Shortest-Path Algorithms – Dijkstra’s Algorithm Explained in a Page

Problem 1

Problem 2 – Shortest Path Variant 1

Write a program to not only find the weighted shortest distances but also count the number of different minimum paths from any vertex to a given source vertex in a digraph. It is guaranteed that all the weights are positive.

Format of functions:

void ShortestDist( MGraph Graph, int dist[], int count[], Vertex S );

where MGraph is defined as the following:

typedef struct GNode *PtrToGNode;
struct GNode{
    int Nv;
    int Ne;
    WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;

The shortest distance from V to the source S is supposed to be stored in dist[V]. If V cannot be reached from S, store -1 instead. The number of different minimum paths from V to the source S is supposed to be stored in count[V] and count[S]=1.

Sample program of judge:

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

typedef enum {false, true} bool;
#define INFINITY 1000000
#define MaxVertexNum 10  /* maximum number of vertices */
typedef int Vertex;      /* vertices are numbered from 0 to MaxVertexNum-1 */
typedef int WeightType;

typedef struct GNode *PtrToGNode;
struct GNode{
    int Nv;
    int Ne;
    WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;

MGraph ReadG(); /* details omitted */

void ShortestDist( MGraph Graph, int dist[], int count[], Vertex S );

int main()
{
    int dist[MaxVertexNum], count[MaxVertexNum];
    Vertex S, V;
    MGraph G = ReadG();

    scanf("%d", &S);
    ShortestDist( G, dist, count, S );

    for ( V=0; V<G->Nv; V++ )
        printf("%d ", dist[V]);
    printf("\n");
    for ( V=0; V<G->Nv; V++ )
        printf("%d ", count[V]);
    printf("\n");

    return 0;
}

/* Your function will be put here */

Sample Input (for the graph shown in the figure):

Sample Output:

40 20 100 0 45 53 -1 50 
1 1 4 1 1 3 0 3

Answer:

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

typedef enum {false, true} bool;
#define INFINITY 1000000
#define MaxVertexNum 10  /* maximum number of vertices */
typedef int Vertex;      /* vertices are numbered from 0 to MaxVertexNum-1 */
typedef int WeightType;

typedef struct GNode *PtrToGNode;
struct GNode{
    int Nv;
    int Ne;
    WeightType G[MaxVertexNum][MaxVertexNum];
};
typedef PtrToGNode MGraph;

MGraph ReadG() /* details omitted */
{
    MGraph graph = (MGraph) malloc (sizeof(struct GNode));
    for(int i = 0; i < MaxVertexNum; i++) {
        for(int j = 0; j < MaxVertexNum; j++) {
            graph->G[i][j] = INFINITY;
        }
    }

    scanf("%d %d", &(graph->Nv), &(graph->Ne));

    for(int i = 0; i < graph->Ne; i++) {
        int v, w, weight;
        scanf("%d %d %d", &v, &w, &weight);
        graph->G[v][w] = weight;
    }

    return graph;
}

void ShortestDist( MGraph Graph, int dist[], int count[], Vertex S );

int main()
{
    int dist[MaxVertexNum], count[MaxVertexNum];
    Vertex S, V;
    MGraph G = ReadG();

    scanf("%d", &S);
    ShortestDist( G, dist, count, S );

    for ( V=0; V<G->Nv; V++ )
        printf("%d ", dist[V]);
    printf("\n");
    for ( V=0; V<G->Nv; V++ )
        printf("%d ", count[V]);
    printf("\n");

    return 0;
}

/* Your function will be put here */
Vertex MinDistance(MGraph Graph, int* count, int* dist, bool* visited)
{
    int minDist = INFINITY;
    // if none returned, should set rest of the count to 0
    Vertex minVertex = -1;
    for(int i = 0; i < Graph->Nv; i++) {
        if(!visited[i] && dist[i] < minDist) {
            minDist = dist[i];
            minVertex = i;
        }
    }

    if(minVertex == -1) {
        for(int i = 0; i < Graph->Nv; i++) {
            if(dist[i] == INFINITY){
                dist[i] = -1;
            }
        }
    } else {
        // Dequeue the vertex
        visited[minVertex] = true;
    }

    return minVertex;
}

void ShortestDist( MGraph Graph, int dist[], int count[], Vertex S )
{
    bool visited[MaxVertexNum] = {false};
    for(int i = 0; i < Graph->Nv; i++) {
        count[i] = 0;
        visited[i] = 0;// not visited
        dist[i] = INFINITY;
    }
    dist[S] = 0;
    count[S] = 1;

    // for all possible solutions
    for(int i = 0; i < Graph->Nv; i++) {
        Vertex u = MinDistance(Graph, count, dist, visited);
        if(u == -1) {
            break;
        }
        // for all neighbors of u
        for(int v = 0; v < Graph->Nv; v++) {
            if(Graph->G[u][v] != INFINITY) {
                if(dist[v] > dist[u] + Graph->G[u][v]){
                    dist[v] = dist[u] + Graph->G[u][v];
                    count[v] = (1 > count[u]) ? 1 : count[u];
                } else if(dist[v] == dist[u] + Graph->G[u][v]) {
                    count[v] += count[u];
                }
            }
        }

    }
}

Problem 3 – Shortest Path Variant 2

Write a program to find the weighted shortest distances from any vertex to a given source vertex in a digraph. If there is more than one minimum path from v to w, a path with the fewest number of edges is chosen. It is guaranteed that all the weights are positive and such a path is unique for any vertex.