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  • 多源最短路径算法

    本博客的代码的思想和图片参考：好大学慕课浙江大学陈越老师、何钦铭老师的《数据结构》

     多源最短路径算法

    1.使用Dijkstra算法对每个顶点运行一次运算，可以得到每个顶点到最图所有顶点的最小值，时间复杂度为：T = O( |V| 3 + |E||V|)。该算法对稀疏图比较好

    2.使用Floyd算法，时间复杂度为：T = O( |V| 3 )，该算法对稠密图比较好



    下面介绍Floyd算法

    D k [i][j] = 路径{ i --> { l --> k } --> j }的最小长度

    D 0 , D 1 , ..., D |V|-1 [i][j]即给出了i到j的真正最短距离

    最初的D -1 是什么?

    当D k-1 已经完成,递推到D k 时:

    或者k 属于最短路径{ i --> { l --> k } --> j },则D k = D k-1

    或者k 属于最短路径{ i --> { l --> k } --> j },则该路径必定由两

    段最短路径组成: D k [i][j]=D k-1 [i][k]+D k-1 [k][j]



    void Floyd()

    { for ( i = 0; i < N; i++ )

    for( j = 0; j < N; j++ ) {

    D[i][j] = G[i][j];

    path[i][j] = -1;

    }

    for( k = 0; k < N; k++ )

    for( i = 0; i < N; i++ )

    for( j = 0; j < N; j++ )

    if( D[i][k] + D[k][j] < D[i][j] ) {

    D[i][j] = D[i][k] + D[k][j];

    path[i][j] = k;

