图，主要内容：深度优先遍历，广度优先遍历，最小生成树。文中代码均已在VS2015上测试，空指针均为nullptr（C++11）。参考来源：慕课网
图
分类：有向图。无向图
顶点，弧，弧头。弧尾，权值，出度，入度，顶点，边，邻接点，连通图，完全图【边数=n(n-1)/2 】，生成树【边数=n-1】
核心：图的遍历，最小生成树
存储结构
图的存储结构：邻接矩阵，邻接表，十字链表，邻接多重表
邻接矩阵--数组存储
struct Node
{
顶点索引;
顶点数据;
};
struct Map
{
顶点数组;
邻接矩阵;
};
邻接表--链式存储
struct Node
{
顶点索引;
该顶点弧链表的头结点;
顶点数据;
}
struct Arc
{
指向的顶点索引;
指向下一条弧的指针;
弧信息;
}
struct Map
{
顶点数组;
}
十字链表--链式存储
struct Node
{
顶点索引;
顶点数据;
第一条入弧结点指针;
第一条出弧结点指针;
}
struct Arc
{
弧尾顶点索引;
弧头顶点索引;
指向下一条弧头相同的弧的指针;
指向下一条弧尾相同的弧的指针;
弧信息;
}
struct Map
{
顶点数组;
}
邻接多重表--链式存储（无向图）
struct Node
{
顶点索引;
顶点数据;
第一条边结点指针;
}
struct Arc
{
顶点A索引;
顶点B索引;
连接A的下一条边的指针;
连接A的下一条边的指针;
边信息;
}
struct Map
{
顶点数组;
}
图的遍历
深度优先搜索（前序遍历，根左右），广度优先搜索（层级搜索）
A
/ \
B D
/ \ / \
C F G - H
\ /
E
深度优先遍历：A B C E F D G H
广度优先遍历：A B D C F G H E
邻接矩阵遍历代码：
【Node.h】
#ifndef NODE_H
#define NODE_H
class Node
{
public:
Node(char data = 0);
char m_cData;
bool m_bIsVisited;
};
#endif // !NODE_H
【Node.cpp】
#include "Node.h"
Node::Node(char data)
{
m_cData = data;
m_bIsVisited = false;
}
【CMap.h】
#ifndef CMAP_H
#define CMAP_H
#include <vector>
#include "Node.h"
#include <iostream>
using namespace std;
class CMap
{
public:
CMap(int capacity);
~CMap();
bool addNode(Node *pNode);//向图中加入顶点
void resetNode();//重置顶点
bool setValueToMatrixForDirectedGraph(int row, int col, int val = 1);//为有向图设置邻接矩阵
bool setValueToMatrixForUndirectedGraph(int row, int col, int val = 1);//为无向图设置邻接矩阵
void printMatrix();//打印邻接矩阵
void depthFirstTraverse(int nodeIndex);//深度优先遍历
void breadthFirstTraverse(int nodeIndex);//广度优先遍历
private:
bool getValueFromMatrix(int row, int col, int &val);//从矩阵中获取权值
void breadthFirstTraverseImpl(vector<int>preVec);//广度优先遍历实现函数
private:
int m_iCapacity;//图中最多可以容纳的顶点数
int m_iNodeCount;//已经添加的顶点个数
Node *m_pNodeArray;//用来存放顶点数组
int *m_pMatrix;//用来存放邻接矩阵
};
#endif // !CMAP_H
【CMap.cpp】
#include "CMap.h"
CMap::CMap(int capacity)
{
m_iCapacity = capacity;
m_iNodeCount = 0;
m_pNodeArray = new Node[m_iCapacity];
m_pMatrix = new int[m_iCapacity *m_iCapacity];
memset(m_pMatrix, 0, m_iCapacity*m_iCapacity * sizeof(int));
/*for (int i = 0; i < m_iCapacity *m_iCapacity; i++)
{
m_pMatrix[i] = 0;
}*/
}
CMap::~CMap()
{
delete[]m_pNodeArray;
delete[]m_pMatrix;
}
bool CMap::addNode(Node * pNode)
{
if (pNode == nullptr)
{
return false;
}
m_pNodeArray[m_iNodeCount].m_cData = pNode->m_cData;
m_iNodeCount++;
return true;
}
void CMap::resetNode()
{
for (int i = 0;i < m_iNodeCount;i++)
{
m_pNodeArray[i].m_bIsVisited = false;
}
}
bool CMap::setValueToMatrixForDirectedGraph(int row, int col, int val)
{
if (row<0||row>=m_iCapacity)
