一.问题分类
二.无权图单源最短路算法
dist[]数组记录的是个个顶点到源点的距离这个数组的下标表示顶点
源点到自己的距离是0,dist[s]=0
path[]数组记录的是这个顶点的前驱，可以同过这个数组找到源点到个个顶点的距离
代码如下
void Unweighted(MGraph Graph, Vertex S) {Vertex V,W;dist[S] = 0;//初始化源点到自己的距离queue<Vertex>q;q.push(S);//压栈while(!q.empty()) {/*弹栈*/V = q.front();q.pop();for (W = 0; W < Graph->Nv; W++) {/*点W到源点的距离没有被确定并且两个顶点有边，*/if (dist[W] == -1 && IsEdge(Graph, V, W) ){dist[W] = dist[V] + 1;/*更新距离*/path[W] = V;//记录W的前驱q.push(W);}}}
}
dist[]和path[]都初始为-1
全部代码
#include<iostream>
#include<queue>
using namespace std;
#define INIFIN 65535/*表示没有边*/
#define MaxVertenNum 100/*最大顶点值*/
typedef int Vertex;/*顶点的下标用整形表示*/
typedef int WeightType;/*权重*/
int dist[MaxVertenNum], path[MaxVertenNum];
/* 边的定义*/
typedef struct ENode* PtrToENode;
struct ENode
{Vertex V1, V2;/*有向图<v1,v2>*///WeightType Weight;/*权重*/
};
typedef PtrToENode Edge;//图的定义
typedef struct GNode* PtrToGNode;
struct GNode {int Nv;/*顶点个数*/int Ne;/*边的个数*/WeightType G[MaxVertenNum][MaxVertenNum];
};
typedef PtrToGNode MGraph;MGraph CreateGraph(int VerNum) {/*创建1个VN个顶点0条边的图*/Vertex V, W;MGraph Graph = new GNode();Graph->Nv = VerNum;Graph->Ne = 0;for (V = 0; V < Graph->Nv; V++)for (W = 0; W < Graph->Nv; W++)Graph->G[V][W] = INIFIN;return Graph;
}
//插入边
void InsertEdge(MGraph Graph, Edge E)
{/* 插入边 <V1, V2> */Graph->G[E->V1][E->V2] = 1;/* 若是无向图，还要插入边<V2, V1> *///Graph->G[E->V2][E->V1] = 1;
}MGraph BuildGraph() {Edge E;MGraph Graph;Vertex Nv;cin >> Nv;/*顶点个数*/Graph = CreateGraph(Nv);cin >> Graph->Ne;/*输入边的个数*/if (Graph->Ne != 0) {for (int i = 0; i < Graph->Ne; i++) {E = new ENode();cin >> E->V1 >> E->V2;InsertEdge(Graph, E);}}return Graph;
}
bool IsEdge(MGraph G, Vertex v1, Vertex v2) {return G->G[v1][v2] < INIFIN ? true : false;
}
void Unweighted(MGraph Graph, Vertex S) {Vertex V,W;dist[S] = 0;//初始化源点到自己的距离queue<Vertex>q;q.push(S);//压栈while(!q.empty()) {/*弹栈*/V = q.front();q.pop();for (W = 0; W < Graph->Nv; W++) {/*点W到源点的距离没有被确定并且两个顶点有边，*/if (dist[W] == -1 && IsEdge(Graph, V, W) ){dist[W] = dist[V] + 1;/*更新距离*/path[W] = V;//记录W的前驱q.push(W);}}}
}
void FindDp(int V) {if (V == -1)return;FindDp(path[V]);cout << V << " ";}
int main()
{MGraph G = BuildGraph();for (int i = 0; i < MaxVertenNum; i++) {dist[i] = path[i] = -1;}Unweighted(G, 0);cout << "请输入终点顶点:";int n;cin >> n;cout << "源点到终点的路径:";FindDp(n);cout << endl;system("pause");return 0;
}
/*
7
12
0 1
0 3
1 3
1 4
2 0
2 5
3 2
3 4
3 5
3 6
4 6
6 5
*/运行结果
三.有权图单源最短路算法
S集合存储的是的已经确定最短距离的顶点，起初集合只有一个顶点源点
若顶点V没有收录到集合,dist[V]表示源点到顶点V的距离，但这个距离不是真正的最短路径，当V顶点被收录到集合的时候，dist[V]才是最短距离
每次从dist[]中找到距离最短的顶点
当有顶点加入到集合中的时候,可能会影响dits[W]的值,及顶点的邻接点
代码如下
Vertex FindMinDist(MGraph Graph) {/*返回未被收入顶点中的最小dist者*/Vertex MinV, V;int MinDist = INFINITY;for (V = 0; V < Graph->Nv; V++) {if (Visted[V] == false && dist[V] < MinDist) {/*若V未被收录，且dist[V]更小*/MinDist = dist[V];/*更新最小距离*/MinV = V;/*更新最小顶点*/}}if (MinDist < INFINITY)/*若找到最小dist*/return MinV;/*返回对应的顶点下标*/else return 0;/*若这样的顶点不存在，返回错误标记*/
