Dynamic constant time parallel graph algorithms with sub-linear work

TL;DR

This paper introduces dynamic parallel algorithms for graph connectivity and bipartiteness that operate in constant time with sub-linear work, advancing the efficiency of graph algorithms on parallel models.

Contribution

It presents the first constant-time dynamic parallel algorithms for connectivity and bipartiteness with near-optimal work on the CRCW PRAM model, utilizing sparsification techniques.

Findings

Algorithms achieve constant time per operation.

Work complexity is $O(n^{1/2+psilon})$, nearly matching sequential bounds.

Sparsification can be adapted for constant time algorithms in CRCW PRAM.

Abstract

The paper proposes dynamic parallel algorithms for connectivity and bipartiteness of undirected graphs that require constant time and $O(n^{1/2+\epsilon})$ work on the CRCW PRAM model. The work of these algorithms almost matches the work of the $O(\log n)$ time algorithm for connectivity by Kopelowitz et al. (2018) on the EREW PRAM model and the time of the sequential algorithm for bipartiteness by Eppstein et al. (1997). In particular, we show that the sparsification technique, which has been used in both mentioned papers, can in principle also be used for constant time algorithms in the CRCW PRAM model, despite the logarithmic depth of sparsification trees.