data] = 1;

            EdgeNode* edge;
            edge = u->next;
            while (edge != NULL)
            {
                if (vextexlist[edge->adjvex] != 1 && edge->weight <= (&G->adjList[edge->adjvex])->key)
                {
                    if (heap_node_list[edge->adjvex] == NULL)
                    {
                        (&G->adjList[edge->adjvex])->key = edge->weight;
                        heap_node_list[edge->adjvex] = Fib_Heap_Insert(H, &G->adjList[edge->adjvex]);
                    }
                    else
                    {
                        Fib_Heap_Decrease_Node(H, heap_node_list[edge->adjvex], edge->weight);
                    }
                }
                edge = edge->next;
            }

            if (u != r)
            {
                edge = u->next;
                while (edge != NULL)
                {
                    if (vextexlist[edge->adjvex] == 1 && edge->weight == u->key) 
                    {
                        break;
                    }
                    edge = edge->next;
                }
                // 将边添加入A中
                AddEdge_GraphAdjList(A, u->data, edge->adjvex, edge->weight);
                AddEdge_GraphAdjList(A, edge->adjvex, u->data, edge->weight);
            }
        }
        Free_Fib_Heap(H);
        free(vextexlist);
        free(heap_node_list);

        return A;
    }
}

// 单源最短路径
// 松弛边 u -> v
void Relax_Edge(VextexNode* u, VextexNode* v, EdgeType weight)
{
    if (u->key != INT_MAX && v->key > u->key + weight)
    {
        v->key = u->key + weight;
        v->predecessor = u;
    }
}

// Bellman_Ford算法
bool Bellman_Ford(GraphAdjList* G, VextexNode* s)
{
    // 初始化
    for (int i = 0; i < G->numNodes; i++)
    {
        G->adjList[i].predecessor = NULL;
        G->adjList[i].key = INT_MAX;
    }
    s->key = 0;

    for (int k = 0; k < G->numNodes - 1; k++)
    {
        for (int i = 0; i < G->numNodes; i++)
        {
            EdgeNode* edge = G->adjList[i].next;
            while (edge != NULL)
            {
                Relax_Edge(&G->adjList[i], &G->adjList[edge->adjvex], edge->weight);
                edge = edge->next;
            }
        }
    }

    // 检查是否有负环路
    for (int i = 0; i < G->numNodes; i++)
    {
        EdgeNode* edge;
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            if(G->adjList[edge->adjvex].key > G->adjList[i].key + edge->weight)
            {
                return false;
            }
            edge = edge->next;
        }
    }
    return true;
}

// Dijkstra算法
void Dijkstra(GraphAdjList* G, VextexNode* s)
{
    // 初始化
    for (int i = 0; i < G->numNodes; i++)
    {
        G->adjList[i].predecessor = NULL;
        G->adjList[i].key = INT_MAX;
    }
    s->key = 0;

    Fib_Heap_Node** heap_node_list;
    heap_node_list = (Fib_Heap_Node**)malloc(sizeof(Fib_Heap_Node*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        heap_node_list[i] = NULL;
    }

    // 与s连接的点集合
    Fib_Heap* H;
    H = Create_Fib_Heap();
    heap_node_list[s->data] = Fib_Heap_Insert(H, s);

    while (H->min_node != NULL)
    {
        VextexNode* u;
        u = Fib_Heap_Extract_Min(H);

        EdgeNode* edge;
        edge = u->next;
        while (edge != NULL)
        {
            if (heap_node_list[edge->adjvex] != NULL)
            {
                Relax_Edge(u, &G->adjList[edge->adjvex], edge->weight);
                if ((&G->adjList[edge->adjvex])->predecessor == u)
                {
                    Fib_Heap_Decrease_Node(H, heap_node_list[edge->adjvex], u->key + edge->weight);
                }
            }
            else
            {
                G->adjList[edge->adjvex].key = u->key + edge->weight;
                G->adjList[edge->adjvex].predecessor = u;
                heap_node_list[edge->adjvex] = Fib_Heap_Insert(H, &G->adjList[edge->adjvex]);
            }
            edge = edge->next;
        }
    }
    Free_Fib_Heap(H);
    free(heap_node_list);
}

// 节点对最短路径
typedef struct Graph_Shortest_Paths
{
    int numNodes;   // 节点个数
    EdgeType** D;   // 路径权重
    int** Pai;        // 前驱图
}Graph_Shortest_Paths;

// 删除节点对最短路径
void Delete_Graph_Shortest_Paths(Graph_Shortest_Paths* P)
{
    for (int i = 0; i < P->numNodes; i++)
    {
        free(P->D[i]);
        free(P->Pai[i]);
    }
    
    free(P);
}

// 输出节点对最短路径
void Print_Graph_Shortest_Paths(Graph_Shortest_Paths* P)
{
    printf("节点对最短路径权重\n");
    for (int i = 0; i < P->numNodes; i++)
    {
        for (int j = 0; j < P->numNodes; j++)
        {
            printf("%d  ", P->D[i][j]);
        }
        printf("\n");
    }
    
    printf("节点对最短路径前驱\n");
    for (int i = 0; i < P->numNodes; i++)
    {
        for (int j = 0; j < P->numNodes; j++)
        {
            printf("%d  ", P->Pai[i][j]);
        }
        printf("\n");
    }
}

