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The Streaming k-Mismatch Problem: Tradeoffs Between Space and Total Time

Authors Shay Golan , Tomasz Kociumaka , Tsvi Kopelowitz , Ely Porat

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Author Details

Shay Golan
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Tomasz Kociumaka
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Tsvi Kopelowitz
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Ely Porat
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

Funding

This work was supported in part by ISF grants no. 1278/16 and 1926/19, by a BSF grant no. 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).

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Shay Golan, Tomasz Kociumaka, Tsvi Kopelowitz, and Ely Porat. The Streaming k-Mismatch Problem: Tradeoffs Between Space and Total Time. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CPM.2020.15

Abstract

We revisit the k-mismatch problem in the streaming model on a pattern of length m and a streaming text of length n, both over a size-σ alphabet. The current state-of-the-art algorithm for the streaming k-mismatch problem, by Clifford et al. [SODA 2019], uses Õ(k) space and Õ(√k) worst-case time per character. The space complexity is known to be (unconditionally) optimal, and the worst-case time per character matches a conditional lower bound. However, there is a gap between the total time cost of the algorithm, which is Õ(n√k), and the fastest known offline algorithm, which costs Õ(n + min(nk/√m, σn)) time. Moreover, it is not known whether improvements over the Õ(n√k) total time are possible when using more than O(k) space.
We address these gaps by designing a randomized streaming algorithm for the k-mismatch problem that, given an integer parameter k≤s≤m, uses Õ(s) space and costs Õ(n+min(nk²/m, nk/√s, σnm/s)) total time. For s=m, the total runtime becomes Õ(n + min(nk/√m, σn)), which matches the time cost of the fastest offline algorithm. Moreover, the worst-case time cost per character is still Õ(√k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • Streaming pattern matching
  • Hamming distance
  • k-mismatch

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References

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