单源点有权图的最短路径算法

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》

Dijkstra 算法

令S={源点s + 已经确定了最短路径的顶点v i }

对任一未收录的顶点v,定义dist[v]为s到v的最

短路径长度,但该路径仅经过S中的顶点。即路径

{s(v i S)v}的最小长度

若路径是按照递增(非递减)的顺序生成的,则

真正的最短路必须只经过S中的顶点(为什么?)

每次从未收录的顶点中选一个dist最小的收录(贪心)

增加一个v进入S,可能影响另外一个w的dist值!

dist[w] = min{dist[w], dist[v] + <v,w>的权重}

伪代码描述：

void Dijkstra( Vertex s )

{ while (1) {

V = 未收录顶点中dist最小者;

if ( 这样的V不存在 )

break;

collected[V] = true;

for ( V 的每个邻接点 W )

if ( collected[W] == false )

if ( dist[V]+E <V,W> < dist[W] ) {

dist[W] = dist[V] + E <V,W> ;

path[W] = V;

}

}

} /* 不能解决有负边的情况 */

算法复杂度分析：

方法1:直接扫描所有未收录顶点 – O( |V| )

T = O( |V| 2 + |E| )

对于稠密图效果好

方法2:将dist存在最小堆中 – O( log|V| )

更新dist[w]的值 – O( log|V| )

T = O( |V| log|V| + |E| log|V| ) = O( |E| log|V| )

/*
 * dijkstra.c
 *
 *  Created on: 2017年5月14日
 *      Author: ygh
 */

#include <stdio.h>
#include <stdlib.h>

#define MAX_VERTEX_NUM 100 /*define the max number of the vertex*/
#define INFINITY 65535     /*define double byte no negitive integer max number is 65535*/
#define ERROR -1

typedef int vertex; /*define the data type of the vertex*/
typedef int weightType; /*define the data type of the weight*/
typedef char dataType; /*define the data type of the vertex value*/

/*define the data structure of the Edge*/
typedef struct eNode *ptrToENode;
typedef struct eNode {
    vertex v1, v2; /*two vertex between the edge <v1,v2>*/
    weightType weight; /*the value of the edge's weigth */
};
typedef ptrToENode edge;

/*define the data structure of the graph*/
typedef struct gNode *ptrToGNode;
typedef struct gNode {
    int vertex_number; /*the number of the vertex*/
    int edge_nunber; /*the number of the edge*/
    weightType g[MAX_VERTEX_NUM][MAX_VERTEX_NUM]; /*define the adjacent matrix of graph*/
    dataType data[MAX_VERTEX_NUM]; /*define the dataType array to store the value of vertex*/
};
typedef ptrToGNode adjacentMatrixGraph; /*a graph show by adjacent matrix*/

/*
 create a graph given the vertex number.
 @param vertexNum The verter number of the graph
 @return a graph with vertex but no any egdgs
 */
adjacentMatrixGraph createGraph(int vertexNum) {
    vertex v, w;
    adjacentMatrixGraph graph;
    graph = (adjacentMatrixGraph) malloc(sizeof(struct gNode));
    graph->vertex_number = vertexNum;
    graph->edge_nunber = 0;
    /*initialize the adjacent matrix*/
    for (v = 0; v < graph->vertex_number; v++) {
        for (w = 0; w < graph->vertex_number; w++) {
            graph->g[v][w] = INFINITY;
        }
    }

    return graph;
}

/*
 insert a edge to graph.We will distinct oriented graph and undirected graph
 @param graph The graph you want to insert edge
 @param e The edge you want to insert the graph
 @param isOriented Whether the graph is oriented graph.If the graph is oriented
 we will set adjacent matrix [n][m]=[m][n]=edge's weight,else we only set
 the adjacent matrix [n][m]=edge's weight
 */
void inserEdge(adjacentMatrixGraph graph, edge e, int isOriented) {
    graph->g[e->v1][e->v2] = e->weight;
    if (!isOriented) {
        graph->g[e->v2][e->v1] = e->weight;
    }
}

