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Compressed Communication Complexity of Longest Common Prefixes

  • Philip Bille
  • Mikko Berggreen Ettienne
  • Roberto GrossiEmail author
  • Inge Li Gørtz
  • Eva Rotenberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11147)

Abstract

We consider the communication complexity of fundamental longest common prefix \(({{\mathrm{\textsc {Lcp}}}})\) problems. In the simplest version, two parties, Alice and Bob, each hold a string, A and B, and we want to determine the length of their longest common prefix \(\ell ={{\mathrm{\textsc {Lcp}}}}(A,B)\) using as few rounds and bits of communication as possible. We show that if the longest common prefix of A and B is compressible, then we can significantly reduce the number of rounds compared to the optimal uncompressed protocol, while achieving the same (or fewer) bits of communication. Namely, if the longest common prefix has an LZ77 parse of z phrases, only \(O(\lg z)\) rounds and \(O(\lg \ell )\) total communication is necessary. We extend the result to the natural case when Bob holds a set of strings \(B_1, \ldots , B_k\), and the goal is to find the length of the maximal longest prefix shared by A and any of \(B_1, \ldots , B_k\). Here, we give a protocol with \(O(\log z)\) rounds and \(O(\lg z \lg k + \lg \ell )\) total communication. We present our result in the public-coin model of computation but by a standard technique our results generalize to the private-coin model. Furthermore, if we view the input strings as integers the problems are the greater-than problem and the predecessor problem.

Keywords

Communication complexity LZ77 Compression Upper bound Output sensitive Longest common prefix Predecessor 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.DTU Compute, AlgoLoGTechnical University of DenmarkKongens LyngbyDenmark
  2. 2.Dipartimento di InformaticaUniversità di PisaPisaItaly

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