// 初始化节点对最短路径
Graph_Shortest_Paths* Init__Graph_Shortest_Paths(EdgeType** W, int numNodes)
{
    Graph_Shortest_Paths* PP;
    PP = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    PP->numNodes = numNodes;

    EdgeType** L;
    int** Pai;
    L = (EdgeType**)malloc(sizeof(EdgeType*) * numNodes);
    Pai = (int**)malloc(sizeof(int*) * numNodes);
    for (int i = 0; i < numNodes; i++)
    {
        L[i] = (EdgeType*)malloc(sizeof(EdgeType) * numNodes);
        Pai[i] = (int*)malloc(sizeof(int) * numNodes);
        for (int j = 0; j < numNodes; j++)
        {
            L[i][j] = W[i][j];
            if (i == j || W[i][j] == INT_MAX)
            {
                Pai[i][j] = -1;
            }
            else
            {
                Pai[i][j] = i;
            }
        }
    }
    
    PP->D = L;
    PP->Pai = Pai;
    return PP;
}

// 自底向上最短路径权重算法
Graph_Shortest_Paths* Extend_Shortest_Paths(EdgeType** W, Graph_Shortest_Paths* P)
{
    EdgeType** LL;
    int** PPai;
    LL = (EdgeType**)malloc(sizeof(EdgeType*) * P->numNodes);
    PPai = (int**)malloc(sizeof(int*) * P->numNodes);
    for (int i = 0; i < P->numNodes; i++)
    {
        LL[i] = (EdgeType*)malloc(sizeof(EdgeType) * P->numNodes);
        PPai[i] = (int*)malloc(sizeof(int) * P->numNodes);
        for (int j = 0; j < P->numNodes; j++)
        {
            PPai[i][j] = P->Pai[i][j];
        }
    }
        
    Graph_Shortest_Paths* PP;
    PP = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    PP->numNodes = P->numNodes;
    PP->D = LL;
    PP->Pai = PPai;
    
    for (int i = 0; i < P->numNodes; i++)
    {
        for (int j = 0; j < P->numNodes; j++)
        {
            if (i == j)
            {
                LL[i][j] = 0;
            }
            else
            {
                LL[i][j] = INT_MAX;
                for (int k = 0; k < P->numNodes; k++)
                {
                    if (i != j && k != j && P->D[i][k] != INT_MAX && W[k][j] != INT_MAX && P->D[i][k] + W[k][j] < LL[i][j])
                    {
                        LL[i][j] = P->D[i][k] + W[k][j];
                        // 快速算法
                        if (P->D == W)
                        {
                            PPai[i][j] = P->Pai[k][j];
                        }
                        else
                        {
                            PPai[i][j] = k;
                        }
                    }
                }
            }
            
        }
    }

    return PP;
}

// 慢速算法
Graph_Shortest_Paths* Slow_All_Pairs_Shortest_Paths(GraphAdjList* G)
{
    //构造边权重矩阵
    EdgeType** W;
    W = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        W[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        for (int j = 0; j < G->numNodes; j++)
        {
            if (i == j)
            {
                W[i][j] = 0;
            }
            else
            {
                W[i][j] = INT_MAX;
            }
        }
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            W[i][edge->adjvex] = edge->weight;
            edge = edge->next;
        }
    }

    Graph_Shortest_Paths* P, *Pm, *t;
    P = Init__Graph_Shortest_Paths(W, G->numNodes);

    for (int m = 1; m < G->numNodes - 1; m++)
    {
        Pm = Extend_Shortest_Paths(W, P);
        t = P;
        P = Pm;
        Delete_Graph_Shortest_Paths(t);
    }

    for (int i = 0; i < G->numNodes; i++)
    {
        free(W[i]);
    }
    free(W);

    return P;
}

// 快速算法
Graph_Shortest_Paths* Faster_All_Pairs_Shortest_Paths(GraphAdjList* G)
{
    //构造边权重矩阵
    EdgeType** W;
    W = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        W[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        for (int j = 0; j < G->numNodes; j++)
        {
            if (i == j)
            {
                W[i][j] = 0;
            }
            else
            {
                W[i][j] = INT_MAX;
            }
        }
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            W[i][edge->adjvex] = edge->weight;
            edge = edge->next;
        }
    }

    Graph_Shortest_Paths* P, *Pm, *t;
    P = Init__Graph_Shortest_Paths(W, G->numNodes);

    int m = 1;
    while (m < G->numNodes)
    {
        Pm = Extend_Shortest_Paths(P->D, P);
        t = P;
        P = Pm;
        Delete_Graph_Shortest_Paths(t);
        m = 2 * m;
    }
    
    for (int i = 0; i < G->numNodes; i++)
    {
        free(W[i]);
    }
    free(W);

    return P;
}

// Floyd_Warshall算法
Graph_Shortest_Paths* Floyd_Warshall(GraphAdjList* G)
{
    //构造边权重矩阵
    EdgeType** D, **Dn;
    int** Pai, ** PPai;
    D = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    Dn = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
    Pai = (int**)malloc(sizeof(int*) * G->numNodes);
    PPai = (int**)malloc(sizeof(int*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        D[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        Dn[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        Pai[i] = (int*)malloc(sizeof(int) * G->numNodes);
        PPai[i] = (int*)malloc(sizeof(int) * G->numNodes);
        for (int j = 0; j < G->numNodes; j++)
        {
            if (i == j)
            {
                D[i][j] = 0;
            }
            else
            {
                D[i][j] = INT_MAX;
            }
            PPai[i][j] = -1;
        }
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            D[i][edge->adjvex] = edge->weight;
            PPai[i][edge->adjvex] = i;
            edge = edge->next;
        }
    }

