無向，強連通圖的所有簡單路徑

Question

我一直在嘗試在業余時間了解有關圖遍歷的更多信息，並且嘗試使用深度優先搜索來查找無向，強連接圖中起點和終點之間的所有簡單路徑。 到目前為止，我一直在使用“ 打印從給定源到目標的所有路徑”中的此代碼，這僅用於有向圖。

使用遞歸DFS的主要算法出現在以下兩個函數中：

void Graph::printAllPaths(int s, int d)
{
    // Mark all the vertices as not visited
    bool *visited = new bool[V];

    // Create an array to store paths
    int *path = new int[V];
    int path_index = 0; // Initialize path[] as empty

    // Initialize all vertices as not visited
    for (int i = 0; i < V; i++)
        visited[i] = false;

    // Call the recursive helper function to print all paths
    printAllPathsUtil(s, d, visited, path, path_index);
}

// A recursive function to print all paths from 'u' to 'd'.
// visited[] keeps track of vertices in current path.
// path[] stores actual vertices and path_index is current
// index in path[]
void Graph::printAllPathsUtil(int u, int d, bool visited[],
                          int path[], int &path_index)
{
    // Mark the current node and store it in path[]
    visited[u] = true;
    path[path_index] = u;
    path_index++;

    // If current vertex is same as destination, then print
    // current path[]
    if (u == d)
    {
        for (int i = 0; i<path_index; i++)
            cout << path[i] << " ";
        cout << endl;
    }
    else // If current vertex is not destination
    {
        // Recur for all the vertices adjacent to current vertex
        list<int>::iterator i;
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
            if (!visited[*i])
                printAllPathsUtil(*i, d, visited, path, path_index);
    }

    // Remove current vertex from path[] and mark it as unvisited
    path_index--;
    visited[u] = false;
}

對於有向圖，但對於無向，強連接圖，該方法很好用。

我想知道他們是否是一種方法來調整此代碼，使其也可用於無向圖？ 我感覺需要更多的回溯來探索更多可能的路徑，但是不確定如何實現這一點。

任何幫助，將不勝感激。

Answer 1

RoadRunner，您確定所顯示的代碼中包含“非定向問題”嗎？ 乍一看似乎還可以。 可能是由於您沒有修復addEdge來使您創建的圖形無向而引起的，例如：

void Graph::addEdge(int u, int v)
{
    adj[u].push_back(v); 
    adj[v].push_back(u); // Fix: add back edge as well!
}

更新 （C代碼，但非常難看）

好的，這是我嘗試將代碼轉換為純C的嘗試。顯然，代碼風格很丑陋，根本沒有錯誤檢查，但是您可以改善它，因為我希望您會精通C。圖節點的自定義鏈接列表，其名稱有點奇怪，即NodeListNode即包含圖節點的ListNode。

圖

#pragma once

#ifdef __cplusplus
extern "C" {  // only need to export C interface if
              // used by C++ source code
#endif

    typedef struct tagNodeListNode {
        struct tagNodeListNode* next;
        int index;
    } NodeListNode;

    typedef struct tagGraph {
        int nodesCount;
        NodeListNode** adjArr;
    } Graph;


    typedef void(*GraphPathVisitorFunc)(NodeListNode const* const path);

    Graph GraphCreate(int nodesCount);
    void GraphDestroy(Graph gr);
    void GraphAddEdge(Graph gr, int u, int v);
    void GraphVisitAllPaths(Graph gr, int s, int d, GraphPathVisitorFunc visitor);
    void GraphPrintAllPaths(Graph gr, int s, int d);


#ifdef __cplusplus
}
#endif

圖

#include "Graph.h"

#include <stdbool.h>
#include <stdlib.h>
#include <stdio.h>


Graph GraphCreate(int nodesCount)
{
    // calloc ensures zeroing array
    NodeListNode** adjArr = (NodeListNode**)calloc(nodesCount, sizeof(NodeListNode*));
    Graph gr = { nodesCount, adjArr };
    return gr;
}

void GraphDestroy(Graph gr)
{
    for (int i = 0; i < gr.nodesCount; i++)
    {
        for (NodeListNode* adj = gr.adjArr[i]; adj != NULL;)
        {
            NodeListNode* tmp = adj;
            adj = adj->next; //first move on the free
            free(tmp);
        }
    }
    free(gr.adjArr);
}

void GraphAddEdgeImplFirst(Graph gr, int from, int to)
{
    NodeListNode* adj = gr.adjArr[from];
    NodeListNode* n = (NodeListNode*)malloc(sizeof(NodeListNode));
    n->next = adj;
    n->index = to;
    gr.adjArr[from] = n;
}

void GraphAddEdgeImplLast(Graph gr, int from, int to)
{
    NodeListNode* adj = gr.adjArr[from];
    NodeListNode* n = (NodeListNode*)malloc(sizeof(NodeListNode));
    n->next = NULL;
    n->index = to;
    if(adj == NULL)
    {
        gr.adjArr[from] = n;
    }
    else
    {
        while (adj->next != NULL)
            adj = adj->next;
        adj->next = n;
    }
}



void GraphAddEdge(Graph gr, int u, int v)
{
    GraphAddEdgeImplFirst(gr, u, v);
    GraphAddEdgeImplFirst(gr, v, u);

