What is the maximum flow problem?

It asks for the greatest total flow that can be routed from a designated source node to a designated sink node through a network of directed edges, each limited by a capacity.

How does the Edmonds-Karp algorithm work?

It repeatedly uses breadth-first search to find the shortest available augmenting path from source to sink in the residual graph, pushes the maximum flow possible along that path (limited by its smallest remaining capacity), and repeats until no path remains.

What is a residual graph?

It's a graph tracking how much additional flow can still be pushed along each edge (remaining capacity) plus a 'reverse' edge for flow already sent, which allows the algorithm to effectively 'undo' a suboptimal earlier choice.

What is the max-flow min-cut theorem?

It states that the maximum flow from source to sink always exactly equals the minimum total capacity of any 'cut' (set of edges that, if removed, would disconnect the source from the sink) — a fundamental guarantee behind why Edmonds-Karp finds the true optimum.

Why use BFS instead of any path (Ford-Fulkerson)?

Plain Ford-Fulkerson can pick arbitrarily bad augmenting paths and run slowly on certain graphs; always choosing the shortest path via BFS (Edmonds-Karp) guarantees the algorithm terminates in polynomial time, O(VE²).

What does the '->' arrow mean in the edge input?

It indicates a directed edge — A->B:10 means up to 10 units of flow can move from A to B, but not from B to A (unless a separate B->A edge is also entered).

How many nodes can this calculator handle?

This calculator supports up to 30 nodes, keeping the Edmonds-Karp computation instant while covering the vast majority of hand-entered network examples.

Where is max flow used in real life?

It models network bandwidth allocation, traffic and transportation routing, bipartite matching (job assignment, dating apps), water/oil pipeline distribution, and image segmentation in computer vision.

Network Flow Calculator — Find the Maximum Flow Through a Network

Our Network Flow Calculator finds the maximum amount of flow that can be pushed from a source node to a sink node through a directed graph where each edge has a limited capacity, using the Edmonds-Karp algorithm (BFS-based Ford-Fulkerson), given your network as a simple text edge list (e.g. A->B:10, B->C:5).

Quick Answer

Enter your capacitated directed graph and a source/sink pair to instantly find the maximum flow via the Edmonds-Karp algorithm.

Directed edges (comma-separated)

Enter edges as NodeA->NodeB:capacity, e.g. A->B:10, B->C:5.

Source

Sink

How to Use the Network Flow Calculator — Max Flow via Edmonds-Karp Online

1
Enter your directed, capacitated graph as a comma-separated edge list, e.g. A->B:10, B->C:5.
2
Enter the source and sink node names.
3
Click 'Calculate' to find the maximum flow via Edmonds-Karp.

Why Use Network Flow Calculator — Max Flow via Edmonds-Karp Online?

Maximum flow asks how much total 'stuff' (data, water, traffic) can move from a source to a sink through a network where every edge has a limited capacity. The Edmonds-Karp algorithm solves this exactly: repeatedly find any shortest augmenting path from source to sink using BFS in the residual graph (the graph of remaining capacity, including 'undo' edges for flow already sent), push the maximum possible flow along that path, and repeat until no augmenting path remains — at which point the total flow pushed equals the maximum possible flow, a fact guaranteed by the max-flow min-cut theorem.

Frequently Asked Questions

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