    Graph_Shortest_Paths* PP;
    PP = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    PP->numNodes = G->numNodes;
    PP->D = D;
    PP->Pai = Pai;

    for (int k = 0; k < G->numNodes; k++)
    {
        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                if (D[i][k] != INT_MAX && D[k][j] != INT_MAX && D[i][k] + D[k][j] < D[i][j])
                {
                    D[i][j] = D[i][k] + D[k][j];
                    Pai[i][j] = PPai[k][j];
                }
                else
                {
                    Pai[i][j] = PPai[i][j];
                }
            }
        }

        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                PPai[i][j] = Pai[i][j];
            }
        }
    }
    
    for (int i = 0; i < G->numNodes; i++)
    {
        free(PPai[i]);
    }
    free(PPai);
    
    return PP;
}

// 传递闭包
bool** Graph_Transitive_Closure(GraphAdjList* G)
{
    // 初始化T
    bool** T, ** TT;
    T = (bool**)malloc(sizeof(bool*) * G->numNodes);
    TT = (bool**)malloc(sizeof(bool*) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        T[i] = (bool*)calloc(G->numNodes, sizeof(bool));
        TT[i] = (bool*)calloc(G->numNodes, sizeof(bool));
        T[i][i] = TT[i][i] = true;
    }

    EdgeNode* edge;
    for (int i = 0; i < G->numNodes; i++)
    {
        edge = G->adjList[i].next;
        while (edge != NULL)
        {
            T[i][edge->adjvex] = TT[i][edge->adjvex] = true;
            edge = edge->next;
        }
    }

    for (int k = 0; k < G->numNodes; k++)
    {
        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                if (T[i][j] == 0 && (TT[i][k] == true && TT[k][j] == true))
                {
                    T[i][j] = true;
                }
            }
        }
        
        for (int i = 0; i < G->numNodes; i++)
        {
            for (int j = 0; j < G->numNodes; j++)
            {
                TT[i][j] = T[i][j];
            }
        }
    }
    
    for (int i = 0; i < G->numNodes; i++)
    {
        free(TT[i]);
    }
    free(TT);

    return T;
}

// Johnson算法
Graph_Shortest_Paths* Graph_Johnson(GraphAdjList* G)
{
    // 增加点
    GraphAdjList* GG;
    GG = Create_GraphAdjList(G->numNodes + 1, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(GG, i, p->adjvex, p->weight);
            p = p->next;
        }
        AddEdge_GraphAdjList(GG, GG->numNodes - 1, i, 0);
    }
    
    Graph_Shortest_Paths* P;
    P = (Graph_Shortest_Paths*)malloc(sizeof(Graph_Shortest_Paths));
    P->numNodes = G->numNodes;
    P->D = NULL;
    P->Pai = NULL;

    if (Bellman_Ford(GG, &GG->adjList[GG->numNodes - 1]) == false)
    {
        printf("有负环路\n");
    }
    else
    {
        EdgeType* h;
        h = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
        for (int i = 0; i < G->numNodes; i++)
        {
            h[i] = GG->adjList[i].key;
        }
        // 更新边
        GraphAdjList* Gh;
        Gh = Create_GraphAdjList(G->numNodes, true);
        EdgeNode* p;
        for (int i = 0; i < Gh->numNodes; i++)
        {
            p = G->adjList[i].next;
            while (p != NULL)
            {
                AddEdge_GraphAdjList(Gh, i, p->adjvex, p->weight + h[i] - h[p->adjvex]);
                p = p->next;
            }
        }

        P->D = (EdgeType**)malloc(sizeof(EdgeType*) * G->numNodes);
        P->Pai = (int**)malloc(sizeof(int*) * G->numNodes);
        for (int i = 0; i < G->numNodes; i++)
        {
            P->D[i] = (EdgeType*)malloc(sizeof(EdgeType) * G->numNodes);
            P->Pai[i] = (int*)malloc(sizeof(int) * G->numNodes);
            for (int j = 0; j < G->numNodes; j++)
            {
                P->D[i][j] = INT_MAX;
            }
            P->Pai[i][i] = -1;
        }

        for (int i = 0; i < Gh->numNodes; i++)
        {
            Dijkstra(Gh, &Gh->adjList[i]);
            for (int j = 0; j < Gh->numNodes; j++)
            {
                P->D[i][j] = Gh->adjList[j].key - h[i] + h[j];
                if (i != j)
                {
                    P->Pai[i][j] = Gh->adjList[j].predecessor->data;
                }
            }
        }
        
        free(h);
        Delete_GraphAdjList(Gh);
    }
    return P;
}

// 最大流算法
// 深度优先遍历寻找增广路径
bool DFS_FindAugmentingPath(GraphAdjList* G, int u, int t, bool* visited, int* path)
{
    if (u == t) 
    {
        return true; // 如果到达汇点，返回成功
    }
    visited[u] = 1; // 标记当前节点为已访问

