Parallel Small Vertex Connectivity in Near-Linear Work and Polylogarithmic Depth
Yonggang Jiang, Changki Yun
TL;DR
The paper addresses the problem of parallelizing k-vertex connectivity in undirected graphs within the PRAM model, aiming for near-linear work and polylogarithmic depth when k is polylogarithmic in n. It introduces a framework that avoids reliance on dense reachability by combining a multiplicative weight updates (MWU) approach for s-t cuts with a sublinear, s-source length sparsifier and a parallel local-cuts subroutine that yields fractional cuts, which can be rounded to integral cuts. The main technical contributions include a scalable local-cuts methodology, a length-preserving sparsifier construction, and a randomized framework that achieves tilde-notation bounds $\widetilde{O}(m\cdot \mathrm{poly}(k))$ work and $\widetilde{O}(\mathrm{poly}(k))$ depth for $k=\mathrm{polylog}(n)$. This work demonstrates that k-vertex connectivity is highly parallelizable for polynomially-bounded k, providing foundations for fast parallel graph analysis and implications for related models like distributed and streaming computation. The results push toward practical parallel algorithms for graph reliability and connectivity problems on large-scale networks by combining localization, sparsification, and probabilistic rounding techniques.
Abstract
We present a randomized parallel algorithm in the {\sf PRAM} model for $k$-vertex connectivity. Given an undirected simple graph, our algorithm either finds a set of fewer than $k$ vertices whose removal disconnects the graph or reports that no such set exists. The algorithm runs in $O(m \cdot \text{poly}(k, \log n))$ work and $O(\text{poly}(k, \log n))$ depth, which is nearly optimal for any $k = \text{poly}(\log n)$. Prior to our work, algorithms with near-linear work and polylogarithmic depth were known only for $k=3$ [Miller, Ramachandran, STOC'87]; for $k=4$, sequential algorithms achieving near-linear time were known [Forster, Nanongkai, Yang, Saranurak, Yingchareonthawornchai, SODA'20], but no algorithm with near-linear work could achieve even sublinear (on $n$) depth.