Parallel Complexity of Depth-First-Search and Maximal path in restricted graph classes

TL;DR

This paper advances the understanding of parallel algorithms for depth-first search (DFS) and maximal path problems in various restricted graph classes, providing new NC algorithms and complexity bounds.

Contribution

It extends NC algorithms for DFS to more general graph classes like bounded genus and H-minor-free graphs, and improves complexity bounds for maximal path in planar graphs.

Findings

NC algorithms for DFS in bounded genus and H-minor-free graphs

Logspace bounds for DFS in graphs of bounded tree-width

Improved parallel complexity for maximal path in planar graphs

Abstract

Constructing a Depth First Search (DFS) tree is a fundamental graph problem, whose parallel complexity is still not settled. Reif showed parallel intractability of lex-first DFS. In contrast, randomized parallel algorithms (and more recently, deterministic quasipolynomial parallel algorithms) are known for constructing a DFS tree in general (di)graphs. However a deterministic parallel algorithm for DFS in general graphs remains an elusive goal. Working towards this, a series of works gave deterministic NC algorithms for DFS in planar graphs and digraphs. We further extend these results to more general graph classes, by providing NC algorithms for (di)graphs of bounded genus, and for undirected H-minor-free graphs where H is a fixed graph with at most one crossing. For the case of (di)graphs of bounded tree-width, we further improve the complexity to a Logspace bound. Constructing a maximal path is a simpler problem (that reduces to DFS) for which no deterministic parallel bounds are known for general graphs. For planar graphs a bound of O(log n) parallel time on a CREW PRAM (thus in NC2) is known. We improve this bound to Logspace.