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Fast Approximation of Search Trees on Trees with Centroid Trees

Authors Benjamin Aram Berendsohn, Ishay Golinsky, Haim Kaplan , László Kozma

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Author Details

Benjamin Aram Berendsohn
  • Institut für Informatik, Freie Universität Berlin, Germany
Ishay Golinsky
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Haim Kaplan
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
László Kozma
  • Institut für Informatik, Freie Universität Berlin, Germany

Funding

  • Berendsohn, Benjamin Aram: DFG grant KO 6140/1-1.
  • Golinsky, Ishay: ISF grant no. 1595/19 and the Blavatnik Research Foundation.
  • Kaplan, Haim: ISF grant no. 1595/19 and the Blavatnik Research Foundation.
  • Kozma, László: DFG grant KO 6140/1-1.

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Benjamin Aram Berendsohn, Ishay Golinsky, Haim Kaplan, and László Kozma. Fast Approximation of Search Trees on Trees with Centroid Trees. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.19

Abstract

Search trees on trees (STTs) generalize the fundamental binary search tree (BST) data structure: in STTs the underlying search space is an arbitrary tree, whereas in BSTs it is a path. An optimal BST of size n can be computed for a given distribution of queries in 𝒪(n²) time [Knuth, Acta Inf. 1971] and centroid BSTs provide a nearly-optimal alternative, computable in 𝒪(n) time [Mehlhorn, SICOMP 1977]. 
By contrast, optimal STTs are not known to be computable in polynomial time, and the fastest constant-approximation algorithm runs in 𝒪(n³) time [Berendsohn, Kozma, SODA 2022]. Centroid trees can be defined for STTs analogously to BSTs, and they have been used in a wide range of algorithmic applications. In the unweighted case (i.e., for a uniform distribution of queries), the centroid tree can be computed in 𝒪(n) time [Brodal, Fagerberg, Pedersen, Östlin, ICALP 2001; Della Giustina, Prezza, Venturini, SPIRE 2019]. These algorithms, however, do not readily extend to the weighted case. Moreover, no approximation guarantees were previously known for centroid trees in either the unweighted or weighted cases. 
In this paper we revisit centroid trees in a general, weighted setting, and we settle both the algorithmic complexity of constructing them, and the quality of their approximation. For constructing a weighted centroid tree, we give an output-sensitive 𝒪(n log h) ⊆ 𝒪(n log n) time algorithm, where h is the height of the resulting centroid tree. If the weights are of polynomial complexity, the running time is 𝒪(n log log n). We show these bounds to be optimal, in a general decision tree model of computation. For approximation, we prove that the cost of a centroid tree is at most twice the optimum, and this guarantee is best possible, both in the weighted and unweighted cases. We also give tight, fine-grained bounds on the approximation-ratio for bounded-degree trees and on the approximation-ratio of more general α-centroid trees.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • centroid tree
  • search trees on trees
  • approximation

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