[Verse 1]
Graph got edges with some weights that might be negative
Shortest path finder when Dijkstra can't be definitive
Bellman-Ford steps up when cycles bring the drama
Detects the negative loops like algorithmic karma
Start with source vertex, distance set to zero
Every other node infinity, that's how we play hero
Relax the edges, that's the key to our success
If distance plus weight is less, update and progress
[Chorus]
Relax relax relax, V minus one times through
Check every single edge, that's what we gotta do
If we can still improve after all those rounds are done
Negative cycle found, the algorithm's won
Bellman-Ford don't quit when weights go below zero
Finding shortest paths like a computational hero
[Verse 2]
V minus one iterations, that's the magic number
Any path that's optimal can't have more to lumber
Each round we're guaranteeing one more edge precision
Building up the answer with mathematical vision
Take an edge from U to V, check the relaxation
If U distance plus weight beats V's calculation
Update V's distance, keep the parent pointer too
Trace back the shortest path when the work is through
[Chorus]
Relax relax relax, V minus one times through
Check every single edge, that's what we gotta do
If we can still improve after all those rounds are done
Negative cycle found, the algorithm's won
Bellman-Ford don't quit when weights go below zero
Finding shortest paths like a computational hero
[Bridge]
One more pass to catch the lies
If distances still dropping that's our warning sign
Negative cycle means no shortest path exists
Infinite improvement, something's been missed
Time complexity O of V times E
Space complexity O of V, that's the key
[Verse 3]
Unlike Dijkstra's greedy approach with priority queue
Bellman-Ford examines every edge, sees the whole view
Works with negative weights but not negative cycles
Dynamic programming vibes, breaking down the riddles
From source to every vertex, find the minimal cost
Even when the edge weights make other algorithms lost
Johnson's algorithm uses us as preprocessing stage
Bellman-Ford's the foundation, turn another page
[Chorus]
Relax relax relax, V minus one times through
Check every single edge, that's what we gotta do
If we can still improve after all those rounds are done
Negative cycle found, the algorithm's won
Bellman-Ford don't quit when weights go below zero
Finding shortest paths like a computational hero
[Outro]
When the graph gets complicated and the weights turn mean
Bellman-Ford's the algorithm keeping pathways clean
Relax those edges, check for cycles, find your way
Shortest path solution at the end of the day