{
return false;
}
if (col<0 || col >= m_iCapacity)
{
return false;
}
m_pMatrix[row *m_iCapacity + col] = val;
return true;
}
bool CMap::setValueToMatrixForUndirectedGraph(int row, int col, int val)
{
if (row<0 || row >= m_iCapacity)
{
return false;
}
if (col<0 || col >= m_iCapacity)
{
return false;
}
m_pMatrix[row *m_iCapacity + col] = val;
m_pMatrix[col *m_iCapacity + row] = val;
return true;
}
void CMap::printMatrix()
{
for (int i = 0; i < m_iCapacity; i++)
{
for (int k = 0; k < m_iCapacity; k++)
{
cout << m_pMatrix[i*m_iCapacity + k] << " ";
}
cout << endl;
}
}
void CMap::depthFirstTraverse(int nodeIndex)
{
int value = 0;
cout << m_pNodeArray[nodeIndex].m_cData << " ";
m_pNodeArray[nodeIndex].m_bIsVisited = true;
for (int i = 0;i < m_iCapacity;i++)
{
getValueFromMatrix(nodeIndex, i, value);
if (value == 1)
{
if (m_pNodeArray[i].m_bIsVisited)
{
continue;
}
else
{
depthFirstTraverse(i);
}
}
else
{
continue;
}
}
}
void CMap::breadthFirstTraverse(int nodeIndex)
{
cout << m_pNodeArray[nodeIndex].m_cData << " ";
m_pNodeArray[nodeIndex].m_bIsVisited = true;
vector<int>curVec;
curVec.push_back(nodeIndex);
breadthFirstTraverseImpl(curVec);
}
bool CMap::getValueFromMatrix(int row, int col, int & val)
{
if (row<0 || row >= m_iCapacity)
{
return false;
}
if (col<0 || col >= m_iCapacity)
{
return false;
}
val = m_pMatrix[row *m_iCapacity + col];
return true;
}
void CMap::breadthFirstTraverseImpl(vector<int> preVec)
{
int value = 0;
vector<int> curVec;
for (int j = 0; j < (int)preVec.size(); j++)
{
for (int i = 0; i < m_iCapacity; i++)
{
getValueFromMatrix(preVec[j], i, value);
if (value!=0)
{
if (m_pNodeArray[i].m_bIsVisited)
{
continue;
}
else
{
cout << m_pNodeArray[i].m_cData << " ";
m_pNodeArray[i].m_bIsVisited = true;
curVec.push_back(i);
}
}
}
}
if (curVec.size()==0)
{
return;
}
else
{
breadthFirstTraverseImpl(curVec);
}
}
【main.cpp】
#include "CMap.h"
int main(void)
{
CMap *pMap = new CMap(8);
Node *pNodeA = new Node('A');
Node *pNodeB = new Node('B');
Node *pNodeC = new Node('C');
Node *pNodeD = new Node('D');
Node *pNodeE = new Node('E');
Node *pNodeF = new Node('F');
Node *pNodeG = new Node('G');
Node *pNodeH = new Node('H');
pMap->addNode(pNodeA);
pMap->addNode(pNodeB);
pMap->addNode(pNodeC);
pMap->addNode(pNodeD);
pMap->addNode(pNodeE);
pMap->addNode(pNodeF);
pMap->addNode(pNodeG);
pMap->addNode(pNodeH);
pMap->setValueToMatrixForUndirectedGraph(0, 1);
pMap->setValueToMatrixForUndirectedGraph(0, 3);
pMap->setValueToMatrixForUndirectedGraph(1, 2);
pMap->setValueToMatrixForUndirectedGraph(1, 5);
