算法导论-25.1-最短路径与矩阵乘法

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一、介绍

二、代码

#include <iostream>#include <algorithm>using namespace std;#define N 6//点的个数#define M 10//边的个数//邻接矩阵struct Graph{int map[N+1][N+1];int row;Graph(int n):row(n){int i, j;for(i = 1; i <= N; i++){for(j = 1; j <= N; j++)if(i == j)map[i][j] = 0;elsemap[i][j] = 0x7fffffff;}}Graph(Graph &G){row = G.row;int i, j;for(i = 1; i <= N; i++){for(j = 1; j <= N; j++)map[i][j] = G.map[i][j];}}};int min(int a, int b){return a < b ? a : b;}void Print(Graph G){int i, j;for(i = 1; i <= N; i++){for(j = 1; j <= N; j++)cout<<G.map[i][j]<<' ';cout<<endl;}cout<<endl;}Graph Extend_Shortest_Paths(Graph L, Graph W){int i, j, k, n = W.row;Graph ret(n);for(i = 1; i <= n; i++){for(j = 1; j <= n; j++){ret.map[i][j] = 0x7fffffff;for(k = 1; k <= n; k++){if(L.map[i][k] != 0x7fffffff && W.map[k][j] != 0x7fffffff)ret.map[i][j] = min(ret.map[i][j], L.map[i][k] + W.map[k][j]);}}}return ret;}Graph Matrix_Multiply(Graph A, Graph B){int i, j, k, n = A.row;Graph ret(n);for(i = 1; i <= n; i++){for(j = 1; j <= n; j++){ret.map[i][j] = 0;for(k = 1; k <= n; k++){ret.map[i][j] = ret.map[i][j] + A.map[i][k] * B.map[k][j];}}}return ret;}Graph Slow_All_Pairs_Shortest_Paths(Graph W){int n = W.row, m;Graph L(W);for(m = 2; m < n; m++){L = Extend_Shortest_Paths(L, W);//Print(L);}return L;}Graph Faster_All_Pairs_Shortest_Paths(Graph W){int n = W.row, m = 1;Graph L(W);while(m < n - 1){L = Extend_Shortest_Paths(L, L);m = 2 * m;Print(L);}return L;}/*1 5 -12 1 12 4 23 2 23 6 -84 1 -44 5 35 2 76 2 56 3 10*/int main(){int i, start, end, value;Graph G(N);for(i = 1;i <= M; i++){cin>>start>>end>>value;G.map[start][end] = value;}Print(G);Slow_All_Pairs_Shortest_Paths(G);Faster_All_Pairs_Shortest_Paths(G);return 0;}

三、练习

25.1-1

SLOW-ALL-PARIS-SHORTEST-PATHS：

FASTER-ALL-PAIRS-SHORTEST-PATHS：

25.1-2

自己到自己不需要代价

25.1-3

单位矩阵E

25.1-5

25.1-6

25.1-7

25.1-8

25.1-9

25.1-10