Listing, Verifying and Counting Lowest Common Ancestors in DAGs: Algorithms and FineGrained Lower Bounds
Abstract
The APLCA 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 APLCA, providing both algorithms and finegrained lower bounds for them. The lower bounds we obtain are the first conditional lower bounds for LCA problems higher than $n^{\omegao(1)}$, 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 $n^{3o(1)}$ time under the popular assumption that 3uniform 5hyperclique detection requires $n^{5o(1)}$ 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 $n^{3o(1)}$ 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, Mühling 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 $n^{2.5o(1)}$ time assuming 3uniform 4hyperclique 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].
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.10932
 arXiv:
 arXiv:2204.10932
 Bibcode:
 2022arXiv220410932M
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 To appear in ICALP 2022. Abstract shortened to fit arXiv requirement