[Verse 1]
Graph's a network, vertices connected tight
Edges carry capacity, flow from left to right
Source pumps the liquid, sink receives the stream
Ford-Fulkerson algorithm maximizes the dream
Start with zero flow, find an augmenting path
Residual graph shows what's left to do the math
Forward edges minus flow, backward edges plus
Calculate the bottleneck, that's the minimum thrust
[Chorus]
Find path, push flow, update the residual
Repeat until no path exists, that's the ritual
Maximum flow equals minimum cut, it's mathematical
Ford-Fulkerson method, optimally practical
Augment, update, iterate the process
Until the network reaches maximum progress
[Verse 2]
Breadth-first search or depth-first exploration
Hunt for paths with positive flow allocation
Residual capacity determines how much we send
Push-relabel variant makes the process transcend
Each iteration boosts the total throughput
Blocking flow when bottlenecks compute
Edmonds-Karp guarantees polynomial time
BFS ensures the complexity stays prime
[Chorus]
Find path, push flow, update the residual
Repeat until no path exists, that's the ritual
Maximum flow equals minimum cut, it's mathematical
Ford-Fulkerson method, optimally practical
Augment, update, iterate the process
Until the network reaches maximum progress
[Bridge]
Min-cut theorem proves the duality
Saturated edges form the boundary
Bipartite matching, network reliability
Transportation problems, resource mobility
When no augmenting path can be traced
Maximum flow has been perfectly placed
[Verse 3]
Implementation needs adjacency representation
Track capacity matrix and flow documentation
While there exists a path with residual space
Push minimum bottleneck value through that place
Update forward flow and backward cancellation
Repeat until you reach flow termination
Complexity depends on pathfinding strategy
Choose wisely for optimal algorithmic mastery
[Outro]
Ford-Fulkerson maximizes every stream
Residual graphs reveal the optimization scheme
From source to sink the liquid gold flows free
Network flow perfection, algorithmic guarantee