    EdgeNode* edge;
    edge = G->adjList[u].next;
    while (edge != NULL)
    {
        int v = edge->adjvex;
        if (!visited[v] && edge->weight > 0)    // 如果未访问且残存容量大于0
        { 
            path[v] = u; // 记录路径
            if (DFS_FindAugmentingPath(G, v, t, visited, path))
            {
                return true; // 如果找到一条路径，返回成功
            }
        }
        edge = edge->next;
    }
    return false; // 未找到路径，返回失败
}

// 增广路径的残存容量
EdgeType Augmenting_Path_Capacity(GraphAdjList* G, int* path, int s, int t)
{
    int path_flow = INT_MAX;
    for (int v = t; v != s; v = path[v])
    {
        // 前驱节点
        int u = path[v];
        // 查找边(u, v)的残存容量
        EdgeNode* edge = G->adjList[u].next;
        while (edge != NULL)
        {
            if (edge->adjvex == v)  //找到节点v
            {
                if (path_flow > edge->weight)
                {
                    path_flow = edge->weight;
                }
                break;
            }
            edge = edge->next;
        }
    }
    return path_flow;
}

// Ford_Fulkerson算法
GraphAdjList* Ford_Fulkerson(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    bool* visit_list;
    visit_list = (bool*)malloc(sizeof(bool) * G->numNodes);
    int* pre_path;
    pre_path = (int*)malloc(sizeof(int) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        visit_list[i] = 0;
        pre_path[i] = -1;
    }
    
    EdgeType path_capacity;
    while (DFS_FindAugmentingPath(Ef, s, t, visit_list, pre_path))
    {
        path_capacity = Augmenting_Path_Capacity(Ef, pre_path, s, t);
        // 更新流网络与残差网络
        for (int v = t; pre_path[v] != -1; v = pre_path[v]) // 遍历path路径上的节点
        {
            int u = pre_path[v];
            // 更新流
            EdgeNode* edge;
            edge = G->adjList[u].next;
            while (edge != NULL)
            {
                if (edge->adjvex == v)
                {
                    ChangeEdge_GraphAdjList(Flow, u, v, path_capacity);
                    break;
                }
                edge = edge->next;
            }
            if (edge == NULL)
            {
                ChangeEdge_GraphAdjList(Flow, v, u, -path_capacity);
            }
            
            // 更新残差网络
            ChangeEdge_GraphAdjList(Ef, u, v, -path_capacity);
            ChangeEdge_GraphAdjList(Ef, v, u, path_capacity);
        }
        for (int i = 0; i < G->numNodes; i++)
        {
            visit_list[i] = 0;
            pre_path[i] = -1;
        }
    }
    
    free(visit_list);
    free(pre_path);
    Delete_GraphAdjList(Ef); // 删除残差网络

    return Flow;
}

// 广度优先遍历寻找增广路径
bool BFS_FindAugmentingPath(GraphAdjList* G, int s, int t, int* path)
{
    LinkQueue* q = InitLinkQueue(); // 初始化队列
    bool visited[G->numNodes]; // 访问标记数组
    for (int i = 0; i < G->numNodes; i++)
    {
        visited[i] = false;
        path[i] = -1; // 初始化父节点数组
    }

    visited[s] = true; // 标记源点为已访问
    EnLinkQueue(q, &G->adjList[s]); // 源点入队

    while (EmptyLinkQueue(q))   // 队列非空
    {
        VextexNode* uNode = DeLinkQueue(q); // 出队
        int u = uNode->data; // 获取当前顶点编号

        EdgeNode* e = uNode->next; // 获取邻接边
        while (e != NULL)
        {
            int v = e->adjvex; // 邻接顶点编号
            if (!visited[v] && e->weight > 0)   // 如果未访问且边权重（容量）大于0
            {
                visited[v] = true;
                path[v] = u; // 记录路径
                EnLinkQueue(q, &G->adjList[v]); // 邻接顶点入队
                if (v == t) // 如果到达汇点
                { 
                    free(q->front); // 清理队列
                    free(q);
                    return true; // 找到增广路径
                }
            }
            e = e->next; // 移动到下一条边
        }
    }

    free(q->front); // 清理队列
    free(q);
    return false; // 未找到增广路径
}

// Edmonds_Karp算法
GraphAdjList* Edmonds_Karp(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    int* pre_path;
    pre_path = (int*)malloc(sizeof(int) * G->numNodes);
    for (int i = 0; i < G->numNodes; i++)
    {
        pre_path[i] = -1;
    }
    
