[Verse 1]
Ford-Fulkerson algorithm needs refinement, that's the truth
Edmonds-Karp stepped in to boost the proof
BFS becomes the pathfinding tool
Maximum flow computed, scientific jewel
Start from source, traverse each edge with care
Breadth-first search eliminates the nightmare
No more exponential time complexity
Polynomial bounds, that's the guarantee
[Chorus]
Breadth-first search finds the path
Shortest route, do the math
Augment flow until complete
O of V times E, can't be beat
Residual graph shows the way
Capacity minus flow today
Maximum flow theorem sealed
Ford-Fulkerson improved and healed
[Verse 2]
Queue up vertices level by level spread
Visit neighbors systematically ahead
Residual capacity guides each decision
Forward edges, backward with precision
When bottleneck discovered, augment the stream
Update residual values, fulfill the dream
Reverse edges get the flow amount added
Forward edges see capacity subtracted
[Chorus]
Breadth-first search finds the path
Shortest route, do the math
Augment flow until complete
O of V times E, can't be beat
Residual graph shows the way
Capacity minus flow today
Maximum flow theorem sealed
Ford-Fulkerson improved and healed
[Bridge]
No more infinite loops or worst-case disasters
BFS guarantees we find paths faster
Each iteration shortens augmenting distance
Network flows solved with mathematical persistence
Cut capacity equals maximum flow
Min-cut max-flow theorem starts to glow
[Verse 3]
Implementation details crystallize the vision
Adjacency lists support the algorithm's mission
Track parent nodes to reconstruct the route
Push minimum residual, that's the attribute
When sink unreachable, algorithm terminates
Optimal solution, the network celebrates
Transportation problems, matching algorithms thrive
Edmonds-Karp keeps maximum flow alive
[Chorus]
Breadth-first search finds the path
Shortest route, do the math
Augment flow until complete
O of V times E, can't be beat
Residual graph shows the way
Capacity minus flow today
Maximum flow theorem sealed
Ford-Fulkerson improved and healed
[Outro]
From chaos came order, from exponential came polynomial time
Edmonds-Karp algorithm, network flow sublime