42 research outputs found
Fully-Functional Suffix Trees and Optimal Text Searching in BWT-runs Bounded Space
Indexing highly repetitive texts - such as genomic databases, software
repositories and versioned text collections - has become an important problem
since the turn of the millennium. A relevant compressibility measure for
repetitive texts is r, the number of runs in their Burrows-Wheeler Transforms
(BWTs). One of the earliest indexes for repetitive collections, the Run-Length
FM-index, used O(r) space and was able to efficiently count the number of
occurrences of a pattern of length m in the text (in loglogarithmic time per
pattern symbol, with current techniques). However, it was unable to locate the
positions of those occurrences efficiently within a space bounded in terms of
r. In this paper we close this long-standing problem, showing how to extend the
Run-Length FM-index so that it can locate the occ occurrences efficiently
within O(r) space (in loglogarithmic time each), and reaching optimal time, O(m
+ occ), within O(r log log w ({\sigma} + n/r)) space, for a text of length n
over an alphabet of size {\sigma} on a RAM machine with words of w =
{\Omega}(log n) bits. Within that space, our index can also count in optimal
time, O(m). Multiplying the space by O(w/ log {\sigma}), we support count and
locate in O(dm log({\sigma})/we) and O(dm log({\sigma})/we + occ) time, which
is optimal in the packed setting and had not been obtained before in compressed
space. We also describe a structure using O(r log(n/r)) space that replaces the
text and extracts any text substring of length ` in almost-optimal time
O(log(n/r) + ` log({\sigma})/w). Within that space, we similarly provide direct
access to suffix array, inverse suffix array, and longest common prefix array
cells, and extend these capabilities to full suffix tree functionality,
typically in O(log(n/r)) time per operation.Comment: submitted version; optimal count and locate in smaller space: O(r log
log_w(n/r + sigma)
The expected profile of digital search trees
AbstractA digital search tree (DST) is a fundamental data structure on words that finds various applications from the popular Lempel–Zivʼ78 data compression scheme to distributed hash tables. The profile of a DST measures the number of nodes at the same distance from the root; it depends on the number of stored strings and the distance from the root. Most parameters of DST (e.g., depth, height, fillup) can be expressed in terms of the profile. We study here asymptotics of the average profile in a DST built from sequences generated independently by a memoryless source. After representing the average profile by a recurrence, we solve it using a wide range of analytic tools. This analysis is surprisingly demanding but once it is carried out it reveals an unusually intriguing and interesting behavior. The average profile undergoes phase transitions when moving from the root to the longest path: at first it resembles a full tree until it abruptly starts growing polynomially and oscillating in this range. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, poissonization and depoissonization, the saddle point method, singularity analysis and uniform asymptotic analysis
Document retrieval on repetitive string collections
Most of the fastest-growing string collections today are repetitive, that is, most of the constituent documents are similar to many others. As these collections keep growing, a key approach to handling them is to exploit their repetitiveness, which can reduce their space usage by orders of magnitude. We study the problem of indexing repetitive string collections in order to perform efficient document retrieval operations on them. Document retrieval problems are routinely solved by search engines on large natural language collections, but the techniques are less developed on generic string collections. The case of repetitive string collections is even less understood, and there are very few existing solutions. We develop two novel ideas, interleaved LCPs and precomputed document lists, that yield highly compressed indexes solving the problem of document listing (find all the documents where a string appears), top-k document retrieval (find the k documents where a string appears most often), and document counting (count the number of documents where a string appears). We also show that a classical data structure supporting the latter query becomes highly compressible on repetitive data. Finally, we show how the tools we developed can be combined to solve ranked conjunctive and disjunctive multi-term queries under the simple model of relevance. We thoroughly evaluate the resulting techniques in various real-life repetitiveness scenarios, and recommend the best choices for each case.Peer reviewe