# Dinic's algorithm

**Dinitz's algorithm** is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli (formerly Soviet) computer scientist Yefim Dinitz.^{[1]} The algorithm runs in time and is similar to the Edmonds–Karp algorithm, which runs in time, in that it uses shortest augmenting paths. The introduction of the concepts of the *level graph* and *blocking flow* enable Dinic's algorithm to achieve its performance.

## Definition

Let be a network with and the capacity and the flow of the edge respectively.

- Define to be the length of the shortest path from to in . Then the
**level graph**of is the graph , where

## Algorithm

**Dinic's Algorithm**

- Set for each .
- Construct from of . If , stop and output .
- Find a blocking flow in .
- Augment flow by and go back to step 2.

## Analysis

It can be shown that the number of edges in each blocking flow increases by at least 1 each time and thus there are at most blocking flows in the algorithm, where is the number of vertices in the network. The level graph can be constructed by Breadth-first search in time and a blocking flow in each level graph can be found in time. Hence, the running time of Dinic's algorithm is .

Using a data structure called dynamic trees, the running time of finding a blocking flow in each phase can be reduced to and therefore the running time of Dinic's algorithm can be improved to .

### Special cases

In networks with unit capacities, a much stronger time bound holds. Each blocking flow can be found in time, and it can be shown that the number of phases does not exceed and . Thus the algorithm runs in time.

In networks arising during the solution of bipartite matching problem, the number of phases is bounded by , therefore leading to the time bound. The resulting algorithm is also known as Hopcroft–Karp algorithm. More generally, this bound holds for any *unit network* — a network in which each vertex, except for source and sink, either has a single entering edge of capacity one, or a single outgoing edge of capacity one, and all other capacities are arbitrary integers.Template:Sfn

## Example

The following is a simulation of the Dinic's algorithm. In the level graph , the vertices with labels in red are the values . The paths in blue form a blocking flow.

## History

Dinic's algorithm was published in 1970 by former Russian Computer Scientist Yefim (Chaim) A. Dinitz, who is today a member of the Computer Science department at Ben-Gurion University of the Negev (Israel), earlier than the Edmonds–Karp algorithm, which was published in 1972 but was discovered earlier. They independently showed that in the Ford–Fulkerson algorithm, if each augmenting path is the shortest one, the length of the augmenting paths is non-decreasing.

## See also

## Notes

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## References

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