/*
 construct a graph according user's input

 @return a graph has been filled good
 */
adjacentMatrixGraph buildGraph(int isOrdered) {
    adjacentMatrixGraph graph;
    edge e;
    vertex i;
    int vertex_num;
    scanf("%d", &vertex_num);
    graph = createGraph(vertex_num);
    scanf("%d", &(graph->edge_nunber));
    if (graph->edge_nunber) {
        e = (edge) malloc(sizeof(struct eNode));
        for (i = 0; i < graph->edge_nunber; i++) {
            scanf("%d %d %d", &e->v1, &e->v2, &e->weight);
            e->v1--;
            e->v2--;
            inserEdge(graph, e, isOrdered);
        }
    }
    return graph;

}

/*
 * Find the the index of point in the graph whose dist is minimal and has not been
 * accessed.
 * @param graph A graph which use adjacent matrix to store
 * @param dist A integer array to store the length from source to destination, it will be initialize with 65535 at first
 * @param collection A integer array to show whether the point has been accessed
 *         0 indicates the point has not been accessed,`1 indicates the point has been accessed
 *         the index in collection is same as the graph
 */
vertex findMinDist(adjacentMatrixGraph graph, int *dist, int *collection) {
    vertex minVertex, v;
    int minDist = INFINITY;
    /*
     * Find the minimal dist
     */
    for (v = 0; v < graph->vertex_number; v++) {
        if (dist[v] < minDist && collection[v] == 0) {
            minDist = dist[v];
            minVertex = v;
        }
    }

    if (minDist < INFINITY) {
        return minVertex;
    } else {
        return ERROR;
    }
}

/*
 * Find the shortest path from source to every point in graph with weight
 *@param graph A graph which use adjacent matrix to store
 *@param dist A integer array to store the length from source to destination, it will be initialize with 65535 at first
 *@param path A integer to store the index of last vertex which is shortest to pass current point,it will be initialize -1
 *@return 1 indicate the algorithms is correct calculate the result,0 indicates there is negative edge in the graph,a error happened
 */
int dijkstar(adjacentMatrixGraph graph, int *dist, int *path, vertex startPoint) {
    int collection[graph->vertex_number];
    vertex v, w;
    for (v = 0; v < graph->vertex_number; v++) {
        dist[v] = graph->g[startPoint][v];
        if (dist[v] < INFINITY) {
            path[v] = startPoint;
        } else {
            path[v] = -1;
            collection[v] = 0;
        }
    }
    collection[startPoint] = 1;
    dist[startPoint] = 0;
    while (1) {
        v = findMinDist(graph, dist, collection);
        if (v == ERROR) {
            break;
        }
        collection[v] = 1;
        for (w = 0; w < graph->vertex_number; w++) {
            if (collection[w] == 0 && graph->g[v][w] < INFINITY) {
                /*
                 * If a edge weight is a negative,Dijkstra will not to solve it,return 0
                 */
                if (graph->g[v][w]<0) {
                    return 0;
                }
                /*
                 * If v make dist[w] get smaller,updata it
                 */
                if (dist[v] + graph->g[v][w] < dist[w]) {
                    dist[w] = dist[v] + graph->g[v][w];
                    path[w] = v;
                }
            }
        }
    }
    return 1;
}

/*============================define a stack to print result=============*/
typedef int stackElement;
typedef struct node3 {
    stackElement element;
    struct node3 *next;
} sta, *pStack;

pStack createEmptyStack() {
    pStack stack;
    stack = (pStack) malloc(sizeof(sta));
    if (stack) {
        stack->next = NULL;
    }
    return stack;
}

int isEmpty(pStack stack) {
    if (stack->next == NULL) {
        return 1;
    } else {
        return 0;
    }
}

void push(pStack stack, stackElement element) {
    pStack node = (pStack) malloc(sizeof(sta));
    node->element = element;
    node->next = stack->next;
    stack->next = node;
}

stackElement pop(pStack stack) {
    stackElement element;
    pStack topHead;
    if (isEmpty(stack)) {
        printf("the stack is empty,can not pop");
        return -65536;
    } else {
        topHead = stack->next;
        stack->next = topHead->next;
        element = topHead->element;
        free(topHead);
        return element;
    }
}

void findPath(int *path, int length, int destination) {
    pStack stack = createEmptyStack();
    vertex v;
    int index = destination;
    push(stack, index);
    while (v != -1) {
        v = path[index];
        push(stack, v);
        index = v;
    }

    pop(stack);
    while (!isEmpty(stack)) {
        stackElement element = pop(stack);
        printf("%d ", element + 1);
    }
}