    }

    }

      1 /*
  2  * floyd.c
  3  *
  4  *  Created on: 2017年5月14日
  5  *      Author: ygh
  6  */
  7 #include <stdio.h>
  8 #include <stdlib.h>
  9 
 10 #define MAX_VERTEX_NUM 100 /*define the max number of the vertex*/
 11 #define INFINITY 65535     /*define double byte no negitive integer max number is 65535*/
 12 #define ERROR -1
 13 
 14 typedef int vertex; /*define the data type of the vertex*/
 15 typedef int weightType; /*define the data type of the weight*/
 16 typedef char dataType; /*define the data type of the vertex value*/
 17 
 18 /*define the data structure of the Edge*/
 19 typedef struct eNode *ptrToENode;
 20 typedef struct eNode {
 21     vertex v1, v2; /*two vertex between the edge <v1,v2>*/
 22     weightType weight; /*the value of the edge's weight */
 23 };
 24 typedef ptrToENode edge;
 25 
 26 /*define the data structure of the graph*/
 27 typedef struct gNode *ptrToGNode;
 28 typedef struct gNode {
 29     int vertex_number; /*the number of the vertex*/
 30     int edge_nunber; /*the number of the edge*/
 31     weightType g[MAX_VERTEX_NUM][MAX_VERTEX_NUM]; /*define the adjacent matrix weight of graph*/
 32     dataType data[MAX_VERTEX_NUM]; /*define the dataType array to store the value of vertex*/
 33 };
 34 typedef ptrToGNode adjacentMatrixGraph; /*a graph show by adjacent matrix*/
 35 
 36 /*
 37  create a graph given the vertex number.
 38  @param vertexNum The verter number of the graph
 39  @return a graph with vertex but no any egdgs
 40  */
 41 adjacentMatrixGraph createGraph(int vertexNum) {
 42     vertex v, w;
 43     adjacentMatrixGraph graph;
 44     graph = (adjacentMatrixGraph) malloc(sizeof(struct gNode));
 45     graph->vertex_number = vertexNum;
 46     graph->edge_nunber = 0;
 47     /*initialize the adjacent matrix*/
 48     for (v = 0; v < graph->vertex_number; v++) {
 49         for (w = 0; w < graph->vertex_number; w++) {
 50             graph->g[v][w] = INFINITY;
 51         }
 52     }
 53 
 54     return graph;
 55 }
 56 
 57 /*
 58  insert a edge to graph.We will distinct oriented graph and undirected graph
 59  @param graph The graph you want to insert edge
 60  @param e The edge you want to insert the graph
 61  @param isOriented Whether the graph is oriented graph.If the graph is oriented
 62  we will set adjacent matrix [n][m]=[m][n]=edge's weight,else we only set
 63  the adjacent matrix [n][m]=edge's weight
 64  */
 65 void inserEdge(adjacentMatrixGraph graph, edge e, int isOriented) {
 66     graph->g[e->v1][e->v2] = e->weight;
 67     if (!isOriented) {
 68         graph->g[e->v2][e->v1] = e->weight;
 69     }
 70 }
 71 
 72 /*
 73  construct a graph according user's input
 74 
 75  @return a graph has been filled good
 76  */
 77 adjacentMatrixGraph buildGraph(int isOrdered) {
 78     adjacentMatrixGraph graph;
 79     edge e;
 80     vertex i;
 81     int vertex_num;
 82     scanf("%d", &vertex_num);
 83     graph = createGraph(vertex_num);
 84     scanf("%d", &(graph->edge_nunber));
 85     if (graph->edge_nunber) {
 86         e = (edge) malloc(sizeof(struct eNode));
 87         for (i = 0; i < graph->edge_nunber; i++) {
 88             scanf("%d %d %d", &e->v1, &e->v2, &e->weight);
 89             e->v1--;
 90             e->v2--;
 91             inserEdge(graph, e, isOrdered);
 92         }
 93     }
 94     return graph;
 95 
 96 }
 97 
 98 /*
 99  * Floyd algorithms:
100  1.D k [i][j] = 路径{ i --> { l -->  k } -->  j }的最小长度
101  2.D 0 , D 1 , ..., D |V|-1 [i][j]即给出了i到j的真正最短距离
102  3.最初的D -1 是什么?
103  4.当D k-1 已经完成,递推到D k 时:
104  或者k 属于最短路径{ i -->  { l -->  k } --> j },则D k = D k-1
105  或者k 属于最短路径{ i -->  { l -->  k } -->  j },则该路径必定由两
106  段最短路径组成: D k [i][j]=D k1 [i][k]+D k1 [k][j]
107  *@param graph A graph store by adjacent matrix
108  *@param d A two-dimensional array to store the distance value
109  *@param path A two-dimensional array to store the path
110  */
111 int floyd(adjacentMatrixGraph graph, weightType d[][MAX_VERTEX_NUM],
112         vertex path[][MAX_VERTEX_NUM]) {
113     vertex i, j, k;
114     /*
115      * Initialize array
116      */
117     for (i = 0; i < graph->vertex_number; i++) {
118         for (j = 0; j < graph->vertex_number; j++) {
119             d[i][j] = graph->g[i][j];
120             path[i][j] = -1;
121         }
122     }
123     for (k = 0; k < graph->vertex_number; k++) {
124         for (i = 0; i < graph->vertex_number; i++) {
125             for (j = 0; j < graph->vertex_number; j++) {
126                 if (d[i][k] + d[k][j] < d[i][j]) {
127                     d[i][j] = d[i][k] + d[k][j];
128                     /*
129                      * Find negative circle
130                      */
131                     if (i == j && d[i][j] < 0) {
132                         return 0;
133                     }
134                     path[i][j] = k;
135                 }
136             }
137         }
138     }
139     return 1;
140 }
141 
142 /*
143  * Print the distance matrix
144  */
145 void toStringDistance(int d[MAX_VERTEX_NUM][MAX_VERTEX_NUM], int length) {
146     vertex i, j;
147     for (i = 0; i < length; i++) {
148         printf("%d:", i);
149         for (j = 0; j < length; j++) {
150             printf("%d ", d[i][j]);
151         }
152         printf("
");
153     }
154 }
155 
156 /*
157  * Print the path from source to destination
158  * we will recursive method to print the path
159  * 1.find the k between source and destination
160  * 2.find point between source and k and recursive this method until there no points between them
161  * 3.find the point between k and destination and recursive this method until there no points between them
162  * @param source The index of the source point
163  * @param destination The index of the destination
164  * @param path A two-dimensional array to store the path
165  */
166 void toStringPath(int source, int destination, vertex path[][MAX_VERTEX_NUM]) {
167     if (destination == -1 || source == -1) {
168         return;
169     } else {
170         toStringPath(source, path[source][destination], path);
171         if (path[source][destination] != -1) {
172             printf("%d ", path[source][destination]);
173         }
174         toStringPath(path[source][destination], destination, path);
175     }
176 
177 }
178 
179 /*
180  * A test method
181  */
182 int main() {
183     adjacentMatrixGraph graph = buildGraph(1);
184     int d[MAX_VERTEX_NUM][MAX_VERTEX_NUM];
185     int path[MAX_VERTEX_NUM][MAX_VERTEX_NUM];
186     int source = 1;
187     int destination = 5;
188     floyd(graph, d, path);
189     printf("toDistance
");
190     toStringDistance(d, graph->vertex_number);
191     printf("toPath:");
192     printf("%d ", source);
193     toStringPath(source, destination, path);
194     printf("%d", destination);
195     return 0;
196 }
    Floyd