    EdgeType path_capacity;
    while (BFS_FindAugmentingPath(Ef, s, t, pre_path))
    {
        path_capacity = Augmenting_Path_Capacity(Ef, pre_path, s, t);
        // 更新流网络与残差网络
        for (int v = t; pre_path[v] != -1; v = pre_path[v]) // 遍历path路径上的节点
        {
            int u = pre_path[v];
            // 更新流
            EdgeNode* edge;
            edge = G->adjList[u].next;
            while (edge != NULL)
            {
                if (edge->adjvex == v)
                {
                    ChangeEdge_GraphAdjList(Flow, u, v, path_capacity);
                    break;
                }
                edge = edge->next;
            }
            if (edge == NULL)
            {
                ChangeEdge_GraphAdjList(Flow, v, u, -path_capacity);
            }
            
            // 更新残差网络
            ChangeEdge_GraphAdjList(Ef, u, v, -path_capacity);
            ChangeEdge_GraphAdjList(Ef, v, u, path_capacity);
        }
        for (int i = 0; i < G->numNodes; i++)
        {
            pre_path[i] = -1;
        }
    }
    
    free(pre_path);
    Delete_GraphAdjList(Ef); // 删除残差网络

    return Flow;
}

// 溢出节点链表
typedef struct OverflowingNode{
    int vertexIndex;         // 存储顶点索引
    OverflowingNode* next;   // 指向下一个活跃节点
}OverflowingNode;

// 创建空头节点
OverflowingNode* Init_OverflowingList()
{
    OverflowingNode* head = (OverflowingNode*)malloc(sizeof(OverflowingNode));
    head->vertexIndex = -1; // 空头节点不存储有效的顶点索引
    head->next = NULL;
    return head;
}

// 判断空链表
bool Empty_OverflowingList(OverflowingNode* head)
{
    if (head->next == NULL)
    {
        return true;
    }
    else
    {
        return false;
    }
}

// 头插法添加节点
void Add_OverflowingNode(OverflowingNode* head, int vertexIndex)
{
    OverflowingNode* newNode;
    newNode = (OverflowingNode*)malloc(sizeof(OverflowingNode));
    newNode->vertexIndex = vertexIndex;
    // 插入到头节点之后
    newNode->next = head->next;
    head->next = newNode;
}

// 移除溢出节点
void Remove_OverflowingNode(OverflowingNode* head, int vertexIndex)
{
    OverflowingNode* prev = head;
    OverflowingNode* current = head->next;
    while (current != NULL)
    {
        if (current->vertexIndex == vertexIndex)
        {
            prev->next = current->next;
            free(current);
            break;
        }
        prev = current;
        current = current->next;
    }
}

// 删除溢出节点链表
void Delete_OverflowingList(OverflowingNode* head)
{
    OverflowingNode* current = head;
    while (current != NULL)
    {
        OverflowingNode* temp = current;
        current = current->next;
        free(temp);
    }
}

// Push算法
void Push(GraphAdjList* G, GraphAdjList* Ef, GraphAdjList* Flow, int u, int v)
{
    EdgeNode* edge;
    edge = Ef->adjList[u].next;
    while (edge != NULL)
    {
        if (edge->adjvex == v)
        {
            break;
        }
        edge = edge->next;
    }
    
    EdgeType delta;
    delta = (edge->weight < Ef->adjList[u].key) ? edge->weight : Ef->adjList[u].key;

    edge = G->adjList[u].next;
    while (edge != NULL)
    {
        if (edge->adjvex == v)
        {
            ChangeEdge_GraphAdjList(Flow, u, v, delta);
            ChangeEdge_GraphAdjList(Ef, u, v, -delta);
            ChangeEdge_GraphAdjList(Ef, v, u, delta);
            break;
        }
        edge = edge->next;
    }
    if (edge == NULL)
    {
        ChangeEdge_GraphAdjList(Flow, v, u, -delta);
        ChangeEdge_GraphAdjList(Ef, u, v, -delta);
        ChangeEdge_GraphAdjList(Ef, v, u, delta);
    }

    Ef->adjList[u].key -= delta;
    Ef->adjList[v].key += delta;
}

// Relabel算法
void Relabel(GraphAdjList* Ef, int u)
{
    int min_height = INT_MAX; // 初始化为最大整数，以确保能找到最小的高度
    for (EdgeNode* edge = Ef->adjList[u].next; edge != NULL; edge = edge->next) // 只考虑残余容量大于0的边
    {
        if (edge->weight > 0)
        {
            int neighbor_height = Ef->adjList[edge->adjvex].f;
            if (neighbor_height < min_height)
            {
                min_height = neighbor_height;
            }
        }
    }
    // 只有当找到至少一个有效的邻接节点时，才更新高度
    if (min_height != INT_MAX)
    {
        Ef->adjList[u].f = min_height + 1;
    }
}

// 判断能否使用Push算法
bool exists_Push(GraphAdjList* Ef, int u, int v)
{
    EdgeNode* edge;
    edge = Ef->adjList[u].next;
    while (edge != NULL)
    {
        if (edge->adjvex == v)
        {
            break;
        }
        edge = edge->next;
    }

    if (edge != NULL && edge->weight > 0)
    {
        if (Ef->adjList[u].f == Ef->adjList[v].f + 1)
        {
            return true;
        }
        else
        {
            return false;
        }
    }
    else
    {
        return false;
    }
}

