# Compressed Communication Complexity of Longest Common Prefixes

Philip Bille, Mikko Berggreen Ettienne, Roberto Grossi, Inge Li Gørtz, Eva Rotenberg

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

### 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.
Original language English String Processing and Information Retrieval Springer 2018 74-87 9783030004798 https://doi.org/10.1007/978-3-030-00479-8_7 Published - 2018 25th International Symposium on String Processing and Information Retrieval - Lima, PeruDuration: 9 Oct 2018 → 11 Oct 2018

### Conference

Conference 25th International Symposium on String Processing and Information Retrieval Peru Lima 09/10/2018 → 11/10/2018
Series Lecture Notes in Computer Science 11147 0302-9743

### Keywords

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

### Cite this

Bille, P., Berggreen Ettienne, M., Grossi, R., Gørtz, I. L., & Rotenberg, E. (2018). Compressed Communication Complexity of Longest Common Prefixes. In String Processing and Information Retrieval (pp. 74-87). Springer. Lecture Notes in Computer Science, Vol.. 11147 https://doi.org/10.1007/978-3-030-00479-8_7
Bille, Philip ; Berggreen Ettienne, Mikko ; Grossi, Roberto ; Gørtz, Inge Li ; Rotenberg, Eva. / Compressed Communication Complexity of Longest Common Prefixes. String Processing and Information Retrieval. Springer, 2018. pp. 74-87 (Lecture Notes in Computer Science, Vol. 11147).
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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",
author = "Philip Bille and {Berggreen Ettienne}, Mikko and Roberto Grossi and G{\o}rtz, {Inge Li} and Eva Rotenberg",
year = "2018",
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Bille, P, Berggreen Ettienne, M, Grossi, R, Gørtz, IL & Rotenberg, E 2018, Compressed Communication Complexity of Longest Common Prefixes. in String Processing and Information Retrieval. Springer, Lecture Notes in Computer Science, vol. 11147, pp. 74-87, 25th International Symposium on String Processing and Information Retrieval, Lima, Peru, 09/10/2018. https://doi.org/10.1007/978-3-030-00479-8_7

Compressed Communication Complexity of Longest Common Prefixes. / Bille, Philip; Berggreen Ettienne, Mikko; Grossi, Roberto; Gørtz, Inge Li; Rotenberg, Eva.

String Processing and Information Retrieval. Springer, 2018. p. 74-87 (Lecture Notes in Computer Science, Vol. 11147).

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

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N2 - 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.

AB - 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.

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KW - Compression Upper bound

KW - Output sensitive

KW - Longest common prefix

KW - Predecessor

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DO - 10.1007/978-3-030-00479-8_7

M3 - Article in proceedings

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T3 - Lecture Notes in Computer Science

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BT - String Processing and Information Retrieval

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Bille P, Berggreen Ettienne M, Grossi R, Gørtz IL, Rotenberg E. Compressed Communication Complexity of Longest Common Prefixes. In String Processing and Information Retrieval. Springer. 2018. p. 74-87. (Lecture Notes in Computer Science, Vol. 11147). https://doi.org/10.1007/978-3-030-00479-8_7