We consider two closely related problems of text indexing in a sub-linear working space. The first problem is the Sparse Suffix Tree (SST) construction, where a text S is given in a read-only memory, along with a set of suffixes B, and the goal is to construct the compressed trie of all these suffixes ordered lexicographically, using only O(|B|) words of space. The second problem is the Longest Common Extension (LCE) problem, where again a text S of length n is given in a read-only memory with some parameter 1 ≤ τ ≤ n, and the goal is to construct a data structure that uses O( n τ ) words of space and can compute for any pair of suffixes their longest common prefix length. We show how to use ideas based on the Locally Consistent Parsing technique, that was introduced by Sahinalp and Vishkin [35], in some non-trivial ways in order to improve the known results for the above problems. We introduce new Las-Vegas and deterministic algorithms for both problems.For the randomized algorithms, we introduce the first Las-Vegas SST construction algorithm that takes O(n) time. This is an improvement over the last result of Gawrychowski and Kociumaka [19] who obtained O(n) time for Monte-Carlo algorithm, and O(n log |B|) time for Las-Vegas algorithm. In addition, we introduce a randomized Las-Vegas construction for a data structure that uses O( n τ ) words of space, can be constructed in linear time and answers LCE queries in O(τ ) time.For the deterministic algorithms, we introduce an SST construction algorithm that takes O(n(log n |B| + log * n)) time (for |B| = Ω(log n log * n)). This is the first almost linear time, O(n · polylog n), deterministic SST construction algorithm, where all previous algorithms take at least Ω min{n|B|, n 2 |B| } time. For the LCE problem, we introduce a data structure that uses O( n τ ) words of space and answers LCE queries in O(τ log * n) time, with O(n(log τ + log * n)) construction time (for τ = O( n log n log * n )). This data structure improves both query time and construction time upon the results of Tanimura et al. [37].

No abstract

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T [i . . i + m − 1] for every shift i, under the standard Word-RAM model with Θ(log n)-bit words. • We provide an O(n √ m) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n √ m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(n √ m(log m log log m) 1/4 ) running time. Our randomized algorithm extends to the k-bounded setting, with running time O n + nk √ m , removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uznański, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020].

In the pattern matching with d wildcards problem one is given a text T of length n and a pattern P of length m that contains d wildcard characters, each denoted by a special symbol ? . A wildcard character matches any other character. The goal is to establish for each m-length substring of T whether it matches P . In the streaming model variant of the pattern matching with d wildcards problem the text T arrives one character at a time and the goal is to report, before the next character arrives, if the last m characters match P while using only o(m) words of space.In this paper we introduce two new algorithms for the d wildcard pattern matching problem in the streaming model. The first is a randomized Monte Carlo algorithm that is parameterized by a constant 0 ≤ δ ≤ 1. This algorithm usesÕ (d 1−δ ) amortized time per character andÕ(d 1+δ ) words of space. The second algorithm, which is used as a black box in the first algorithm, is a randomized Monte Carlo algorithm which uses O(d + log m) worst-case time per character and O(d log m) words of space.. * Part of this work took place while the second author was at University of Michigan. 1 We assume the RAM model where each word has size of O(log n) bits.

No abstract