/*
 * Fill a array with value
 * @param arr The array need to be filled
 * @param length The length of the array
 * @param filledValue The value the array will be filled
 */
void fullArray(int *arr, int length, int filledValue) {
    int i;
    for (i = 0; i < length; i++) {
        arr[i] = filledValue;
    }
}
int main() {
    adjacentMatrixGraph graph = buildGraph(1);
    int dist[graph->vertex_number];
    int path[graph->vertex_number];
    dijkstar(graph, dist, path, 0);
    findPath(path, graph->edge_nunber, 5);
    return 0;
}

测试数据：

a data of graph:

7 12
1 2 2
1 4 1
2 5 10
2 4 3
3 1 4
3 6 5
4 3 2
4 6 8
4 7 4
4 5 2
5 7 6
7 6 1

from 1 to 6 shortest path
1 4 7 6

下面列举一道题目：

有了一张自驾旅游路线图，你会知道城市间的高速公路长度、以及该公路要收取的过路费。现在需要你写一个程序，帮助前来咨询的游客找一条出发地和目的地之间的最短路径。如果有若干条路径都是最短的，那么需要输出最便宜的一条路径。

输入格式:

输入说明：输入数据的第1行给出4个正整数NNN、MMM、SSS、DDD，其中NNN（2≤N≤5002le Nle 5002≤N≤500）是城市的个数，顺便假设城市的编号为0~(N−1N-1N−1)；MMM是高速公路的条数；SSS是出发地的城市编号；DDD是目的地的城市编号。随后的MMM行中，每行给出一条高速公路的信息，分别是：城市1、城市2、高速公路长度、收费额，中间用空格分开，数字均为整数且不超过500。输入保证解的存在。

输出格式:

在一行里输出路径的长度和收费总额，数字间以空格分隔，输出结尾不能有多余空格。

输入样例:

4 5 0 3
0 1 1 20
1 3 2 30
0 3 4 10
0 2 2 20
2 3 1 20

输出样例:

3 40

算法思想：其实很简单，我们只需要在Dijkstra算法的基础上，除了吧路径的长度(权重)作为最短路径以外，还需要把价格按照路径的相同方法进行

更新和记录。当路径不等时，我们优先考虑路径，并根据路径的长短里记录上一个节点的角标。如果路径相等，我们需要根据价格来判断上一个节点的

角标。这样就可以实现这道题目了。在插入城市节点的时候，注意是无向图，不然代码会不通过的。下面是具体的代码

/*
 * travelPlan.c
 *
 *  Created on: 2017年5月14日
 *      Author: ygh
 */
#include <stdio.h>
#include <stdlib.h>

#define MAX_VERTEX_NUM 500 /*define the max number of the vertex*/
#define INFINITY 65535     /*define double byte no negitive integer max number is 65535*/
#define ERROR -1

typedef int vertex; /*define the data type of the vertex*/
typedef int weightType; /*define the data type of the weight*/
typedef char dataType; /*define the data type of the vertex value*/
typedef int priceType;

/*define the data structure of the Edge*/
typedef struct eNode *ptrToENode;
typedef struct eNode {
    vertex v1, v2; /*two vertex between the edge <v1,v2>*/
    weightType weight; /*the value of the edge's weight */
    priceType price; /*The value of price of the price*/
};
typedef ptrToENode edge;

/*define the data structure of the graph*/
typedef struct gNode *ptrToGNode;
typedef struct gNode {
    vertex source; /*The index of the source*/
    vertex destination; /*The index of the destination*/
    int vertex_number; /*the number of the vertex*/
    int edge_nunber; /*the number of the edge*/
    weightType g[MAX_VERTEX_NUM][MAX_VERTEX_NUM]; /*define the adjacent matrix weight of graph*/
    priceType p[MAX_VERTEX_NUM][MAX_VERTEX_NUM]; /*define the adjacent matrix price of graph*/
    dataType data[MAX_VERTEX_NUM]; /*define the dataType array to store the value of vertex*/
};
typedef ptrToGNode adjacentMatrixGraph; /*a graph show by adjacent matrix*/

