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Faster Deterministic Modular Subset Sum

Author Krzysztof Potępa

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Krzysztof Potępa
  • Jagiellonian University, Kraków, Poland

Acknowledgements

I would like to thank Lech Duraj, Krzysztof Pióro, and Adam Polak for their help with preparing the paper, and the anonymous reviewers for their useful suggestions.

Cite AsGet BibTex

Krzysztof Potępa. Faster Deterministic Modular Subset Sum. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.76

Abstract

We consider the Modular Subset Sum problem: given a multiset X of integers from ℤ_m and a target integer t, decide if there exists a subset of X with a sum equal to t (mod m). Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in 𝒪(m log m) time, while Axiotis et al. gave a deterministic algorithm that runs in 𝒪(m polylog m) time. Both results work by reduction to a text problem, which is solved using a dynamic strings data structure. In this work, we develop a simple data structure, designed specifically to handle the text problem that arises in the algorithms for Modular Subset Sum. Our data structure, which we call the shift-tree, is a simple variant of a segment tree. We provide both a hashing-based and a deterministic variant of the shift-trees. We then apply our data structure to the Modular Subset Sum problem and obtain two algorithms. The first algorithm is Monte-Carlo randomized and matches the 𝒪(m log m) runtime of the Las-Vegas algorithm by Cardinal and Iacono. The second algorithm is fully deterministic and runs in 𝒪(m log m ⋅ α(m)) time, where α is the inverse Ackermann function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Algorithm design techniques
Keywords
  • Modular Subset Sum
  • String Problem
  • Segment Tree
  • Data Structure

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