pMap->setValueToMatrixForUndirectedGraph(3, 6);
pMap->setValueToMatrixForUndirectedGraph(3, 7);
pMap->setValueToMatrixForUndirectedGraph(6, 7);
pMap->setValueToMatrixForUndirectedGraph(2, 4);
pMap->setValueToMatrixForUndirectedGraph(2, 5);
pMap->printMatrix();
cout << endl;
pMap->depthFirstTraverse(0);
cout << endl;
pMap->resetNode();
pMap->breadthFirstTraverse(0);
cout << endl;
return 0;
}
最小生成树
普里姆（Prim）算法；克鲁斯卡尔（Kruskal）算法
普里姆算法
基本思想
普里姆算法的基本思想：普里姆算法是一种构造最小生成树的算法，它是按逐个将顶点连通的方式来构造最小生成树的。
从连通网络N = { V, E }中的某一顶点u0出发，选择与它关联的具有最小权值的边(u0, v)，将其顶点加入到生成树的顶点集合U中。以后每一步从一个顶点在U中，而另一个顶点不在U中的各条边中选择权值最小的边(u, v),把该边加入到生成树的边集TE中，把它的顶点加入到集合U中。如此重复执行，直到网络中的所有顶点都加入到生成树顶点集合U中为止。
假设G=(V，E)是一个具有n个顶点的带权无向连通图，T(U，TE)是G的最小生成树，其中U是T的顶点集，TE是T的边集，则构造G的最小生成树T的步骤如下：
（1）初始状态，TE为空，U={v0}，v0∈V；
（2）在所有u∈U，v∈V-U的边(u，v)∈E中找一条代价最小的边(u′，v′)并入TE，同时将v′并入U；
重复执行步骤（2）n-1次，直到U=V为止。
在普里姆算法中，为了便于在集合U和(V-U)之间选取权值最小的边，需要设置两个辅助数组closest和lowcost，分别用于存放顶点的序号和边的权值。
对于每一个顶点v∈V-U，closest[v]为U中距离v最近的一个邻接点，即边(v，closest[v])是在所有与顶点v相邻、且其另一顶点j∈U的边中具有最小权值的边，其最小权值为lowcost[v]，即lowcost[v]=cost[v][closest[v]]，采用邻接表作为存储结构：
设置一个辅助数组closedge[]：
lowcost域存放生成树顶点集合内顶点到生成树外各顶点的各边上的当前最小权值；
adjvex域记录生成树顶点集合外各顶点距离集合内哪个顶点最近(即权值最小)。
克鲁斯卡尔算法
基本思想
克鲁斯卡尔算法是在剩下的所有未选取的边中，找最小边，如果和已选取的边构成回路，则放弃，选取次小边。
克鲁斯卡尔算法的时间复杂度为O（eloge）(e为网中边的数目)，因此它相对于普里姆算法而言，适合于求边稀疏的网的最小生成树。
实例
【Node.h】
#ifndef NODE_H
#define NODE_H
class Node
{
public:
Node(char data = 0);
char m_cData;
bool m_bIsVisited;
};
#endif // !NODE_H
【Node.cpp】
#include "Node.h"
Node::Node(char data)
{
m_cData = data;
m_bIsVisited = false;
}
【Edge.h】
#ifndef EDGE_H
#define EDGE_H
class Edge
{
public:
Edge(int nodeIndexA = 0, int nodeIndexB = 0, int weightValue = 0);
int m_iNodeIndexA;
int m_iNodeIndexB;
int m_iWeightValue;
bool m_bSelected;
};
#endif // !EDGE_H
【Edge.cpp】
#include "Edge.h"
Edge::Edge(int nodeIndexA, int nodeIndexB, int weightValue)
{
m_iNodeIndexA = nodeIndexA;
m_iNodeIndexB = nodeIndexB;
m_iWeightValue = weightValue;
m_bSelected = false;
}
【CMap.h】
#ifndef CMAP_H
#define CMAP_H
#include <vector>
#include "Node.h"
#include "Edge.h"
#include <iostream>
using namespace std;
class CMap
{
public:
CMap(int capacity);
~CMap();
bool addNode(Node *pNode);//向图中加入顶点
void resetNode();//重置顶点
bool setValueToMatrixForDirectedGraph(int row, int col, int val = 1);//为有向图设置邻接矩阵
bool setValueToMatrixForUndirectedGraph(int row, int col, int val = 1);//为无向图设置邻接矩阵
void printMatrix();//打印邻接矩阵
void depthFirstTraverse(int nodeIndex);//深度优先遍历
void breadthFirstTraverse(int nodeIndex);//广度优先遍历
void primTree(int nodeIndex);//普里姆生成树
void kruskalTree();
private:
