Hamiton图系列文章 (1) Hamilton图证明算法的道路矩阵基础数据结构与类的设计

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已于 2022-09-10 17:41:40 修改

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分类专栏：计算复杂度 C++11 文章标签：算法矩阵数据结构 Hamilton

于 2022-09-10 16:59:32 首次发布

本文链接：https://blog.youkuaiyun.com/zkmrobot/article/details/126797640

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本文介绍了道路矩阵在求解简单图Hamilton圈问题中的应用，详细阐述了PathArray4SimpleGraph类的设计，包括邻接矩阵的随机初始化、矩阵转换为路径数组的方法，以及用于生成和打印调试信息的辅助函数。后续章节将探讨矩阵运算及Hamilton图的判定算法。

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1.简介：

多年前就有老外提到用道路矩阵算Hamilton圈的算法以及判定方法，后来国内也有多人吃透原理，发表了类似文章。结论就是简单图G（V，E）为Hamilton图的充要条件是：

如果G对应的道路矩阵是P，|V| = n，则

G是H图等价于 P^n(P*P*P..*P) !=0;

这里的乘法是基于道路矩阵定义的乘法，具体请参考相关文献。

算法复杂度暂且不论，这篇文章先写道路矩阵的基础数据结构与类设计，后续再补充。

2. 类的设计

//头文件patharray4simplegraph.h

#ifndef PATHARRAY4SIMPLEGRAPH_H

#define PATHARRAY4SIMPLEGRAPH_H

#include <iostream>

#include <vector>

using namespace std;

typedef vector<int> PATH;

typedef vector<PATH> PATHS;

typedef vector<vector<bool> > GRAPH_ADJ;

typedef vector<vector<PATH> > SINGLE_PATH_ARRAY;

typedef vector<vector<PATHS> > MULTI_PATHS_ARRAY;

class PathArray4SimpleGraph

public:

    explicit PathArray4SimpleGraph(const int ndimen);

private:

    int m_ndimension;

    GRAPH_ADJ m_srcAdjArray;

    SINGLE_PATH_ARRAY m_srcPathArray;

private:

    void initSrcAdjArrayRandom(const int ndimen);

    void printAdjArray(const GRAPH_ADJ &adjArray);

    void printPathArray(const SINGLE_PATH_ARRAY &pathArray);

public:

    void adjArray2PathArray(const GRAPH_ADJ &srcAdjArray, SINGLE_PATH_ARRAY &dstPathArray);

};

#endif // PATHARRAY4SIMPLEGRAPH_H

//源文件patharray4simplegraph.cpp

#include "patharray4simplegraph.h"

#include <iostream>

#include<QRandomGenerator>

const int MAXRANDOM=1000;

PathArray4SimpleGraph::PathArray4SimpleGraph(const int ndimen):

m_ndimension(ndimen)

    initSrcAdjArrayRandom(ndimen);

    printAdjArray(m_srcAdjArray);

    adjArray2PathArray(m_srcAdjArray,m_srcPathArray);

    printPathArray(m_srcPathArray);

//initial the source adjacency matrices with random number 0 or 1

void PathArray4SimpleGraph::initSrcAdjArrayRandom(const int ndimen)

    m_srcAdjArray.resize(ndimen);

    for (int i = 0; i < ndimen; ++i){

        m_srcAdjArray[i].resize(ndimen);

    for (int i = 0; i < ndimen; ++i){

        m_srcAdjArray[i][i] = 0;

    for (int i = 0; i < ndimen-1; ++i){

        for(int j= i+1; j< ndimen; ++j)

            qint16 rand= QRandomGenerator::global()->bounded(MAXRANDOM);

            bool brand = (rand >= MAXRANDOM / 2)? true: false;

            m_srcAdjArray[i][j] = brand;

            m_srcAdjArray[j][i] = brand;

//print the source adjacency matrices to debug

void PathArray4SimpleGraph::printAdjArray(const GRAPH_ADJ &adjArray)

    if(adjArray.size() == 0)

        cout << "Adjacency Array vector is null\n";

        return;

    for (auto& line : adjArray) {

        for (const auto& v : line) {

            cout << v << " ";

        cout << endl;

//print the source path Array to debug

void PathArray4SimpleGraph::printPathArray(const SINGLE_PATH_ARRAY &pathArray)

    if(pathArray.size() == 0)

        cout << "Path Array vector is null\n";

        return;

    for (auto& line : pathArray) {

        for (const auto& path : line) {

            if (path.size() == 1) continue;

            cout << "[";

            for (size_t i = 0; i < path.size(); ++i) {

                cout << path[i];

                if (i != path.size() - 1)

                    cout << "->";

            cout << "]";

        cout << endl;

void PathArray4SimpleGraph::adjArray2PathArray(const GRAPH_ADJ &srcAdjArray, SINGLE_PATH_ARRAY &dstPathArray)

    dstPathArray.resize(srcAdjArray.size());

    for (int i = 0; i < srcAdjArray.size(); ++i){

        dstPathArray[i].resize(srcAdjArray.size());

    for (int i = 0; i < srcAdjArray.size(); ++i){

        for(int j= 0; j< srcAdjArray.size(); ++j)

            if(srcAdjArray[i][j] == 0)

                dstPathArray[i][j].push_back(0);

            } else {

                dstPathArray[i][j].push_back(i);

                dstPathArray[i][j].push_back(j);

3. 应用

//main.cpp

#include "patharray4simplegraph.h"

#include <iostream>

#include <vector>

using namespace std;

int main(int argc, char *argv[])

    PathArray4SimpleGraph pathArr4SimpleGraph(8);

    return 1;

4.运行结果：

下图显示用随机生成的邻接矩阵，可以正常生成道路矩阵，下篇文章设计矩阵的运算以及Hamilton图的判定。