Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Mathialagan, Surya; Vassilevska Williams, Virginia; Xu, Yinzhan https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-164359
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Listing, Verifying and Counting Lowest Common Ancestors in DAGs: Algorithms and Fine-Grained Lower Bounds

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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, i.e. a node that is an ancestor of both u and v but no proper descendent of it is their common ancestor. Recently [Grandoni et al. SODA'21] obtained the first sub-n^{2.5} time algorithm for AP-LCA running in O(n^{2.447}) time. Meanwhile, the only known conditional lower bound for AP-LCA is that the problem requires n^{ω-o(1)} time where ω is the matrix multiplication exponent.
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^{ω-o(1)}. 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^ω) time. This algorithm extends a result of [Kowaluk and Lingas ESA'07] which showed an Õ(n^ω) 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 ω = 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^{ω(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^{2.5-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.

BibTeX - Entry

@InProceedings{mathialagan_et_al:LIPIcs.ICALP.2022.94,
  author =	{Mathialagan, Surya and Vassilevska Williams, Virginia and Xu, Yinzhan},
  title =	{{Listing, Verifying and Counting Lowest Common Ancestors in DAGs: Algorithms and Fine-Grained Lower Bounds}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{94:1--94:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16435},
  URN =		{urn:nbn:de:0030-drops-164359},
  doi =		{10.4230/LIPIcs.ICALP.2022.94},
  annote =	{Keywords: All-Pairs Lowest Common Ancestor, Fine-Grained Complexity}
}

Keywords: All-Pairs Lowest Common Ancestor, Fine-Grained Complexity
Seminar: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue date: 2022
Date of publication: 28.06.2022


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