bool getValueFromMatrix(int row, int col, int &val);//从矩阵中获取权值
void breadthFirstTraverseImpl(vector<int>preVec);//广度优先遍历实现函数
int getMinEdge(vector<Edge>edgeVec);
bool isInSet(vector<int>nodeSet, int target);
void mergeNodeSet(vector<int>&nodeSetA, vector<int>nodeSetB);
private:
int m_iCapacity;//图中最多可以容纳的顶点数
int m_iNodeCount;//已经添加的顶点个数
Node *m_pNodeArray;//用来存放顶点数组
int *m_pMatrix;//用来存放邻接矩阵
Edge *m_pEdge;
};
#endif // !CMAP_H
【CMap.cpp】
#include "CMap.h"
CMap::CMap(int capacity)
{
m_iCapacity = capacity;
m_iNodeCount = 0;
m_pNodeArray = new Node[m_iCapacity];
m_pMatrix = new int[m_iCapacity *m_iCapacity];
memset(m_pMatrix, 0, m_iCapacity*m_iCapacity * sizeof(int));
/*for (int i = 0; i < m_iCapacity *m_iCapacity; i++)
{
m_pMatrix[i] = 0;
}*/
m_pEdge = new Edge[m_iCapacity - 1];
}
CMap::~CMap()
{
delete[]m_pNodeArray;
delete[]m_pMatrix;
}
bool CMap::addNode(Node * pNode)
{
if (pNode == nullptr)
{
return false;
}
m_pNodeArray[m_iNodeCount].m_cData = pNode->m_cData;
m_iNodeCount++;
return true;
}
void CMap::resetNode()
{
for (int i = 0;i < m_iNodeCount;i++)
{
m_pNodeArray[i].m_bIsVisited = false;
}
}
bool CMap::setValueToMatrixForDirectedGraph(int row, int col, int val)
{
if (row<0||row>=m_iCapacity)
{
return false;
}
if (col<0 || col >= m_iCapacity)
{
return false;
}
m_pMatrix[row *m_iCapacity + col] = val;
return true;
}
bool CMap::setValueToMatrixForUndirectedGraph(int row, int col, int val)
{
if (row<0 || row >= m_iCapacity)
{
return false;
}
if (col<0 || col >= m_iCapacity)
{
return false;
}
m_pMatrix[row *m_iCapacity + col] = val;
m_pMatrix[col *m_iCapacity + row] = val;
return true;
}
void CMap::printMatrix()
{
for (int i = 0; i < m_iCapacity; i++)
{
for (int k = 0; k < m_iCapacity; k++)
{
cout << m_pMatrix[i*m_iCapacity + k] << " ";
}
cout << endl;
}
}
void CMap::depthFirstTraverse(int nodeIndex)
{
int value = 0;
cout << m_pNodeArray[nodeIndex].m_cData << " ";
m_pNodeArray[nodeIndex].m_bIsVisited = true;
for (int i = 0;i < m_iCapacity;i++)
{
getValueFromMatrix(nodeIndex, i, value);
if (value == 1)
{
if (m_pNodeArray[i].m_bIsVisited)
{
continue;
}
else
{
depthFirstTraverse(i);
}
}
else
{
continue;
}
}
}
void CMap::breadthFirstTraverse(int nodeIndex)
{
cout << m_pNodeArray[nodeIndex].m_cData << " ";
m_pNodeArray[nodeIndex].m_bIsVisited = true;
vector<int>curVec;
curVec.push_back(nodeIndex);
breadthFirstTraverseImpl(curVec);
}
void CMap::primTree(int nodeIndex)
{
int value = 0;
int edgeCount = 0;
vector<int>nodeVec;
vector<Edge>edgeVec;
cout << m_pNodeArray[nodeIndex].m_cData << endl;
m_pNodeArray[nodeIndex].m_bIsVisited = true;