// 判断能否使用Relabel算法
bool exists_Relabel(GraphAdjList* Ef, int u)
{
    for (EdgeNode* edge = Ef->adjList[u].next; edge != NULL; edge = edge->next)
    {
        if (Ef->adjList[edge->adjvex].f <= Ef->adjList[u].f && edge->weight > 0)
        {
            return false;
        }
    }
    return true;
}

// Push_Relabel算法
GraphAdjList* Generic_Push_Relabel(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    // 创建残差网络
    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    // 初始化
    for (int i = 0; i < Ef->numNodes; i++)
    {
        Ef->adjList[i].f = 0;
        Ef->adjList[i].key = 0;
    }
    Ef->adjList[s].f = Ef->numNodes;

    // 溢出节点链表
    OverflowingNode* overflowinglist;
    overflowinglist = Init_OverflowingList();

    // 初始化预流
    EdgeNode* edge;
    edge = Ef->adjList[s].next;
    for (EdgeNode* edge = Ef->adjList[s].next; edge != NULL; edge = edge->next)
    {
        EdgeType flow = edge->weight;
        AddEdge_GraphAdjList(Flow, s, edge->adjvex, flow);
        ChangeEdge_GraphAdjList(Ef, s, edge->adjvex, -flow);
        ChangeEdge_GraphAdjList(Ef, edge->adjvex, s, flow);

        Ef->adjList[edge->adjvex].key = flow;
        Ef->adjList[s].key -= flow;
        Add_OverflowingNode(overflowinglist, edge->adjvex);
    }

    while (!Empty_OverflowingList(overflowinglist))
    {
        OverflowingNode* currentNode;
        currentNode = overflowinglist->next;
        while (currentNode != NULL)
        {
            int u = currentNode->vertexIndex;

            // 尝试对所有邻接节点v进行Push操作
            bool pushed = false;
            for (EdgeNode* edge = Ef->adjList[u].next; edge != NULL; edge = edge->next)
            {
                int v = edge->adjvex;
                if (exists_Push(Ef, u, v))
                {
                    Push(G, Ef, Flow, u, v);
                    pushed = true;

                    // 如果v变成了新的活跃节点，添加到活跃节点列表
                    if (Ef->adjList[v].key > 0 && v != s && v != t)
                    {
                        Add_OverflowingNode(overflowinglist, v);
                    }
                    break; // 只要成功进行一次Push，就跳出循环
                }
            }

            // 如果没有成功进行Push操作，则尝试Relabel
            if (!pushed)
            {
                Relabel(Ef, u);
            }

            // 移动到下一个活跃节点
            currentNode = currentNode->next;

            // 检查u是否仍然是活跃节点
            if (Ef->adjList[u].key <= 0)
            {
                Remove_OverflowingNode(overflowinglist, u); // 移除不再活跃的节点u
            }
        }
    }

    Delete_GraphAdjList(Ef);
    free(overflowinglist);
    return Flow;
}

// 释放溢出节点
void Discharge(GraphAdjList* G, GraphAdjList* Ef, GraphAdjList* Flow, int u)
{
    EdgeNode* currentNode;
    currentNode = Ef->adjList[u].next;
    while (Ef->adjList[u].key > 0)
    {
        if (currentNode == NULL)    // 重贴标签
        {
            Ef->adjList[u].f += 1;
            currentNode = Ef->adjList[u].next;
        }
        else if (Ef->adjList[u].f == Ef->adjList[currentNode->adjvex].f + 1 && currentNode->weight > 0)   // push许可边
        {
            Push(G, Ef, Flow, u, currentNode->adjvex);
        }
        else
        {
            currentNode = currentNode->next;
        }
    }
}

// 前置重贴标签算法
GraphAdjList* Relabel_To_Front(GraphAdjList* G, int s, int t)
{
    // 初始化流函数
    GraphAdjList* Flow;
    Flow = Create_GraphAdjList(G->numNodes, true);

    // 创建残差网络
    GraphAdjList * Ef;
    Ef = Create_GraphAdjList(G->numNodes, true);
    EdgeNode* p;
    for (int i = 0; i < G->numNodes; i++)
    {
        p = G->adjList[i].next;
        while (p != NULL)
        {
            AddEdge_GraphAdjList(Ef, i, p->adjvex, p->weight);
            AddEdge_GraphAdjList(Ef, p->adjvex, i, 0);
            p = p->next;
        }
    }

    // 初始化
    for (int i = 0; i < Ef->numNodes; i++)
    {
        Ef->adjList[i].f = 0;
        Ef->adjList[i].key = 0;
    }
    Ef->adjList[s].f = Ef->numNodes;
    
    // 溢出节点链表
    OverflowingNode* OverflowingList;
    OverflowingList = Init_OverflowingList();
    for (int i = 0; i < G->numNodes; i++)
    {
        if (i != s && i != t)
        {
            Add_OverflowingNode(OverflowingList, i);
        }
    }
    
    // 初始化预流
    EdgeNode* edge;
    edge = Ef->adjList[s].next;
    for (EdgeNode* edge = Ef->adjList[s].next; edge != NULL; edge = edge->next)
    {
        EdgeType flow = edge->weight;
        AddEdge_GraphAdjList(Flow, s, edge->adjvex, flow);
        ChangeEdge_GraphAdjList(Ef, s, edge->adjvex, -flow);
        ChangeEdge_GraphAdjList(Ef, edge->adjvex, s, flow);