    // closer to https://ideone.com/u3WoIJ but slower and thus makes no sense
    //GraphAddEdgeImplLast(gr, u, v);
    //GraphAddEdgeImplLast(gr, v, u);
}


void GraphVisitAllPathsImpl(Graph gr, int cur, int dst, GraphPathVisitorFunc visitor, NodeListNode* pathFst, NodeListNode* pathLst, bool* visited)
{
    if (cur == dst)
    {
        visitor(pathFst);
        return;
    }

    NodeListNode* adj = gr.adjArr[cur];
    for (NodeListNode const* tmp = adj; tmp != NULL; tmp = tmp->next)
    {
        int next = tmp->index;
        if (visited[next])
            continue;
        visited[next] = true;
        NodeListNode nextNode = { NULL,next };
        pathLst->next = &nextNode;

        GraphVisitAllPathsImpl(gr, next, dst, visitor, pathFst, &nextNode, visited);

        pathLst->next = NULL;
        visited[next] = false;
    }
}

void GraphVisitAllPaths(Graph gr, int start, int dst, GraphPathVisitorFunc visitor)
{
    bool* visited = calloc(gr.nodesCount, sizeof(bool));
    visited[start] = true;
    NodeListNode node = { NULL,start };
    GraphVisitAllPathsImpl(gr, start, dst, visitor, &node, &node, visited);
    free(visited);
}


void PrintPath(NodeListNode const* const path)
{
    for (NodeListNode const* tmp = path; tmp != NULL; tmp = tmp->next)
    {
        printf("%d ", tmp->index);
    }
    printf("\n");
}

void GraphPrintAllPaths(Graph gr, int s, int d)
{
    GraphVisitAllPaths(gr, s, d, PrintPath);
}

和用法示例，其圖形與您的ideaone示例相同。 請注意，要獲得匹配的輸出，應使用GraphAddEdgeImplLast而不是GraphAddEdgeImplFirst否則結果將以相反的順序進行。

void testGraph()
{
    Graph gr = GraphCreate(20);

    GraphAddEdge(gr, 0, 1);
    GraphAddEdge(gr, 0, 7);

    GraphAddEdge(gr, 1, 2);
    GraphAddEdge(gr, 1, 6);
    GraphAddEdge(gr, 1, 5);

    GraphAddEdge(gr, 2, 3);
    GraphAddEdge(gr, 2, 5);

    GraphAddEdge(gr, 3, 4);
    GraphAddEdge(gr, 3, 5);

    GraphAddEdge(gr, 4, 5);
    GraphAddEdge(gr, 4, 10);
    GraphAddEdge(gr, 4, 11);

    GraphAddEdge(gr, 5, 6);
    GraphAddEdge(gr, 5, 10);
    GraphAddEdge(gr, 5, 11);

    GraphAddEdge(gr, 6, 7);
    GraphAddEdge(gr, 6, 8);
    GraphAddEdge(gr, 6, 9);
    GraphAddEdge(gr, 6, 10);

    GraphAddEdge(gr, 7, 8);

    GraphAddEdge(gr, 8, 9);
    GraphAddEdge(gr, 8, 13);

    GraphAddEdge(gr, 9, 10);
    GraphAddEdge(gr, 9, 13);
    GraphAddEdge(gr, 9, 12);

    GraphAddEdge(gr, 10, 12);

    GraphAddEdge(gr, 11, 12);

    GraphAddEdge(gr, 12, 13);
    GraphAddEdge(gr, 12, 14);
    GraphAddEdge(gr, 12, 16);

    GraphAddEdge(gr, 13, 14);

    GraphAddEdge(gr, 14, 15);

    GraphAddEdge(gr, 16, 17);

    GraphAddEdge(gr, 15, 17);
    GraphAddEdge(gr, 15, 19);

    GraphAddEdge(gr, 17, 18);
    GraphAddEdge(gr, 17, 19);

    GraphAddEdge(gr, 18, 19);

    GraphPrintAllPaths(gr, 12, 4);

    GraphDestroy(gr);
}

希望這可以幫助