Bellman Ford algorithm

Bellman Ford algorithm is also simpler than Dijkstra and suites well for distributed systems. But time complexity of Bellman Ford algorithm is O(VE), which is more than Dijkstra.

Algorithm
Following are the detailed steps.

Input: Graph and a source vertex src
Output: Shortest distance to all vertices from src. If there is a negative weight cycle, then shortest distances are not calculated, negative weight cycle is reported.

1) This step initializes distances from source to all vertices as infinite and distance to source itself as 0. Create an array dist[] of size |V| with all values as infinite except dist[src] where src is source vertex.

2) This step calculates shortest distances. Do following |V|-1 times where |V| is the number of vertices in given graph.
a) Do following for each edge u-v
If dist[v] > dist[u] + weight of edge uv, then update dist[v]
dist[v] = dist[u] + weight of edge uv

3) This step reports if there is a negative weight cycle in graph. Do following for each edge u-v

If dist[v] > dist[u] + weight of edge uv, then “Graph contains negative weight cycle”

The idea of step 3 is, step 2 guarantees shortest distances if graph doesn’t contain negative weight cycle. If we iterate through all edges one more time and get a shorter path for any vertex, then there is a negative weight cycle

How does this work? Like other Dynamic Programming Problems, the algorithm calculate shortest paths in bottom-up manner. It first calculates the shortest distances for the shortest paths which have at-most one edge in the path. Then, it calculates shortest paths with at-nost 2 edges, and so on. After the ith iteration of outer loop, the shortest paths with at most i edges are calculated. There can be maximum |V| – 1 edges in any simple path, that is why the outer loop runs |v| – 1 times. The idea is, assuming that there is no negative weight cycle, if we have calculated shortest paths with at most i edges, then an iteration over all edges guarantees to give shortest path with at-most (i+1) edges.

Example
Let us understand the algorithm with following example graph.

Let the given source vertex be 0. Initialize all distances as infinite, except the distance to source itself. Total number of vertices in the graph is 5, so all edges must be processed 4 times.

Example Graph

Let all edges are processed in following order: (B,E), (D,B), (B,D), (A,B), (A,C), (D,C), (B,C), (E,D). We get following distances when all edges are processed first time. The first row in shows initial distances. The second row shows distances when edges (B,E), (D,B), (B,D) and (A,B) are processed. The third row shows distances when (A,C) is processed. The fourth row shows when (D,C), (B,C) and (E,D) are processed.

The first iteration guarantees to give all shortest paths which are at most 1 edge long. We get following distances when all edges are processed second time (The last row shows final values).

The second iteration guarantees to give all shortest paths which are at most 2 edges long. The algorithm processes all edges 2 more times. The distances are minimized after the second iteration, so third and fourth iterations don’t update the distances.

Implementation:

#include <stdio.h>

#include <stdlib.h>

#include <string.h>

#include <limits.h>

// a structure to represent a weighted edge in graph

struct Edge

{

int src, dest, weight;

};

// a structure to represent a connected, directed and weighted graph

struct Graph

{

// V-> Number of vertices, E-> Number of edges

int V, E;

// graph is represented as an array of edges.

struct Edge* edge;

};

// Creates a graph with V vertices and E edges

struct Graph* createGraph(int V, int E)

{

struct Graph* graph = (struct Graph*) malloc( sizeof(struct Graph) );

graph->V = V;

graph->E = E;

graph->edge = (struct Edge*) malloc( graph->E * sizeof( struct Edge ) );

return graph;

}

// A utility function used to print the solution

void printArr(int dist[], int n)

{

printf(“Vertex   Distance from Source\n”);

for (int i = 0; i < n; ++i)

printf(“%d \t\t %d\n”, i, dist[i]);

}

// The main function that finds shortest distances from src to all other

// vertices using Bellman-Ford algorithm. The function also detects negative

// weight cycle

void BellmanFord(struct Graph* graph, int src)

{

int V = graph->V;

int E = graph->E;

int dist[V];

// Step 1: Initialize distances from src to all other vertices as INFINITE

for (int i = 0; i < V; i++)

dist[i]   = INT_MAX;

dist[src] = 0;

// Step 2: Relax all edges |V| – 1 times. A simple shortest path from src

// to any other vertex can have at-most |V| – 1 edges

for (int i = 1; i <= V-1; i++)

{

for (int j = 0; j < E; j++)

{

int u = graph->edge[j].src;

int v = graph->edge[j].dest;

int weight = graph->edge[j].weight;

if (dist[u] != INT_MAX && dist[u] + weight < dist[v])

dist[v] = dist[u] + weight;

}

}

// Step 3: check for negative-weight cycles. The above step guarantees

// shortest distances if graph doesn’t contain negative weight cycle.

// If we get a shorter path, then there is a cycle.

for (int i = 0; i < E; i++)

{

int u = graph->edge[i].src;

int v = graph->edge[i].dest;

int weight = graph->edge[i].weight;

if (dist[u] != INT_MAX && dist[u] + weight < dist[v])

printf(“Graph contains negative weight cycle”);

}

printArr(dist, V);

return;

}

// Driver program to test above functions

int main()

{

/* Let us create the graph given in above example */

int V = 5; // Number of vertices in graph

int E = 8; // Number of edges in graph

struct Graph* graph = createGraph(V, E);

// add edge 0-1 (or A-B in above figure)

graph->edge[0].src = 0;

graph->edge[0].dest = 1;

graph->edge[0].weight = -1;

// add edge 0-2 (or A-C in above figure)

graph->edge[1].src = 0;

graph->edge[1].dest = 2;

graph->edge[1].weight = 4;

// add edge 1-2 (or B-C in above figure)

graph->edge[2].src = 1;

graph->edge[2].dest = 2;

graph->edge[2].weight = 3;

// add edge 1-3 (or B-D in above figure)

graph->edge[3].src = 1;

graph->edge[3].dest = 3;

graph->edge[3].weight = 2;

// add edge 1-4 (or A-E in above figure)

graph->edge[4].src = 1;

graph->edge[4].dest = 4;

graph->edge[4].weight = 2;

// add edge 3-2 (or D-C in above figure)

graph->edge[5].src = 3;

graph->edge[5].dest = 2;

graph->edge[5].weight = 5;

// add edge 3-1 (or D-B in above figure)

graph->edge[6].src = 3;

graph->edge[6].dest = 1;

graph->edge[6].weight = 1;

// add edge 4-3 (or E-D in above figure)

graph->edge[7].src = 4;

graph->edge[7].dest = 3;

graph->edge[7].weight = -3;

BellmanFord(graph, 0);

return 0;

}

Output:

Vertex   Distance from Source
0                0
1                -1
2                2
3                -2
4                1

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