/*
 create a graph given the vertex number.
 @param vertexNum The verter number of the graph
 @return a graph with vertex but no any egdgs
 */
adjacentMatrixGraph createGraph(int vertexNum) {
    vertex v, w;
    adjacentMatrixGraph graph;
    graph = (adjacentMatrixGraph) malloc(sizeof(struct gNode));
    graph->vertex_number = vertexNum;
    graph->edge_nunber = 0;
    /*initialize the adjacent matrix*/
    for (v = 0; v < graph->vertex_number; v++) {
        for (w = 0; w < graph->vertex_number; w++) {
            graph->g[v][w] = INFINITY;
            graph->p[v][w] = INFINITY;
        }
    }

    return graph;
}

/*
 insert a edge to graph.We will distinct oriented graph and undirected graph
 @param graph The graph you want to insert edge
 @param e The edge you want to insert the graph
 @param isOriented Whether the graph is oriented graph.If the graph is oriented
 we will set adjacent matrix [n][m]=[m][n]=edge's weight,else we only set
 the adjacent matrix [n][m]=edge's weight
 */
void inserEdge(adjacentMatrixGraph graph, edge e, int isOriented) {
    graph->g[e->v1][e->v2] = e->weight;
    graph->p[e->v1][e->v2] = e->price;
    if (!isOriented) {
        graph->g[e->v2][e->v1] = e->weight;
        graph->p[e->v2][e->v1] = e->price;
    }
}

/*
 construct a graph according user's input

 @return a graph has been filled good
 */
adjacentMatrixGraph buildGraph(int isOrdered) {
    adjacentMatrixGraph graph;
    edge e;
    vertex i;
    vertex source, destination;
    int vertex_num;
    scanf("%d", &vertex_num);
    graph = createGraph(vertex_num);
    scanf("%d", &(graph->edge_nunber));
    scanf("%d %d", &source, &destination);
    graph->source = source;
    graph->destination = destination;
    if (graph->edge_nunber) {
        e = (edge) malloc(sizeof(struct eNode));
        for (i = 0; i < graph->edge_nunber; i++) {
            scanf("%d %d %d %d", &e->v1, &e->v2, &e->weight, &e->price);
            inserEdge(graph, e, isOrdered);
        }
    }
    return graph;

}

/*
 * Find the the index of point in the graph whose dist is minimal and has not been
 * accessed.
 * @param graph A graph which use adjacent matrix to store
 * @param dist A integer array to store the length from source to destination, it will be initialize with 65535 at first
 * @param collection A integer array to show whether the point has been accessed
 *         0 indicates the point has not been accessed,`1 indicates the point has been accessed
 *         the index in collection is same as the graph
 */
vertex findMinDist(adjacentMatrixGraph graph, int *dist, int *collection) {
    vertex minVertex, v;
    int minDist = INFINITY;
    /*
     * Find the minimal dist
     */
    for (v = 0; v < graph->vertex_number; v++) {
        if (dist[v] < minDist && collection[v] == 0) {
            minDist = dist[v];
            minVertex = v;
        }
    }

    if (minDist < INFINITY) {
        return minVertex;
    } else {
        return ERROR;
    }
}