}
bool Dijkstra(MGraph Graph, Vertex S) {Vertex V, W;/*初始化：此处默认邻接矩阵中不存在的边用INFINITY表示*/for (V = 0; V < Graph->Nv; V++) {dist[V] = Graph->G[S][V];//与S结点邻接把边值放到dist中if (dist[V] < INFINITY)//有边记录它的前驱path[V] = S; //记录V的前驱V的前驱是Selsepath[V] = -1;Visted[V] = false;//全部顶点都没有收录}/*先将收起点收入集合*/dist[S] = 0;Visted[S] = true;//放到集合中while (1) {/*找到未被收入顶点中dist最小者*/V = FindMinDist(Graph);if (!V)/*如不存在*/break;//算法结束Visted[V] = true;//收录Vfor (W = 0; W < Graph->Nv; W++)if (Visted[W] == false && Graph->G[V][W] < INFINITY) {if (Graph->G[V][W] < 0)//若有负边return false;//if (dist[V] + Graph->G[V][W] < dist[W]) {dist[W] = dist[V] + Graph->G[V][W];path[W] = V;//跟新S到W的路径}}}return true;
}
全部代码
#include<iostream>
using namespace std;
#define INFINITY 65535/*设为最大值*/
#define MaxVertexNum 100 /*最大顶点树设为100*/
typedef int Vertex; /*用顶点下标表示为整型*/
typedef int WeightType;/*边的权值设为整型*/
typedef char DataType;/*顶点存储的数据类型设为字符型*/int Visted[MaxVertexNum];//集合等于false的都属于未被找的最短路径的结点
int dist[MaxVertexNum];//路径长度//
int path[MaxVertexNum];/*边的定义*/
typedef struct ENode* PtrToENode;
struct ENode {Vertex V1, V2;/*有向边<v1,v2>*/WeightType Weight;/*权重 */}; typedef PtrToENode Edge;/*图的定义*/
typedef struct GNode* PtrToGNode;
struct GNode
{int Nv;/*顶点个数*/int Ne;/*边的个数*/WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;/*以邻接矩阵存储的图的类型*/MGraph CreateGraph(int VertexNum) {/*初始化一个有VN个顶点但没有边的图*/Vertex V, W;MGraph Graph;Graph = new GNode();Graph->Nv = VertexNum;Graph->Ne = 0;/*初始化邻接矩阵*/for (V = 0; V < Graph->Nv; V++)for (W = 0; W < Graph->Nv; W++)Graph->G[V][W] = INFINITY;return Graph;
}
void Insert(MGraph Graph, Edge E) {Graph->G[E->V1][E->V2] = E->Weight;
}
bool IsEdge(MGraph Graph, Vertex V, Vertex W) {return Graph->G[V][W] < INFINITY ? true : false;
}
/*邻接矩阵存储*/Vertex FindMinDist(MGraph Graph) {/*返回未被收入顶点中的最小dist者*/Vertex MinV, V;int MinDist = INFINITY;for (V = 0; V < Graph->Nv; V++) {if (Visted[V] == false && dist[V] < MinDist) {/*若V未被收录，且dist[V]更小*/MinDist = dist[V];/*更新最小距离*/MinV = V;/*更新最小顶点*/}}if (MinDist < INFINITY)/*若找到最小dist*/return MinV;/*返回对应的顶点下标*/else return 0;/*若这样的顶点不存在，返回错误标记*/
}
bool Dijkstra(MGraph Graph, Vertex S) {Vertex V, W;/*初始化：此处默认邻接矩阵中不存在的边用INFINITY表示*/for (V = 0; V < Graph->Nv; V++) {dist[V] = Graph->G[S][V];//与S结点邻接把边值放到dist中if (dist[V] < INFINITY)//有边记录它的前驱path[V] = S; //记录V的前驱V的前驱是Selsepath[V] = -1;Visted[V] = false;//全部顶点都没有收录}/*先将收起点收入集合*/dist[S] = 0;Visted[S] = true;//放到集合中while (1) {/*找到未被收入顶点中dist最小者*/V = FindMinDist(Graph);if (!V)/*如不存在*/break;//算法结束Visted[V] = true;//收录Vfor (W = 0; W < Graph->Nv; W++)if (Visted[W] == false && Graph->G[V][W] < INFINITY) {if (Graph->G[V][W] < 0)//若有负边return false;//if (dist[V] + Graph->G[V][W] < dist[W]) {dist[W] = dist[V] + Graph->G[V][W];path[W] = V;//跟新S到W的路径}}}return true;
}
