Bellman–Ford Algorithm | DP-23
Given a graph and a source vertex src in the graph, find the shortest paths from src to all vertices in the given graph. The graph may contain negative weight edges.
We have discussed Dijkstra’s algorithm for this problem. Dijkstra’s algorithm is a Greedy algorithm and the time complexity is O((V+E)LogV) (with the use of the Fibonacci heap). Dijkstra doesn’t work for Graphs with negative weights, Bellman-Ford works for such graphs. Bellman-Ford is also simpler than Dijkstra and suites well for distributed systems. But time complexity of Bellman-Ford is O(V * E), which is more than Dijkstra.
Steps for finding the shortest distance to all vertices from the source using the Bellman-Ford algorithm:
- This step initializes distances from the source to all vertices as infinite and distance to the source itself as 0. Create an array dist of size |V| with all values as infinite except dist[src] where src is source vertex.
- This step calculates shortest distances. Do following |V|-1 times where |V| is the number of vertices in given graph. Do following for each edge u-v
- If dist[v] > dist[u] + weight of edge uv, then update dist[v] to
- dist[v] = dist[u] + weight of edge uv
- This step reports if there is a negative weight cycle in the graph. Again traverse every edge and 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 the shortest distances if the graph doesn’t contain a 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 calculates the shortest paths in a bottom-up manner. It first calculates the shortest distances which have at most one edge in the path. Then, it calculates the shortest paths with at-most 2 edges, and so on. After the i-th iteration of the 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 the shortest path with at-most (i+1) edges
Below is the illustration of the above algorithm:
Step 1: Let the given source vertex be 0. Initialize all distances as infinite, except the distance to the source itself. Total number of vertices in the graph is 5, so all edges must be processed 4 times.
Step 2: Let all edges are processed in the following order: (B, E), (D, B), (B, D), (A, B), (A, C), (D, C), (B, C), (E, D). We get the following distances when all edges are processed the first time. The first row 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.
Step 3: The first iteration guarantees to give all shortest paths which are at most 1 edge long. We get the following distances when all edges are processed second time (The last row shows final values).
Step 4: 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.
Below is the implementation of the above approach:
Vertex Distance from Source 0 0 1 -1 2 2 3 -2 4 1
Time Complexity: O(V * E), where V is the number of vertices in the graph and E is the number of edges in the graph
Auxiliary Space: O(E)
- Negative weights are found in various applications of graphs. For example, instead of paying the cost for a path, we may get some advantage if we follow the path.
- Bellman-Ford works better (better than Dijkstra’s) for distributed systems. Unlike Dijkstra’s where we need to find the minimum value of all vertices, in Bellman-Ford, edges are considered one by one.
- Bellman-Ford does not work with an undirected graph with negative edges as it will be declared as a negative cycle.
- The standard Bellman-Ford algorithm reports the shortest path only if there are no negative weight cycles. Modify it so that it reports minimum distances even if there is a negative weight cycle.
- Can we use Dijkstra’s algorithm for shortest paths for graphs with negative weights – one idea can be, to calculate the minimum weight value, add a positive value (equal to the absolute value of minimum weight value) to all weights and run the Dijkstra’s algorithm for the modified graph. Will this algorithm work?