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The Trie Measure, Revisited

Authors Jarno N. Alanko , Ruben Becker , Davide Cenzato , Travis Gagie , Sung-Hwan Kim , Bojana Kodric , Nicola Prezza

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ACM Subject Classification
  • Theory of computation → Data compression
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Dynamic programming
  • Theory of computation → Discrete optimization
Keywords
  • Succinct data structures
  • degenerate strings
  • integer encoding

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Abstract

In this paper, we study the following problem: given n subsets S₁, … , S_n of an integer universe U = {0,… , u-1}, having total cardinality N = ∑_{i = 1}ⁿ |S_i|, find a prefix-free encoding enc : U → {0,1}^+ minimizing the so-called trie measure, i.e., the total number of edges in the n binary tries T₁, … , T_n, where T_i is the trie packing the encoded integers {enc(x):x ∈ S_i}. We first observe that this problem is equivalent to that of merging u sets with the cheapest sequence of binary unions, a problem which in [Ghosh et al., ICDCS 2015] is shown to be NP-hard. Motivated by the hardness of the general problem, we focus on particular families of prefix-free encodings. We start by studying the fixed-length shifted encoding of [Gupta et al., Theoretical Computer Science 2007]. Given a parameter 0 ≤ a < u, this encoding sends each x ∈ U to (x + a) mod u, interpreted as a bit-string of log u bits. We develop the first efficient algorithms that find the value of a minimizing the trie measure when this encoding is used. Our two algorithms run in O(u + Nlog u) and O(Nlog² u) time, respectively. We proceed by studying ordered encodings (a.k.a. monotone or alphabetic), and describe an algorithm finding the optimal such encoding in O(N+u³) time. Within the same running time, we show how to compute the best shifted ordered encoding, provably no worse than both the optimal shifted and optimal ordered encodings. We provide implementations of our algorithms and discuss how these encodings perform in practice.

Cite As Get BibTex

Jarno N. Alanko, Ruben Becker, Davide Cenzato, Travis Gagie, Sung-Hwan Kim, Bojana Kodric, and Nicola Prezza. The Trie Measure, Revisited. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CPM.2025.19

Author Details

Jarno N. Alanko
  • University of Helsinki, Finland
Ruben Becker
  • Ca' Foscari University of Venice, Italy
Davide Cenzato
  • Ca' Foscari University of Venice, Italy
Travis Gagie
  • Dalhousie University, Halifax, Canada
Sung-Hwan Kim
  • Ca' Foscari University of Venice, Italy
Bojana Kodric
  • Ca' Foscari University of Venice, Italy
Nicola Prezza
  • Ca' Foscari University of Venice, Italy

Funding

Ruben Becker, Davide Cenzato, Sung-Hwan Kim, Bojana Kodric, Nicola Prezza: Funded by the European Union (ERC, REGINDEX, 101039208). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
  • Alanko, Jarno N.: Funded by Academy of Finland grant 339070.
  • Gagie, Travis: Funded by NSERC Discovery Grant RGPIN-07185-2020.

Supplementary Materials

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