Time–space trade-offs for longest common extensions

Abstract

We revisit the longest common extension (LCE) problem, that is, preprocess a string T into a compact data structure that supports fast LCE queries. An LCE query takes a pair (i, j) of indices in T and returns the length of the longest common prefix of the suffixes of T starting at positions i and j. We study the time-space trade-offs for the problem, that is, the space used for the data structure vs. the worst-case time for answering an LCE query. Let n be the length of T . Given a parameter τ , 1 ≤ τ ≤ n, we show how to achieve either O(n/ √ τ ) space and O(τ ) query time, or O(n/τ ) space and O(τ log(|LCE(i, j)|/τ )) query time, where |LCE(i, j)| denotes the length of the LCE returned by the query. These bounds provide the first smooth trade-offs for the LCE problem and almost match the previously known bounds at the extremes when τ = 1 or τ = n. We apply the result to obtain improved bounds for several applications where the LCE problem is the computational bottleneck, including approximate string matching and computing palindromes. We also present an efficient technique to reduce LCE queries on two strings to one string. Finally, we give a lower bound on the time-space product for LCE data structures in the non-uniform cell probe model showing that our second trade-off is nearly optimal.

Towards a Definitive Measure of Repetitiveness

“…Given integer parameters r and s, the root of the block tree divides S into s equalsized (that is, with the same number of characters) blocks (assume for simplicity that n = s • r t for some integer t). 7 By storing further data associated with marked and unmarked blocks, the block tree offers the following functionality [5]:…”

“…. n], in O(n log n) expected time [7]. Our index will need to compute Karp-Rabin fingerprints κ(T [i .…”

Time-Space Trade-Offs for the Longest Common Substring Problem

“…Alternatively, we can build a data structure that for any pair of suffixes can be queried for the value of their longest common prefix. Building such a data structure is known as the Longest Common Extension (LCE) Problem and it has several known solutions [2,7]. If a data structure for a string of length n with query time q(n) and space usage s(n) can be built in time p(n), then this implies a solution for the LCS problem using O(q(n)n 2 + p(n)) time and O(s(n)) space.…”

“…If a data structure for a string of length n with query time q(n) and space usage s(n) can be built in time p(n), then this implies a solution for the LCS problem using O(q(n)n 2 + p(n)) time and O(s(n)) space. For example using the deterministic data structure of Bille et al [2], the LCS problem can be solved in O(n 2(1+ε) ) time and O(n 1−ε ) space for any 0 ≤ ε ≤ 1/2.…”

Sparse Suffix Tree Construction in Optimal Time and Space

“…Another natural problem in the model of read-only random access to the text is LCE queries, where we are to preprocess a text subject to queries LCE(i, j) returning the longest common prefix of two suffixes T [i..] and T [j..]. Several trade-off between query time, data structure size, construction time and space usage have been obtained [5,6,24]. The queries are typically deterministic, but construction algorithms range from Monte Carlo randomization via Las Vegas randomization to deterministic solutions.…”