        Ef->adjList[edge->adjvex].key = flow;
        Ef->adjList[s].key -= flow;
    }
    
    OverflowingNode* u,* pre;
    u = OverflowingList->next;
    pre = OverflowingList;
    int old_height;
    while (u != NULL)
    {
        if (Ef->adjList[u->vertexIndex].key > 0)
        {
            old_height = Ef->adjList[u->vertexIndex].f;
            Discharge(G, Ef, Flow, u->vertexIndex);
            int old_index = u->vertexIndex;

            if (Ef->adjList[u->vertexIndex].f > old_height)
            {
                // 移动u到表头
                pre->next = u->next;
                u->next = OverflowingList->next;
                OverflowingList->next = u;

                pre = u;
                u = u->next;
            }
        }
        else
        {
            u = u->next;
            pre = pre->next;
        }
    }
    
    Delete_GraphAdjList(Ef);
    Delete_OverflowingList(OverflowingList);

    return Flow;
}

int main()
{
    // 创建一个有向图的邻接表
    GraphAdjList* graphadjlist = Create_GraphAdjList(8, 0);
    int edges1[16][2] = {
        {0, 1}, {0, 2},
        {1, 3}, {1, 4},
        {2, 3}, {2, 5},
        {3, 4}, {3, 7}, 
        {4, 5}, 
        {5, 0}, {5, 6}, 
        {6, 2}, {6, 7}, 
        {7, 0}, {7, 6}
        };
    for (int i = 0; i < 16; i++) {
        AddEdge_GraphAdjList(graphadjlist, edges1[i][0], edges1[i][1], 1);
    }

    // 将邻接表转换为邻接矩阵
    GraphAdjMatrix* graphadjmatrix = GraphAdjList_Transfor_GraphMatrix(graphadjlist);

    // 输出邻接表
    Print_GraphAdjList(graphadjlist);
    // 输出邻接矩阵
    Print_GraphAdjMatrix(graphadjmatrix);

    // 释放邻接矩阵空间
    Delete_GraphAdjMatrix(graphadjmatrix);

    // 执行广度优先遍历
    BFS_GraphAdjList(graphadjlist, &graphadjlist->adjList[0]); // 假设从节点0开始遍历
    // 打印广度优先树
    Print_BFTree(graphadjlist, &graphadjlist->adjList[0]);

    // 执行深度优先遍历
    DFS_GraphAdjList(graphadjlist);
    // 打印深度优先树
    Print_DFTree(graphadjlist);

    // 释放邻接表空间
    Delete_GraphAdjList(graphadjlist);

    // 定义加权无向连通图
    GraphAdjList* G = Create_GraphAdjList(8, 1);

    // 定义边的数组
    int edges2[10][3] = {
        {0, 1, 4}, {0, 2, 3},
        {1, 2, 1}, {1, 3, 2},
        {2, 3, 4},
        {3, 4, 2},
        {4, 5, 6},
        {5, 6, 2}, {5, 7, 5},
        {6, 7, 1}
    };
    for (int i = 0; i < 10; i++) {
        int u = edges2[i][0];
        int v = edges2[i][1];
        int weight = edges2[i][2];
        AddEdge_GraphAdjList(G, u, v, weight);
        AddEdge_GraphAdjList(G, v, u, weight); // 由于是无向图，添加反向边
    }

    // Kruskal算法MST
    Print_GraphAdjList(G);
    GraphAdjList* MST1 = MST_Kruskal(G);
    Print_GraphAdjList(MST1);
    Delete_GraphAdjList(MST1);

    // Prim算法
    GraphAdjList* MST2 = MST_Prim(G, &G->adjList[0]);
    Print_GraphAdjList(MST2);
    Delete_GraphAdjList(MST2);

    Delete_GraphAdjList(G);


    // 创建一个包含8个顶点的图
    G = Create_GraphAdjList(8, true);

    // 定义边和权重
    int edges3[][3] = {
        {0, 1, 5}, {0, 2, 3},
        {1, 3, 6}, {1, 2, 2}, {1, 7, 11}, 
        {2, 4, 4}, {2, 5, 2},
        {3, 4, 7}, {3, 0, 10}, {3, 2, 1}, {3, 5, 3}, {3, 6, 1},
        {4, 5, 3},
        {5, 6, 1},
        {6, 7, 2}, 
        {7, 5, 9}, {7, 3, 6}
    };

    // 添加边
    int numEdges = sizeof(edges3) / sizeof(edges3[0]);
    for (int i = 0; i < numEdges; i++) {
        int u = edges3[i][0];
        int v = edges3[i][1];
        int weight = edges3[i][2];
        AddEdge_GraphAdjList(G, u, v, weight);
    }

    Print_GraphAdjList(G);
    if (Bellman_Ford(G, &G->adjList[0]))
    {
        Print_DFTree(G);
    }
    else
    {
        printf("存在负权重环路\n");
    }

