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Succinct Data Structures for Segments

Authors Philip Bille , Inge Li Gørtz , Simon R. Tarnow

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ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Succinct
  • Data structures
  • Selection

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Abstract

We consider succinct data structures for representing a set of n horizontal line segments in the plane given in rank space to support segment access, segment selection, and segment rank queries. A segment access query finds the segment (x₁, x₂, y) given its y-coordinate (y-coordinates of the segments are distinct), a segment selection query finds the jth smallest segment (the segment with the jth smallest y-coordinate) among the segments crossing the vertical line for a given x-coordinate, and a segment rank query finds the number of segments crossing the vertical line through x-coordinate i with y-coordinate at most y, for a given x and y. This problem is a central component in compressed data structures for persistent strings supporting random access. 
Our main result is a data structure using 2n lg n + O(n lg n / lg lg n) bits of space and O(lg n / lg lg n) query time for all operations. We show that this space bound is optimal up to lower-order terms. We will also show that the query time for segment rank is optimal. The query time for segment selection is also optimal by a previous bound. 
To obtain our results, we present a novel segment wavelet tree data structure of independent interest. This structure is inspired by and extends the classic wavelet tree for sequences. This leads to a simple, succinct solution with O(log n) query times. We then extend this solution to obtain optimal query time. Our space lower bound follows from a simple counting argument, and our lower bound for segment rank is obtained by a reduction from 2-dimensional counting.

Cite As Get BibTex

Philip Bille, Inge Li Gørtz, and Simon R. Tarnow. Succinct Data Structures for Segments. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CPM.2025.27

Author Details

Philip Bille
  • Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
Inge Li Gørtz
  • DTU Compute, Technical University of Denmark, Lyngby, Denmark
Simon R. Tarnow
  • DTU Compute, Technical University of Denmark, Lyngby, Denmark

Funding

  • Bille, Philip: Danish Research Council grant DFF-8021-002498.
  • Gørtz, Inge Li: Danish Research Council grant DFF-8021-002498.

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