A Note on (Parallel) Depth- and Breadth-First Search by Arc Elimination

TL;DR

This paper presents a parallel algorithm for depth- and breadth-first search in directed graphs that achieves linear speed-up on shared-memory machines, extending previous work on ordered DFS to BFS.

Contribution

It introduces a parallel approach for ordered DFS and extends it to BFS, achieving linear speed-up with minimal synchronization in non-sparse graphs.

Findings

Parallel DFS and BFS run in O(m/p + n) time with p processors.

Both algorithms require n synchronization steps.

Algorithms achieve linear speed-up for p ≤ m/n.

Abstract

This note recapitulates an algorithmic observation for ordered Depth-First Search (DFS) in directed graphs that immediately leads to a parallel algorithm with linear speed-up for a range of processors for non-sparse graphs. The note extends the approach to ordered Breadth-First Search (BFS). With $p$ processors, both DFS and BFS algorithms run in $O(m/p+n)$ time steps on a shared-memory parallel machine allowing concurrent reading of locations, e.g., a CREW PRAM, and have linear speed-up for $p\leq m/n$. Both algorithms need $n$ synchronization steps.