Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

TL;DR

This paper introduces faster algorithms for finding 2-edge and 2-vertex strongly connected components in directed graphs, achieving quadratic time complexity and improving upon decades-old methods.

Contribution

The authors develop the first quadratic-time algorithms for 2-edge and 2-vertex strongly connected components in directed graphs, extending to constant k with improved efficiency.

Findings

First quadratic-time algorithms for 2-edge and 2-vertex SCCs in directed graphs

Improved over 20-year-old algorithms for 2-edge SCCs

Extended approach to k-edge and k-vertex SCCs with efficient running times

Abstract

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices.