Optimal Heaviest Induced Ancestors

TL;DR

This paper introduces a near-linear space data structure for the Heaviest Induced Ancestors problem that achieves optimal query time, improving previous bounds and enabling efficient longest common substring maintenance.

Contribution

The authors present a novel data structure for HIA with $ ilde{ ext{O}}(n)$ size and $ ext{O}(rac{ ext{log} n}{ ext{log} ext{log} n})$ query time, resolving a key complexity gap.

Findings

Achieves optimal query time for HIA in near-linear space

Enables efficient LCS maintenance for static and dynamic strings

Uses fractional cascading and tree decomposition techniques

Abstract

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 $T_1$ and $T_2$ 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 $(u, v)\in T_1 \times T_2$ is \emph{induced} if and only if there is a label shared by leaf-descendants of $u$ and $v$. In an HIA query, given nodes $x \in T_1$ and $y \in T_2$, 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 $\tilde{\mathcal{O}}(n)$ can answer HIA queries in $o(\log n / \log \log n)$ time [Charalampopoulos, Gawrychowski, Pokorski; ICALP 2020]. This (unconditional) lower bound is a $\operatorname{polyloglog} n$ factor away from the query time of the fastest $\tilde{\mathcal{O}}(n)$-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 $\tilde{\mathcal{O}}(n)$ time and answers HIA queries in $\mathcal{O}(\log n/\log\log n)$ time. As a direct corollary, we obtain an $\tilde{\mathcal{O}}(n)$-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 $\mathcal{O}(\log n/ \log\log n)$-depth tree decomposition.