Listing, Verifying and Counting Lowest Common Ancestors in DAGs: Algorithms and Fine-Grained Lower Bounds

TL;DR

This paper explores algorithms and lower bounds for the Lowest Common Ancestor (LCA) problem in DAGs, providing new conditional lower bounds and efficient algorithms for various LCA variants, advancing understanding of computational complexity in this area.

Contribution

It introduces the first conditional lower bounds for LCA problems beyond $n^{ ext{omega}-o(1)}$, and develops algorithms for detecting, listing, and verifying LCAs with proven complexity bounds.

Findings

Detecting all pairs with at most two LCAs in $O(n^\omega)$ time.

Listing 7 LCAs per pair requires $n^{3-o(1)}$ time under hyperclique assumptions.

Counting all LCAs per pair takes $n^{3-o(1)}$ time under ETH and $n^{\omega(1,2,1)}$ under 4-Clique hypothesis.

Abstract

The AP-LCA problem asks, given an $n$-node directed acyclic graph (DAG), to compute for every pair of vertices $u$ and $v$ in the DAG a lowest common ancestor (LCA) of $u$ and $v$ if one exists. In this paper we study several interesting variants of AP-LCA, providing both algorithms and fine-grained lower bounds for them. The lower bounds we obtain are the first conditional lower bounds for LCA problems higher than $n^{\omega-o(1)}$, where $\omega$ is the matrix multiplication exponent. Some of our results include: - In any DAG, we can detect all vertex pairs that have at most two LCAs and list all of their LCAs in $O(n^\omega)$ time. This algorithm extends a result of [Kowaluk and Lingas ESA'07] which showed an $\tilde{O}(n^\omega)$ time algorithm that detects all pairs with a unique LCA in a DAG and outputs their corresponding LCAs. - Listing $7$ LCAs per vertex pair in DAGs requires $n^{3-o(1)}$ time under the popular assumption that 3-uniform 5-hyperclique detection requires $n^{5-o(1)}$ time. This is surprising since essentially cubic time is sufficient to list all LCAs (if $\omega=2$). - Counting the number of LCAs for every vertex pair in a DAG requires $n^{3-o(1)}$ time under the Strong Exponential Time Hypothesis, and $n^{\omega(1,2,1)-o(1)}$ time under the $4$-Clique hypothesis. This shows that the algorithm of [Echkardt, M\"{u}hling and Nowak ESA'07] for listing all LCAs for every pair of vertices is likely optimal. - Given a DAG and a vertex $w_{u,v}$ for every vertex pair $u,v$, verifying whether all $w_{u,v}$ are valid LCAs requires $n^{2.5-o(1)}$ time assuming 3-uniform 4-hyperclique requires $n^{4 - o(1)}$ time. This defies the common intuition that verification is easier than computation since returning some LCA per vertex pair can be solved in $O(n^{2.447})$ time [Grandoni et al. SODA'21].