Faster Algorithms for Edge Connectivity via Random $2$-Out Contractions

TL;DR

This paper introduces a simple randomized contraction method using 2-out edge sampling to improve algorithms for computing edge connectivity in graphs across sequential, distributed, and parallel models, achieving better complexities and new capabilities.

Contribution

The paper presents a novel 2-out contraction approach that leads to faster algorithms for edge connectivity, including optimal sequential algorithms, sublinear distributed algorithms, and constant-round parallel algorithms.

Findings

Sequential algorithms with complexities O(m log n) and O(m + n log^3 n)

First sublinear distributed algorithm with ~O(n^{0.8} D^{0.2} + n^{0.9}) rounds

First constant-round parallel algorithm with linear memory

Abstract

We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity with high probability: -- Two sequential algorithms with complexities $O(m \log n)$ and $O(m+n \log^3 n)$. These improve on a long line of developments including a celebrated $O(m \log^3 n)$ algorithm of Karger [STOC'96] and the state of the art $O(m \log^2 n (\log\log n)^2)$ algorithm of Henzinger et al. [SODA'17]. Moreover, our $O(m+n \log^3 n)$ algorithm is optimal whenever $m = \Omega(n \log^3 n)$. Within our new time bounds, whp, we can also construct the cactus representation of all minimal cuts. -- An $\~O(n^{0.8} D^{0.2} + n^{0.9})$ round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC'19], which achieved a round complexity of $\~O(n^{1-1/353}D^{1/353} + n^{1-1/706})$, hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. -- The first $O(1)$ round algorithm for the massively parallel computation setting with linear memory per machine.