### Compressed Communication Complexity of Longest Common Prefixes [chapter]

Philip Bille, Mikko Berggreen Ettienne, Roberto Grossi, Inge Li Gørtz, Eva Rotenberg
2018 Lecture Notes in Computer Science
We consider the communication complexity of fundamental longest common prefix (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 ℓ = 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
more » ... ng 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 ℓ) total communication is necessary. We extend the result to the natural case when Bob holds a set of strings B1, . . . , B k , and the goal is to find the length of the maximal longest prefix shared by A and any of B1, . . . , B k . Here, we give a protocol with O(log z) rounds and O(lg z lg k + lg ℓ) 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. Compressed Communication Complexity of Longest Common Prefixes 3 at a server. To efficiently handle many queries we want to reduce both communication and rounds for each search. If we again view the strings as integers this is the predecessor problem. We generalize Theorem 1 to this scenario. Theorem 2. The Lcp k problem has a randomized public-coin O(lg z) round communication protocol with O(lg z lg k + lg ℓ) communication complexity, where ℓ is the maximal common prefix between A and any one of B 1 , . . . , B k , and z is the number of phrases in the LZ77 parse of this prefix. Compared to Theorem 1 we obtain the same number of rounds and only increase the total communication by an additive O(lg z lg k) term. As z ≤ ℓ the total communication increases by at most a factor lg k. The mentioned results hold only for LZ77 parses without self-references (see Sec. 2). We also show how to handle self-referential LZ77 parses and obtain the following bounds, where we add either extra O(lg lg ℓ) rounds or extra O(lg lg lg |A|) communication.