/*
 * Find the shortest path from source to every point in graph with weight
 *@param graph A graph which use adjacent matrix to store
 *@param dist A integer array to store the length from source to destination, it will be initialize with 65535 at first
 *@param path A integer to store the index of last vertex which is shortest to pass current point,it will be initialize -1
 *@return 1 indicate the algorithms is correct calculate the result,0 indicates there is negative edge in the graph,a error happened
 */
int dijkstar(adjacentMatrixGraph graph, int *dist, int *path, int *totalPrice,
        vertex startPoint) {
    int collection[graph->vertex_number];
    vertex v, w;
    for (v = 0; v < graph->vertex_number; v++) {
        dist[v] = graph->g[startPoint][v];
        totalPrice[v] = graph->p[startPoint][v];
        if (dist[v] < INFINITY) {
            path[v] = startPoint;
        } else {
            path[v] = -1;
            collection[v] = 0;
        }
    }
    collection[startPoint] = 1;
    dist[startPoint] = 0;
    totalPrice[startPoint] = 0;
    while (1) {
        v = findMinDist(graph, dist, collection);
        if (v == ERROR) {
            break;
        }
        collection[v] = 1;
        for (w = 0; w < graph->vertex_number; w++) {
            if (collection[w] == 0 && graph->g[v][w] < INFINITY) {
                /*
                 * If a edge weight is a negative,Dijkstra will not to solve it,return 0
                 */
                if (graph->g[v][w] < 0) {
                    return 0;
                }
                /*
                 * If v make dist[w] get smaller,updata it
                 */
                if (dist[v] + graph->g[v][w] < dist[w]) {
                    dist[w] = dist[v] + graph->g[v][w];
                    totalPrice[w] = totalPrice[v] + graph->p[v][w];
                    path[w] = v;
                } else if ((dist[v] + graph->g[v][w] == dist[w])
                        && (totalPrice[w] > totalPrice[v] + graph->p[v][w])) {
                    totalPrice[w] = totalPrice[v] + graph->p[v][w];
                    path[w] = v;
                }
            }
        }
    }
    return 1;
}

/*============================define a stack to print result=============*/
typedef int stackElement;
typedef struct node3 {
    stackElement element;
    struct node3 *next;
} sta, *pStack;

pStack createEmptyStack() {
    pStack stack;
    stack = (pStack) malloc(sizeof(sta));
    if (stack) {
        stack->next = NULL;
    }
    return stack;
}

int isEmpty(pStack stack) {
    if (stack->next == NULL) {
        return 1;
    } else {
        return 0;
    }
}

void push(pStack stack, stackElement element) {
    pStack node = (pStack) malloc(sizeof(sta));
    node->element = element;
    node->next = stack->next;
    stack->next = node;
}

stackElement pop(pStack stack) {
    stackElement element;
    pStack topHead;
    if (isEmpty(stack)) {
        printf("the stack is empty,can not pop");
        return -65536;
    } else {
        topHead = stack->next;
        stack->next = topHead->next;
        element = topHead->element;
        free(topHead);
        return element;
    }
}

void calculateDistanceAndFee(adjacentMatrixGraph graph, int *path, int length,
        int destination) {
    pStack stack = createEmptyStack();
    vertex v;
    int index = destination;
    int totalDistace = 0;
    int totalPrice = 0;
    int arr[graph->vertex_number];
    int counter = 0, i;
    push(stack, index);
    while (v != -1) {
        v = path[index];
        push(stack, v);
        index = v;
    }

    pop(stack);
    while (!isEmpty(stack)) {
        stackElement element = pop(stack);
        arr[counter++] = element;
    }

    for (i = 1; i < counter; i++) {
        totalDistace = totalDistace + graph->g[arr[i - 1]][arr[i]];
        totalPrice = totalPrice + graph->p[arr[i - 1]][arr[i]];
    }

    printf("%d %d", totalDistace, totalPrice);
}

/*
 * Fill a array with value
 * @param arr The array need to be filled
 * @param length The length of the array
 * @param filledValue The value the array will be filled
 */
void fullArray(int *arr, int length, int filledValue) {
    int i;
    for (i = 0; i < length; i++) {
        arr[i] = filledValue;
    }
}
int main() {
    /*
     * This is a no-direct graph
     */
    adjacentMatrixGraph graph = buildGraph(0);
    int dist[graph->vertex_number];
    int path[graph->vertex_number];
    int totalPrice[graph->vertex_number];
    dijkstar(graph, dist, path, totalPrice, 0);
    calculateDistanceAndFee(graph, path, graph->edge_nunber,
            graph->destination);
    return 0;
}

但是这个代码也不能通过所以的测试，系统设置价格和最短路径都相同的路径，好像要列出所以的路径，但是我为了赶课程进度，就没有深究了

如果你找到了问题，可以在在评论区回复我。谢谢！