nodeVec.push_back(nodeIndex);
while (edgeCount < m_iCapacity - 1)
{
int temp = nodeVec.back();
for (int i = 0; i < m_iCapacity; i++)
{
getValueFromMatrix(temp, i, value);
if (value != 0)
{
if (m_pNodeArray[i].m_bIsVisited)
{
continue;
}
else
{
Edge edge(temp, i, value);
edgeVec.push_back(edge);
}
}
}
int edgeIndex = getMinEdge(edgeVec);
edgeVec[edgeIndex].m_bSelected = true;
cout << edgeVec[edgeIndex].m_iNodeIndexA << "---" << edgeVec[edgeIndex].m_iNodeIndexB << " ";
cout << edgeVec[edgeIndex].m_iWeightValue << endl;
m_pEdge[edgeCount] = edgeVec[edgeIndex];
edgeCount++;
int nextNodeIndex = edgeVec[edgeIndex].m_iNodeIndexB;
nodeVec.push_back(nextNodeIndex);
m_pNodeArray[nextNodeIndex];
m_pNodeArray[nextNodeIndex].m_bIsVisited = true;
cout << m_pNodeArray[nextNodeIndex].m_cData << endl;
}
}
void CMap::kruskalTree()
{
int value = 0;
int edgeCount = 0;
//定义存放结点集合的数组
vector<vector<int>>nodeSets;
//第一步:取出所有边
vector<Edge>edgeVec;
for (int i=0;i<m_iCapacity;i++)
{
for (int k = i+1;k<m_iCapacity;k++)
{
getValueFromMatrix(i, k, value);
if (value!=0)
{
Edge edge(i, k, value);
edgeVec.push_back(edge);
}
}
}
//第二步:从所有边中取出组成最小生成树的边
//1.找到算法结束条件
while (edgeCount<m_iCapacity-1)
{
//2.从边集合中找到最小边
int minEdgeIndex = getMinEdge(edgeVec);
edgeVec[minEdgeIndex].m_bSelected = true;
//3.找到最小边连接的点
int nodeAIndex = edgeVec[minEdgeIndex].m_iNodeIndexA;
int nodeBIndex = edgeVec[minEdgeIndex].m_iNodeIndexB;
//4.找到点所在的点集合
bool nodeAIsSet = false;
bool nodeBIsSet = false;
int nodeAInSetLabel = -1;
int nodeBInSetLabel = -1;
for (int i = 0;i<(int)nodeSets.size();i++)
{
nodeAIsSet = isInSet(nodeSets[i], nodeAIndex);
if (nodeAIsSet)
{
nodeAInSetLabel = i;
}
}
for (int i = 0;i < (int)nodeSets.size();i++)
{
nodeBIsSet = isInSet(nodeSets[i], nodeBIndex);
if (nodeBIsSet)
{
nodeBInSetLabel = i;
}
}
//5.根据点所在集合的不同做出不同处理
if (nodeAInSetLabel==-1&&nodeBInSetLabel==-1)
{
vector<int>vec;
vec.push_back(nodeAIndex);
vec.push_back(nodeBIndex);
nodeSets.push_back(vec);
}
else if (nodeAInSetLabel!=-1&&nodeBInSetLabel==-1)
{
nodeSets[nodeAInSetLabel].push_back(nodeBIndex);
}
else if (nodeAInSetLabel != -1 && nodeBInSetLabel != -1 && nodeAInSetLabel != nodeBInSetLabel)
{
mergeNodeSet(nodeSets[nodeAInSetLabel], nodeSets[nodeBInSetLabel]);
for (int k = nodeBInSetLabel;k < (int)nodeSets.size() - 1;k++)
{
nodeSets[k] = nodeSets[k + 1];
}
}
else if (nodeAInSetLabel!=-1&&nodeBInSetLabel!=-1&&nodeAInSetLabel==nodeBInSetLabel)
{
continue;
}
m_pEdge[edgeCount] = edgeVec[minEdgeIndex];
edgeCount++;
cout << edgeVec[minEdgeIndex].m_iNodeIndexA << "---" << edgeVec[minEdgeIndex].m_iNodeIndexB << " ";
