eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-06-21
8:1
8:18
10.4230/LIPIcs.CPM.2023.8
article
Optimal Near-Linear Space Heaviest Induced Ancestors
Charalampopoulos, Panagiotis
1
https://orcid.org/0000-0002-6024-1557
Dudek, Bartłomiej
2
https://orcid.org/0000-0003-2652-995X
Gawrychowski, Paweł
2
https://orcid.org/0000-0002-6993-5440
Pokorski, Karol
2
https://orcid.org/0000-0002-2140-8641
Birkbeck, University of London, UK
Institute of Computer Science, University of Wrocław, Poland
We revisit the Heaviest Induced Ancestors (HIA) problem that was introduced by Gagie, Gawrychowski, and Nekrich [CCCG 2013] and has a number of applications in string algorithms. Let T₁ and T₂ be two rooted trees whose nodes have weights that are increasing in all root-to-leaf paths, and labels on the leaves, such that no two leaves of a tree have the same label. A pair of nodes (u, v) ∈ T₁ × T₂ is induced if and only if there is a label shared by leaf-descendants of u and v. In an HIA query, given nodes x ∈ T₁ and y ∈ T₂, the goal is to find an induced pair of nodes (u, v) of the maximum total weight such that u is an ancestor of x and v is an ancestor of y.
Let n be the upper bound on the sizes of the two trees. It is known that no data structure of size 𝒪̃(n) can answer HIA queries in o(log n / log log n) time [Charalampopoulos, Gawrychowski, Pokorski; ICALP 2020]. This (unconditional) lower bound is a polyloglog n factor away from the query time of the fastest 𝒪̃(n)-size data structure known to date for the HIA problem [Abedin, Hooshmand, Ganguly, Thankachan; Algorithmica 2022]. In this work, we resolve the query-time complexity of the HIA problem for the near-linear space regime by presenting a data structure that can be built in 𝒪̃(n) time and answers HIA queries in 𝒪(log n/log log n) time. As a direct corollary, we obtain an 𝒪̃(n)-size data structure that maintains the LCS of a static string and a dynamic string, both of length at most n, in time optimal for this space regime.
The main ingredients of our approach are fractional cascading and the utilization of an 𝒪(log n/ log log n)-depth tree decomposition. The latter allows us to break through the Ω(log n) barrier faced by previous works, due to the depth of the considered heavy-path decompositions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol259-cpm2023/LIPIcs.CPM.2023.8/LIPIcs.CPM.2023.8.pdf
data structures
string algorithms
fractional cascading