Maximum FlowPush relabel algorithm By Waqas Shehzad Fast NU Pakistan

Push-RelabelPush-relabel algorithm is described by Dr. Andrew

Goldberg and Robert E.Tarjan.Leads to better running times, both in theory and

practice. All previous Network Flow related algorithms are

related to augmentation. A draw back of aug-based schemes: each

augmentation takes O(n) time. Multiple augmentations may partially share paths, but it’s not exploited by algorithms.

Not very hard to code.Push relabel is consider as the Fastest maximum-flow

algorithm.

Preflow-Basics• Preflow algorithm allow temporary imbalance

(excess) at nodes, and work towards feasibility.• push-relabel algorithms work on one vertex at a

time, looking only at the vertex’s neighbors.• “pushing” preflow and “relabeling” a vertex

relabeling• Formally, a preflow is a function ƒ: V×V → R that

satisfies capacity constraints and

• In preflow algorithm a graph is consider to have two special nodes Source and Sink

Cont..

PPreflow :

A labeling h : V Z+ is compatible with a preflow if

Code for Preflow-Push

Run time• Generic Preflow-Push algorithm runs in worst-case

time 0 ).• The total number of saturating push operations is

at most m.• The total number of nonsaturating push

operations is at most 2nm.• The total number of relabeling operations is less

than 2.• This running time is farther improved with

Highest-label rule, Excess scaling,

References

• Lecturer: Julian Mestre Optimization II Lecture 3• Andrew V. Goldberg and Robert E. Tarjan, A new

approach to the maximum-flow problem. • Thomas H. Cormen, Charles E. Leiserson, Ronald

L. Rivest. Introduction to Algorithms. • Algorithm Tutorial: MaximumFlow• Algorithm Tutorial: Introduction to graphs and

their data structures• http://iss.ices.utexas.edu/?p=projects/galois/

benchmarks/preflow_push