Longest Common Extensions in Sublinear Space

Abstract

The longest common extension problem (LCE problem) is to construct a data structure for an input string T of length n that supports LCE(i, j) queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions i and j in T . This classic problem has a well-known solution that uses O(n) space and O(1) query time. In this paper we show that for any trade-off parameter 1 ≤ τ ≤ n, the problem can be solved in O( n τ ) space and O(τ ) query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.

Locally Consistent Parsing for Text Indexing in Small Space

“…However, we design a new way to combine Karp-Rabin fingerprints together with approximately min-wise hash functions, in order to use this method using only small amount of space. In previous results for the LCE problem [3,4,19,37], a set of positions of size O( n τ ) was also considered by the data structures. In contrast to these algorithms, where the selected positions were dependent only on the length of the text, we introduce the first algorithm that exploit the actual text using local properties in order to decide which positions to select.…”

“…Using the SST construction, we show how to combine the deterministic selection with the difference covers [29,8] technique, to get almost optimal trade-off for the LCE data structure of O( n τ ) space and O(τ log * n) query time with O(n(log τ + log * n)) deterministic construction time. We mention that although the technique of difference covers already been used for the LCE problem (see [34,4,19]), the usage in our algorithm is different. While in all previous work the considered positions were all the positions in a given range, in our algorithm the positions are subset of the locally selected positions.…”

“…Kosolobov also conjectured that for data structure who used S(n) < o(n) bits of space a similar lower bound of T (n) · S(n) = Ω(n log S(n)) holds. A trade-off of O(τ ) query time and O( n τ ) words of space is already known, both in randomized and deterministic construction algorithms [4], and therefore the research efforts are invested in reducing the construction time. Tanimura et al [37] introduced a deterministic data structure with O(nτ ) construction time algorithm, and with a slightly increasing of the query time to O(τ min{log τ, log n τ }).…”

Tight lower bounds for the longest common extension problem

“…For S(n) = Ω(n), this new trade-off improves by log n factor the trade-off S(n)T (n) = Ω(n) established by Bille et al [2], who used a simple reduction to a lower bound obtained by Brodal et al [3] for the so-called range minimum queries problem. 1 For brevity, log denotes the logarithm with base 2.…”

“…Prezza [14] described a Monte Carlo version of his "in-place" data structure that answers the LCE queries in O(log ℓ) time and has a construction algorithm working in O( n log σ n ) expected time using the same memory, i.e., also "in-place". Bille et al [1] presented a Monte Carlo version of their data structure for read-only inputs that answers the LCE queries in O(τ ) time using O( n log n τ ) bits of additional space and has O(n log n τ ) construction time (within the same space), where 1 ≤ τ ≤ n. Gawrychowski and Kociumaka [7, Th. 3.3] described a modification of this Monte Carlo solution for read-only inputs that has the same optimal space and query time bounds but can be constructed in optimal O(n) time.…”