Abstract
A Longest Common Extension (LCE) query on a text T of length N asks for the length of the longest common prefix of suffixes starting at given two positions. We show that the signature encoding G of size w = O(min(z log N log^* M, N)) [Mehlhorn et al., Algorithmica 17(2):183198, 1997] of T, which can be seen as a compressed representation of T, has a capability to support LCE queries in O(log N + log ell log^* M) time, where ell is the answer to the query, z is the size of the LempelZiv77 (LZ77) factorization of T, and M >= 4N is an integer that can be handled in constant time under word RAM model. In compressed space, this is the fastest deterministic LCE data structure in many cases. Moreover, G can be enhanced to support efficient update operations: After processing G in O(w f_A) time, we can insert/delete any (sub)string of length y into/from an arbitrary position of T in O((y + log Nlog^* M) f_A) time, where f_A = O(min{ (loglog M loglog w)/(logloglog M), sqrt(log w/loglog w)}). This yields the first fully dynamic LCE data structure working in compressed space. We also present efficient construction algorithms from various types of inputs: We can construct G in O(N f_A) time from uncompressed string T; in O(n loglog (n log^* M) log N log^* M) time from grammarcompressed string T represented by a straightline program of size n; and in O(z f_A log N log^* M) time from LZ77compressed string T with z factors. On top of the above contributions, we show several applications of our data structures which improve previous best known results on grammarcompressed string processing.
BibTeX  Entry
@InProceedings{nishimoto_et_al:LIPIcs:2016:6504,
author = {Takaaki Nishimoto and Tomohiro I and Shunsuke Inenaga and Hideo Bannai and Masayuki Takeda},
title = {{Fully Dynamic Data Structure for LCE Queries in Compressed Space}},
booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
pages = {72:172:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770163},
ISSN = {18688969},
year = {2016},
volume = {58},
editor = {Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6504},
URN = {urn:nbn:de:0030drops65045},
doi = {10.4230/LIPIcs.MFCS.2016.72},
annote = {Keywords: dynamic texts, longest common extension (LCE) queries, straightline program}
}
Keywords: 

dynamic texts, longest common extension (LCE) queries, straightline program 
Collection: 

41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) 
Issue Date: 

2016 
Date of publication: 

19.08.2016 