Dinic's algorithm

Dinic's algorithm or 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.

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.

Dinitz invented the algorithm in January 1969, as a master's student in Georgy Adelson-Velsky's group.

A few decades later, he would recall:[2] In Adel'son-Vel'sky's Algorithms class, the lecturer had a habit of giving the problem to be discussed at the next meeting as an exercise to students.

The DA was invented in response to such an exercise.

At that time, the author was not aware of the basic facts regarding [the Ford–Fulkerson algorithm]….

Very probably, DA would not have been invented then, if the idea of possible saturated edge desaturation had been known to the author.In 1970, Dinitz published a description of the algorithm in Doklady Akademii Nauk SSSR.

In 1974, Shimon Even and (his then Ph.D. student) Alon Itai at the Technion in Haifa were very curious and intrigued by Dinitz's algorithm as well as Alexander V. Karzanov's related idea of blocking flow.

However it was hard for them to decipher these two papers, each being limited to four pages to meet the restrictions of journal Doklady Akademii Nauk SSSR.

Even did not give up, and after three days of effort managed to understand both papers except for the layered network maintenance issue.

Over the next couple of years, Even gave lectures on "Dinic's algorithm", mispronouncing the name of the author while popularizing it.

Even and Itai also contributed to this algorithm by combining BFS and DFS, which is how the algorithm is now commonly presented.

[2] For about 10 years of time after the Ford–Fulkerson algorithm was invented, it was unknown if it could be made to terminate in polynomial time in the general case of irrational edge capacities.

This caused a lack of any known polynomial-time algorithm to solve the max flow problem in generic cases.

Dinic's Algorithm It can be shown that the number of layers in each blocking flow increases by at least 1 each time and thus there are at most

As a consequence, the running time of Dinic's algorithm is

[2] 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

In networks with unit capacities, a much stronger time bound holds.

time, and it can be shown that the number of phases does not exceed

[4] In networks that arise from the bipartite matching problem, the number of phases is bounded by

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.

, the vertices with labels in red are the values

The paths in blue form a blocking flow.

Note that each augmenting path in the blocking flow has 3 edges.

Note that each augmenting path has 4 edges.

, the algorithm terminates and returns a flow with maximum value of 19.

Note that in each blocking flow, the number of edges in the augmenting path increases by at least 1.