    // 测试Dijkstra算法
    Dijkstra(G, &G->adjList[0]);
    Print_DFTree(G);

    // 释放图内存
    Delete_GraphAdjList(G);

    G = Create_GraphAdjList(5, 1); // 5个顶点的图

    // 定义边和权重
    int edges4[][3] = {
        {0, 1, 3}, {0, 2, 8}, {0, 4, -4},
        {1, 3, 1}, {1, 4, 7},
        {2, 1, 4},
        {3, 0, 2}, {3, 2, -5},
        {4, 3, 6}
    };

    // 添加边
    numEdges = sizeof(edges4) / sizeof(edges4[0]);
    for (int i = 0; i < numEdges; i++) {
        int u = edges4[i][0];
        int v = edges4[i][1];
        int weight = edges4[i][2];
        AddEdge_GraphAdjList(G, u, v, weight);
    }

    // 运行 Floyd-Warshall 算法
    Graph_Shortest_Paths* result;
    result = Slow_All_Pairs_Shortest_Paths(G);
    Print_Graph_Shortest_Paths(result);
    Delete_Graph_Shortest_Paths(result);

    result = Faster_All_Pairs_Shortest_Paths(G);
    Print_Graph_Shortest_Paths(result);
    Delete_Graph_Shortest_Paths(result);

    // 运行 Floyd-Warshall 算法
    result = Floyd_Warshall(G);
    Print_Graph_Shortest_Paths(result);
    Delete_Graph_Shortest_Paths(result);

    Delete_GraphAdjList(G);

    G = Create_GraphAdjList(4, 1); // 4个顶点的图

    // 定义边和权重
    int edges5[][3] = {
        {0, 1, 3},
        {1, 2, 1},
        {2, 3, 4},
        {3, 1, 2}
    };

    // 添加边
    numEdges = sizeof(edges5) / sizeof(edges5[0]);
    for (int i = 0; i < numEdges; i++) {
        int u = edges5[i][0];
        int v = edges5[i][1];
        int weight = edges5[i][2];
        AddEdge_GraphAdjList(G, u, v, weight);
    }

    // 计算传递闭包
    bool** T = Graph_Transitive_Closure(G);

    Print_GraphAdjList(G);
    printf("图的传递闭包\n");
    for (int i = 0; i < G->numNodes; i++)
    {
        for (int j = 0; j < G->numNodes; j++)
        {
            printf("%d ", T[i][j]);
        }
        printf("\n");
    }
    
    // 清理资源
    for (int i = 0; i < G->numNodes; i++) {
        free(T[i]);
    }
    free(T);
    Delete_GraphAdjList(G);

    G = Create_GraphAdjList(5, 1); // 5个顶点的图

    // 定义边和权重
    int edges6[][3] = {
        {0, 1, 3}, {0, 2, 8}, {0, 4, -4},
        {1, 3, 1}, {1, 4, 7},
        {2, 1, 4},
        {3, 0, 2}, {3, 2, -5},
        {4, 3, 6}
    };

    // 添加边
    numEdges = sizeof(edges6) / sizeof(edges6[0]);
    for (int i = 0; i < numEdges; i++) {
        int u = edges6[i][0];
        int v = edges6[i][1];
        int weight = edges6[i][2];
        AddEdge_GraphAdjList(G, u, v, weight);
    }
    Print_GraphAdjList(G);
    result = Graph_Johnson(G);
    Print_Graph_Shortest_Paths(result);
    Delete_Graph_Shortest_Paths(result);
    result = Floyd_Warshall(G);
    Print_Graph_Shortest_Paths(result);
    Delete_Graph_Shortest_Paths(result);
    Delete_GraphAdjList(G);


    G = Create_GraphAdjList(6, 1); // 6个顶点的图

    // 定义边和权重
    int edges7[][3] = {
        {0, 1, 16}, {0, 2, 13},
        {1, 3, 12},
        {2, 1, 4}, {2, 4, 14},
        {3, 2, 9}, {3, 5, 20},
        {4, 3, 7}, {4, 5, 4}
    };

    // 添加边
    numEdges = sizeof(edges7) / sizeof(edges7[0]);
    for (int i = 0; i < numEdges; i++) {
        int u = edges7[i][0];
        int v = edges7[i][1];
        int weight = edges7[i][2];
        AddEdge_GraphAdjList(G, u, v, weight);
    }
    Print_GraphAdjList(G);

    GraphAdjList* flow;
    printf("Ford_Fulkerson 算法\n");
    flow = Ford_Fulkerson(G, 0, 5);
    Print_GraphAdjList(flow);
    Delete_GraphAdjList(flow);
    printf("Edmonds_Karp 算法\n");
    flow = Edmonds_Karp(G, 0, 5);
    Print_GraphAdjList(flow);
    Delete_GraphAdjList(flow);
    printf("重贴标签算法\n");
    flow = Generic_Push_Relabel(G, 0, 5);
    Print_GraphAdjList(flow);
    Delete_GraphAdjList(flow);
    printf("前置重贴标签算法\n");
    flow = Relabel_To_Front(G, 0, 5);
    Print_GraphAdjList(flow);
    Delete_GraphAdjList(flow);
    Delete_GraphAdjList(G);

    return 0;
}