cout << edgeVec[minEdgeIndex].m_iWeightValue << endl;
}
}
bool CMap::getValueFromMatrix(int row, int col, int & val)
{
if (row<0 || row >= m_iCapacity)
{
return false;
}
if (col<0 || col >= m_iCapacity)
{
return false;
}
val = m_pMatrix[row *m_iCapacity + col];
return true;
}
void CMap::breadthFirstTraverseImpl(vector<int> preVec)
{
int value = 0;
vector<int> curVec;
for (int j = 0; j < (int)preVec.size(); j++)
{
for (int i = 0; i < m_iCapacity; i++)
{
getValueFromMatrix(preVec[j], i, value);
if (value!=0)
{
if (m_pNodeArray[i].m_bIsVisited)
{
continue;
}
else
{
cout << m_pNodeArray[i].m_cData << " ";
m_pNodeArray[i].m_bIsVisited = true;
curVec.push_back(i);
}
}
}
}
if (curVec.size()==0)
{
return;
}
else
{
breadthFirstTraverseImpl(curVec);
}
}
int CMap::getMinEdge(vector<Edge> edgeVec)
{
int minWeight = 0;
int edgeIndex = 0;
int i = 0;
for (;i<(int)edgeVec.size();i++)
{
if (!edgeVec[i].m_bSelected)
{
minWeight = edgeVec[i].m_iWeightValue;
edgeIndex = 1;
break;
}
}
if (minWeight == 0)
{
return -1;
}
for (;i<(int)edgeVec.size();i++)
{
if (edgeVec[i].m_bSelected)
{
continue;
}
else
{
if (minWeight>edgeVec[i].m_iWeightValue)
{
minWeight = edgeVec[i].m_iWeightValue;
edgeIndex = i;
}
}
}
return edgeIndex;
}
bool CMap::isInSet(vector<int> nodeSet, int target)
{
for (int i = 0;i<(int)nodeSet.size();i++)
{
if (nodeSet[i] == target)
{
return true;
}
}
return false;
}
void CMap::mergeNodeSet(vector<int>& nodeSetA, vector<int> nodeSetB)
{
for (int i = 0; i < (int)nodeSetB.size(); i++)
{
nodeSetA.push_back(nodeSetB[i]);
}
}
【main.cpp】
#include "CMap.h"
int main(void)
{
CMap *pMap = new CMap(6);
Node *pNodeA = new Node('A');
Node *pNodeB = new Node('B');
Node *pNodeC = new Node('C');
Node *pNodeD = new Node('D');
Node *pNodeE = new Node('E');
Node *pNodeF = new Node('F');
pMap->addNode(pNodeA);
pMap->addNode(pNodeB);
pMap->addNode(pNodeC);
pMap->addNode(pNodeD);
pMap->addNode(pNodeE);
pMap->addNode(pNodeF);
pMap->setValueToMatrixForUndirectedGraph(0, 1, 6);
pMap->setValueToMatrixForUndirectedGraph(0, 4, 5);
pMap->setValueToMatrixForUndirectedGraph(0, 5, 1);
pMap->setValueToMatrixForUndirectedGraph(1, 2, 3);
pMap->setValueToMatrixForUndirectedGraph(1, 5, 2);
pMap->setValueToMatrixForUndirectedGraph(2, 5, 8);
pMap->setValueToMatrixForUndirectedGraph(2, 3, 7);
pMap->setValueToMatrixForUndirectedGraph(3, 5, 4);
pMap->setValueToMatrixForUndirectedGraph(3, 4, 2);
pMap->setValueToMatrixForUndirectedGraph(4, 5, 9);
cout <<"普利姆算法" << endl;
pMap->primTree(0);
cout << "克鲁斯卡尔算法"<< endl;
pMap->kruskalTree();
return 0;
}
运行结果
