Bellman-Ford Algorithm

What is Bellman-Ford Algorithm?

Bellman-ford algorithm will find the shortest path to all of the vertices from the source node.

Dijkstra's shortest path algorithm can not work around graphs with negative weight. Bellman-ford can.

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Analysis

Time Complexity:
Generally:
$O(|E|(|V|-1))$ which is $O(|E||V|)$

Worst Case: graph is a complete graph
$|E| = {n(n-1) \over 2} = {n^2-n \over 2} {\displaystyle \approx } O(n^2)$
Therefore,
$O(|E||V|)=O(|V^2||V|){\displaystyle \approx }O(|V^3|)$

Space Complexity:
$O(|V|)$

Conclusion:
If we were to say the number of verticies is $n$. At worst case,

• Time Complexity: $O(n^3)$
• Space Complexity: $O(n)$

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Evaluation

Advantages

• Can be applied to graphs w/ negative weight

Disadvantages

• As Dijkstra's is greedy and Bellman-Ford is dynamic, it is slower than Dijkstra's.
• Does not work if there is a negative cycle in the graph. We can find out if there is a negative cycle, but not work through it.

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Implementation

1) Initialize dist list with length of n with $float("inf")$
2) We will relax the graph n-1 times

• Go through each edge to check if $dist[v] > dist[u] + w$ where $u$ will be the source node and $v$ will be the target node of that edge.
• If $dist[v] > dist[u] + w$ update $dist[v] = dist[u] + w$

3) If necessary: We will relax the graph one more time making it relaxed n times. If the values of dist for any nodes change on the nth relaxation, the graph contains negative cycle.

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Code

def bellmanFord(self, times: List[List[int]], n: int, k: int) -> int:
    # Initializing dist list with float("inf")
    dist = [float("inf")] * (n)
    dist[k-1] = 0


    # Relaxation
    for _ in range(n-1):
        for u,v,w in times:
            # Updating shortest path to target node
            if dist[u-1] != float("inf") and dist[u-1] + w < dist[v-1]:
                dist[v-1] = dist[u-1] + w

    
    # Checking if there are unreached nodes
    if float("inf") in dist:
        return -1
    else:
        return max(dist)

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