void FindDp(int V) {if (V == -1)return;FindDp(path[V]);cout << V << endl;}
int main()
{MGraph Graph = CreateGraph(7);Graph->Ne = 12;Vertex a[12] = { 0,0,1,1,2,2,3,3,3,3,4,6 };Vertex b[12] = { 1,3,3,4,0,5,2,4,5,6,6,5 };WeightType w[12] = { 2,1,3,10,4,5,2,2,8,4,6,1 };/* 读入边，格式为"起点 终点 权重"，插入邻接矩阵 */for (int i = 0; i < Graph->Ne; i++) {Edge E = (Edge)malloc(sizeof(struct ENode)); /* 建立边结点 */E->V1 = a[i]; E->V2 = b[i]; E->Weight = w[i];/* 注意：如果权重不是整型，Weight的读入格式要改 */Insert(Graph, E);}for (int i = 0; i < MaxVertexNum; i++) { Visted[i] = false; dist[i] = INFINITY; path[i] = -1; }Dijkstra(Graph, 0);int n;cin >> n;FindDp(n);return 0;
}运行结果
四.多源最短路算法
D^k[i][j]表示i到j经过k的最小长度
依次将每个点作为中间点去做更新
k=V-1是D[i][j]将最小路径更新完成
bool Floyd(MGraph Graph) {for(int i=0;i<Graph->Nv;i++)for (int j = 0; j < Graph->Nv; j++) {D[i][j] = Graph->G[i][j];path[i][j] = -1;}/*k表示中间点*/for(int k=0;k<Graph->Nv;k++)for (int i = 0; i < Graph->Nv; i++)for (int j = 0; j < Graph->Nv; j++)if (D[i][k] + D[k][j] < D[i][j]) {D[i][j] = D[i][k] + D[k][j];if (i == j && D[i][j] < 0)/*若出现负边无法解决*/return false;path[i][j] = k;}return true;
}全部代码
#include<iostream>
using namespace std;
#define MaxvertexNum 100/*最大顶点值*/
#define INF 65535/*无穷大表示没有边直接相连*/
typedef int Vertex;/*顶点下标*/
typedef int WeightType;/*权重*/
WeightType D[MaxvertexNum][MaxvertexNum];/*D[i][j]，顶点i到顶点j的最短距离*/
Vertex path[MaxvertexNum][MaxvertexNum];/*记录路径*//*边的定义*/
typedef struct ENode* PtrToENode;
struct ENode
{Vertex V1, V2;WeightType Weigtht;
};
typedef PtrToENode Edge;/*图的定义*/
typedef struct GNode* PtrToGNode;
struct GNode {int Nv;/*顶点个数*/int Ne; /*边的个数*/WeightType G[MaxvertexNum][MaxvertexNum];
};
typedef PtrToGNode MGraph;MGraph CreateGraph(int Vertexnum) {MGraph G = new GNode();G->Nv = Vertexnum;G->Ne = 0;for (int i = 0; i < G->Nv; i++)for (int j = 0; j < G->Nv; j++)G->G[i][j] = INF;return G;
}
void InsertEdeg(MGraph Graph, Edge E) {Graph->G[E->V1][E->V2] = E->Weigtht;Graph->G[E->V2][E->V1] = E->Weigtht;
}bool Floyd(MGraph Graph) {for(int i=0;i<Graph->Nv;i++)for (int j = 0; j < Graph->Nv; j++) {D[i][j] = Graph->G[i][j];path[i][j] = -1;}/*k表示中间点*/for(int k=0;k<Graph->Nv;k++)for (int i = 0; i < Graph->Nv; i++)for (int j = 0; j < Graph->Nv; j++)if (D[i][k] + D[k][j] < D[i][j]) {D[i][j] = D[i][k] + D[k][j];if (i == j && D[i][j] < 0)/*若出现负边无法解决*/return false;path[i][j] = k;}return true;
}
//void FindDp(int i, int j) {
// if (path[i][j] == -1)
// return;
// FindDp(i, path[i][j]);
// cout << j;
//}
MGraph BuildGraph() {Edge E;MGraph Graph;Vertex Nv;cin >> Nv;/*顶点个数*/Graph = CreateGraph(Nv);cin >> Graph->Ne;/*输入边的个数*/if (Graph->Ne != 0) {for (int i = 0; i < Graph->Ne; i++) {E = new ENode();cin >> E->V1 >> E->V2>>E->Weigtht;E->V1--;E->V2--;InsertEdeg(Graph, E);}}return Graph;
}
int main()
{MGraph Graph = BuildGraph();Floyd(Graph);int i,j;
// cin >> i>>j;
// FindDp(i,j);return 0;
}
怎么样打印两点之间的路径还没有写好
