DAA

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Design & Analysis of
Algorithm

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Bellman Ford’s Algorithm
S u b m i t t e d B y :
 H a s n a i n K h a l i d .
 M u s t a n s a r G u l .
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Table of contents
• Introduction
• Comparison
• Working Mechanism
• Example Problem
• Algorithm
• Pseudocode
• Algorithm Complexity
• Applications
• Conclusion
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Introduction
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What is Bellman Ford’s Algorithm?
• Bellman Ford is an algorithm that is used to find the shortest path from a vertex to
all other vertices of a weighted graph.
• It is similar to Dijkstra's algorithm but it can work with graphs in which edges can
have negative weights.
• If it has not converged after V(G) -1 iterations, then there cannot be a shortest path
tree.
• So, there must be a negative weight cycle

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Dijkstras Vs Bellman Ford
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• Dijkstra doesn’t work for Graphs with negative weight edges.
• Bellman-Ford works for such graphs.
• Both Algorithms are quite similar but the time complexity of bellman ford algorithm is more
than Dijkstra.

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Working Mechanism
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• It works by overestimating the length of the path from the starting vertex to all other
vertices.
• Then it iteratively relaxes those estimates by finding new paths that are shorter than the
previously overestimated paths.
• By doing this repeatedly for all vertices, we can guarantee that the result is optimized.

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Example Problem
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• Step No. 1: Starting with the weighted graph.

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Cont..
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• Step No. 02: Choose a starting vertex and assign infinity path values to all other
vertices.

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Cont..
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• Step No. 03: Visit each edge and relax path distances if they are inaccurate.

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Cont..
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• Step No. 04: We need to do this V times because in the worst case, a vertex’s path length
might need to be readjusted V times.

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Cont..
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• Step No. 05: Notice how the vertex at the top right corner had its path length
adjusted.

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Cont..
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• Step No. 06: After all the vertices have their path lengths, we check if a negative cycle is
present.

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Algorithm
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• For Bellman Ford’s Algorithm, We need to maintain the path distance of every
vertex.
• We can store that in an array of size v, where v is the number of vertices.
• We also want to be able to get the shortest path, not only know the length of the
shortest path
• For this, we map each vertex to the vertex that last updated its path length.
• Once the algorithm is over, we can backtrack from the destination vertex to the
source vertex to find the path.

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Pseudocode
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function bellmanFord(G, S)
for each vertex V in G
distance[V] <- infinite
previous[V] <- NULL
distance[S] <- 0
for each vertex V in G
for each edge (U,V) in G
tempDistance <- distance[U] +
edge_weight(U, V)
if tempDistance < distance[V]
distance[V] <- tempDistance
previous[V] <- U
for each edge (U,V) in G
If distance[U] + edge_weight(U, V) <
distance[V}
Error: Negative Cycle Exists
return distance[ ], previous[ ]

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Bellman Ford's Complexity
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Time Complexity Space Complexity
• Best Case Complexity is O(E)
• Average Case Complexity is O(VE)
• Worst Case Complexity is O(VE)
• And, the space complexity is O(V).

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Applications
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• A version of Bellman-Ford is used in the distance-vector routing protocol. This protocol
decides how to route packets of data on a network.
• And for finding the shortest path.

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Conclusion
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• After analyzing the both algorithm we have concluded that the main advantage of the Bellman-
Ford algorithm is its capability to handle negative weights. However, it has a considerably larger
complexity than Dijkstra's algorithm.

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Thank You !