109. Mathematical Colloquium

Mikkel Thorup

University of Copenhagen

DETERMINISTIC EDGE CONNECTIVITY IN NEAR-LINEAR TIME

Abstract

We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time.

The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m + k^2 n), where k is the edge connectivity, but k could be as big as n-1.

At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.

Our main technical contribution is a near-linear time algorithm that contract vertex sets of a simple input graph G with minimum degree d, producing a multigraph with O(m/d) edges which preserves all minimum cuts of G with at least 2 vertices on each side.

In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.

About the speaker

Mikkel Thorup is a Fellow of the ACM and of AT&T, and a Member of the Royal Danish Academy of Sciences and Letters. He is co-winner of the 2011 MAA Robbins Award and winner of the 2015 Villum Kann Rasmussen Award for Technical and Scientific Research, which is Denmark's biggest individual prize for research. His main work is in algorithms and data structures. Recently one of his main focusses has been on hash functions unifying theory and practice. One of his best-known early results is a linear-time algorithm for the single-source shortest paths problem in undirected graphs. This is also related to his Prague colloquium.