Average-Case Analysis of Online Topological Ordering

TL;DR

This paper provides the first average-case analysis of online topological ordering algorithms, showing they perform significantly better on average than worst-case bounds suggest, especially for random edge insertions in DAGs.

Contribution

It introduces the first average-case runtime bounds for online topological ordering algorithms, improving understanding of their typical performance.

Findings

Expected runtime of O(n^2 polylog(n)) for certain algorithms under random edge insertions.

Average-case performance is substantially better than worst-case bounds.

Applicable to dynamic DAGs in applications like pointer analysis and incremental compilation.

Abstract

Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worst-case insertion sequences or only evaluated experimentally on random DAGs. We present the first average-case analysis of online topological ordering algorithms. We prove an expected runtime of O(n^2 polylog(n)) under insertion of the edges of a complete DAG in a random order for the algorithms of Alpern et al. (SODA, 1990), Katriel and Bodlaender (TALG, 2006), and Pearce and Kelly (JEA, 2006). This is much less than the best known worst-case bound O(n^{2.75}) for this problem.