    下面是一个练习题：

    哈利·波特要考试了，他需要你的帮助。这门课学的是用魔咒将一种动物变成另一种动物的本事。例如将猫变成老鼠的魔咒是haha，将老鼠变成鱼的魔咒是hehe等等。反方向变化的魔咒就是简单地将原来的魔咒倒过来念，例如ahah可以将老鼠变成猫。另外，如果想把猫变成鱼，可以通过念一个直接魔咒lalala，也可以将猫变老鼠、老鼠变鱼的魔咒连起来念：hahahehe。

    现在哈利·波特的手里有一本教材，里面列出了所有的变形魔咒和能变的动物。老师允许他自己带一只动物去考场，要考察他把这只动物变成任意一只指定动物的本事。于是他来问你：带什么动物去可以让最难变的那种动物（即该动物变为哈利·波特自己带去的动物所需要的魔咒最长）需要的魔咒最短？例如：如果只有猫、鼠、鱼，则显然哈利·波特应该带鼠去，因为鼠变成另外两种动物都只需要念4个字符；而如果带猫去，则至少需要念6个字符才能把猫变成鱼；同理，带鱼去也不是最好的选择。

    输入格式:

    输入说明：输入第1行给出两个正整数NNN (≤100le 100≤100)和MMM，其中NNN是考试涉及的动物总数，MMM是用于直接变形的魔咒条数。为简单起见，我们将动物按1~NNN编号。随后MMM行，每行给出了3个正整数，分别是两种动物的编号、以及它们之间变形需要的魔咒的长度(≤100le 100≤100)，数字之间用空格分隔。

    输出格式:

    输出哈利·波特应该带去考场的动物的编号、以及最长的变形魔咒的长度，中间以空格分隔。如果只带1只动物是不可能完成所有变形要求的，则输出0。如果有若干只动物都可以备选，则输出编号最小的那只。

    输入样例:

    6 11
3 4 70
1 2 1
5 4 50
2 6 50
5 6 60
1 3 70
4 6 60
3 6 80
5 1 100
2 4 60
5 2 80

    输出样例:

    4 70

      1 /*
  2  * harrypot.c
  3  *
  4  *  Created on: 2017年5月14日
  5  *      Author: ygh
  6  */
  7 #include <stdio.h>
  8 #include <stdlib.h>
  9 
 10 #define MAX_VERTEX_NUM 100 /*define the max number of the vertex*/
 11 #define INFINITY 65535     /*define double byte no negitive integer max number is 65535*/
 12 #define ERROR -1
 13 
 14 typedef int vertex; /*define the data type of the vertex*/
 15 typedef int weightType; /*define the data type of the weight*/
 16 typedef char dataType; /*define the data type of the vertex value*/
 17 
 18 /*define the data structure of the Edge*/
 19 typedef struct eNode *ptrToENode;
 20 typedef struct eNode {
 21     vertex v1, v2; /*two vertex between the edge <v1,v2>*/
 22     weightType weight; /*the value of the edge's weight */
 23 };
 24 typedef ptrToENode edge;
 25 
 26 /*define the data structure of the graph*/
 27 typedef struct gNode *ptrToGNode;
 28 typedef struct gNode {
 29     int vertex_number; /*the number of the vertex*/
 30     int edge_nunber; /*the number of the edge*/
 31     weightType g[MAX_VERTEX_NUM][MAX_VERTEX_NUM]; /*define the adjacent matrix weight of graph*/
 32     dataType data[MAX_VERTEX_NUM]; /*define the dataType array to store the value of vertex*/
 33 };
 34 typedef ptrToGNode adjacentMatrixGraph; /*a graph show by adjacent matrix*/
 35 
 36 /*
 37  create a graph given the vertex number.
 38  @param vertexNum The verter number of the graph
 39  @return a graph with vertex but no any egdgs
 40  */
 41 adjacentMatrixGraph createGraph(int vertexNum) {
 42     vertex v, w;
 43     adjacentMatrixGraph graph;
 44     graph = (adjacentMatrixGraph) malloc(sizeof(struct gNode));
 45     graph->vertex_number = vertexNum;
 46     graph->edge_nunber = 0;
 47     /*initialize the adjacent matrix*/
 48     for (v = 0; v < graph->vertex_number; v++) {
 49         for (w = 0; w < graph->vertex_number; w++) {
 50             graph->g[v][w] = INFINITY;
 51         }
 52     }
 53 
 54     return graph;
 55 }
 56 
 57 /*
 58  insert a edge to graph.We will distinct oriented graph and undirected graph
 59  @param graph The graph you want to insert edge
 60  @param e The edge you want to insert the graph
 61  @param isOriented Whether the graph is oriented graph.If the graph is oriented
 62  we will set adjacent matrix [n][m]=[m][n]=edge's weight,else we only set
 63  the adjacent matrix [n][m]=edge's weight
 64  */
 65 void inserEdge(adjacentMatrixGraph graph, edge e, int isOriented) {
 66     graph->g[e->v1][e->v2] = e->weight;
 67     if (!isOriented) {
 68         graph->g[e->v2][e->v1] = e->weight;
 69     }
 70 }
 71 
 72 /*
 73  construct a graph according user's input
 74 
 75  @return a graph has been filled good
 76  */
 77 adjacentMatrixGraph buildGraph(int isOrdered) {
 78     adjacentMatrixGraph graph;
 79     edge e;
 80     vertex i;
 81     int vertex_num;
 82     scanf("%d", &vertex_num);
 83     graph = createGraph(vertex_num);
 84     scanf("%d", &(graph->edge_nunber));
 85     if (graph->edge_nunber) {
 86         e = (edge) malloc(sizeof(struct eNode));
 87         for (i = 0; i < graph->edge_nunber; i++) {
 88             scanf("%d %d %d", &e->v1, &e->v2, &e->weight);
 89             e->v1--;
 90             e->v2--;
 91             inserEdge(graph, e, isOrdered);
 92         }
 93     }
 94     return graph;
 95 
 96 }
 97 
 98 int floyd(adjacentMatrixGraph graph, weightType d[][MAX_VERTEX_NUM],
 99         vertex path[][MAX_VERTEX_NUM]) {
100     vertex i, j, k;
101     /*
102      * Initialize array
103      */
104     for (i = 0; i < graph->vertex_number; i++) {
105         for (j = 0; j < graph->vertex_number; j++) {
106             d[i][j] = graph->g[i][j];
107             path[i][j] = -1;
108         }
109     }
110     for (k = 0; k < graph->vertex_number; k++) {
111         for (i = 0; i < graph->vertex_number; i++) {
112             for (j = 0; j < graph->vertex_number; j++) {
113                 if (d[i][k] + d[k][j] < d[i][j]) {
114                     d[i][j] = d[i][k] + d[k][j];
115                     /*
116                      * Find negative circle
117                      */
118                     if (i == j) {
119                         d[i][j] = INFINITY;
120                     }
121                     path[i][j] = k;
122                 }
123             }
124         }
125     }
126     return 1;
127 }
128 
129 void toStringDistance(int d[MAX_VERTEX_NUM][MAX_VERTEX_NUM], int length) {
130     vertex i, j;
131     for (i = 0; i < length; i++) {
132         printf("%d:", i);
133         for (j = 0; j < length; j++) {
134             printf("%d ", d[i][j]);
135         }
136         printf("
");
137     }
138 }
139 
140 /*
141  * Algorithms thoughts:
142  * choose the max value expect for 65535 then sore a array,then find the
143  * minimal value from the array and the index
144  */
145 void selectAnnimal(int d[MAX_VERTEX_NUM][MAX_VERTEX_NUM], int length) {
146     vertex i, j;
147     int maxValue = -1;
148     int minValue = 65535;
149     int key = 0;
150     int arr[length];
151     for (i = 0; i < length; i++) {
152         maxValue = -1;
153         for (j = 0; j < length; j++) {
154             if (d[i][j] > maxValue && i != j) {
155                 maxValue = d[i][j];
156             }
157         }
158         if (maxValue == INFINITY) {
159             printf("0
");
160             return;
161         } else {
162             arr[i] = maxValue;
163         }
164     }
165 
166     for (i = 0; i < length; i++) {
167         if (arr[i] < minValue) {
168             key = i;
169             minValue = arr[i];
170         }
171     }
172 
173     printf("%d %d", key + 1, minValue);
174 }
175 
176 int main() {
177     adjacentMatrixGraph graph = buildGraph(0);
178     int d[MAX_VERTEX_NUM][MAX_VERTEX_NUM];
179     int path[MAX_VERTEX_NUM][MAX_VERTEX_NUM];
180     floyd(graph, d, path);
181     selectAnnimal(d, graph->vertex_number);
182     return 0;
183 }
    HarryPot

    测试数据：

    6 11
    3 4 70
    1 2 1
    5 4 50
    2 6 50
    5 6 60
    1 3 70
    4 6 60
    3 6 80
    5 1 100
    2 4 60
    5 2 80

    输出样例:

    4 70
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  • 原文地址：https://www.cnblogs.com/yghjava/p/6853790.html
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