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Space-Efficient Algorithms for Longest Increasing Subsequence

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Abstract

Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in \(O\left (n \log n\right )\) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For \(\sqrt {n} \le s \le n\), we present algorithms that use \(O\left (s \log n\right )\) bits and \(O\left (\frac {1}{s} \cdot n^{2} \cdot \log n\right )\) time for computing the length of a longest increasing subsequence, and \(O\left (\frac {1}{s} \cdot n^{2} \cdot \log ^{2} n\right )\) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space.

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Notes

  1. This algorithm outputs a longest increasing subsequence in the reversed order. One can access the input in the reversed order and find a longest decreasing subsequence to avoid this issue.

  2. Again this output is reversed. We can also compute the output in nonreversed order as discussed before.

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Acknowledgments

The authors are grateful to William S. Evans for bringing the reduction from LCS to LIS mentioned in the concluding remarks to their attention.

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Authors and Affiliations

  1. Yokohama City University, Yokohama, Japan

    Masashi Kiyomi

  2. Nagoya University, Nagoya, Japan

    Hirotaka Ono

  3. Kumamoto University, Kumamoto, Japan

    Yota Otachi

  4. TU Kaiserslautern, Kaiserslautern, Germany

    Pascal Schweitzer

  5. The University of Electro-Communications, Chofu, Japan

    Jun Tarui

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  1. Masashi Kiyomi
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  2. Hirotaka Ono
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Correspondence to Yota Otachi.

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This article is part of the Topical Collection on Special Issue on Theoretical Aspects of Computer Science (2018)

Partially supported by MEXT/JSPS KAKENHI grant numbers JP17H01698, JP18H04091, JP18K11168, JP18K11169, JP24106004. A preliminary version appeared in the proceedings of the 35th International Symposium on Theoretical Aspects of Computer Science (STACS 2018), vol. 96 of Leibniz International Proceedings in Informatics, pp. 44:1-44:15, 2018.

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Kiyomi, M., Ono, H., Otachi, Y. et al. Space-Efficient Algorithms for Longest Increasing Subsequence. Theory Comput Syst 64, 522–541 (2020). https://doi.org/10.1007/s00224-018-09908-6

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