Randomized efficient algorithms for compressed strings: the finger-print approach

Abstract

Denote by LZ(ω) the coded form of a string ω produced by Lempel-Ziv encoding algorithm. We consider several classical algorithmic problems for texts in the compressed setting. The first of them is the equality-testing: given LZ(ω) and integers i, j, k test the equality: ω[i...i+k]=ω[j...j+k]. We give a simple and efficient randomized algorithm for this problem using the finger-printing idea. The equality testing is reduced to the equivalence of certain context-free grammars generating single strings. The equality-testing is the bottleneck in other algorithms for compressed texts. We relate the time complexity of several classical problems for texts to the complexity Eq(n) of equality-testing. Assume n=¦LZ(T)¦, m=¦LZ(P)¦ and U=¦T¦. Then we can compute the compressed representations of the sets of occurrences of P in T, periods of T, palindromes of T, and squares of T respectively in times O(n log² U · Eq(m)+n ² log U), O(n log² U · Eq(n)+n ² log U), O(n log² U · Eq(n)+n ² log U) and O(n ² log³ U · Eq(n)+n ³ log² U), where Eq(n)=O(n log log n). The randomization improves considerably upon the known deterministic algorithms ([7] and [8]).

On leave from Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097, Warszawa, Poland. WWW: http://zaa.mimuw.edu.pl/@lechu/lechu.html.

This research was partially supported by the DFG Grant KA 673/4-1, and by the ESPRIT BR Grant 7097 and the ECUS 030.

Supported partially by the grant KBN 8T11C01208.

Supported partially by the grant KBN 8T11C01208.