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A Simple Representation of Tree Covering Utilizing Balanced Parentheses and Efficient Implementation of Average-Case Optimal RMQs

Authors Kou Hamada , Sankardeep Chakraborty, Seungbum Jo , Takuto Koriyama, Kunihiko Sadakane , Srinivasa Rao Satti

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Kou Hamada
  • The University of Tokyo, Japan
Sankardeep Chakraborty
  • The University of Tokyo, Japan
Seungbum Jo
  • Chungnam National University, Daejeon, South Korea
Takuto Koriyama
  • The University of Tokyo, Japan
Kunihiko Sadakane
  • The University of Tokyo, Japan
Srinivasa Rao Satti
  • Norwegian University of Science and Technology, Trondheim, Norway

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Kou Hamada, Sankardeep Chakraborty, Seungbum Jo, Takuto Koriyama, Kunihiko Sadakane, and Srinivasa Rao Satti. A Simple Representation of Tree Covering Utilizing Balanced Parentheses and Efficient Implementation of Average-Case Optimal RMQs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 64:1-64:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.64

Abstract

Tree covering is a technique for decomposing a tree into smaller sized trees with desirable properties, and has been employed in various succinct data structures. However, significant hurdles stand in the way of a practical implementation of tree covering: a lot of pointers are used to maintain the tree-covering hierarchy and many indices for tree navigational queries consume theoretically negligible yet practically vast space. To tackle these problems, we propose a simple representation of tree covering using a balanced-parenthesis representation. The key to the proposal is the observation that every micro tree splits into at most two intervals on the BP representation. Utilizing the representation, we propose several data structures that represent a tree and its tree cover, which consequently allow micro tree compression with arbitrary coding and efficient tree navigational queries. We also applied our data structure to average-case optimal RMQ by Munro et al. [ESA 2021] and implemented the RMQ data structure. Our RMQ data structures spend less than 2n bits and process queries in a practical time on several settings of the performance evaluation, reducing the gap between theoretical space complexity and actual space consumption. For example, our implementation consumes 1.822n bits and processes queries in 5µs on average for random queries and in 13µs on average for the worst query widths. We also implement tree navigational operations while using the same amount of space as the RMQ data structures. We believe the representation can be widely utilized for designing practically memory-efficient data structures based on tree covering.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
Keywords
  • Hypersuccinct trees
  • Succinct data structures
  • Range minimum queries
  • Binary trees

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References

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