Algorithms for Jumbled Pattern Matching in Strings

Abstract

The Parikh vector p(s) of a string s over a finite ordered alphabet Σ = {a 1 , . . . , aσ} is defined as the vector of multiplicities of the characters, p(s) = (p 1 , . . . , pσ), where p i = |{j | s j = a i }|. Parikh vector q occurs in s if s has a substring t with p(t) = q. The problem of searching for a query q in a text s of length n can be solved simply and worst-case optimally with a sliding window approach in O(n) time. We present two novel algorithms for the case where the text is fixed and many queries arrive over time.The first algorithm only decides whether a given Parikh vector appears in a binary text. It uses a linear size data structure and decides each query in O(1) time. The preprocessing can be done trivially in Θ(n 2 ) time.The second algorithm finds all occurrences of a given Parikh vector in a text over an arbitrary alphabet of size σ ≥ 2 and has sub-linear expected time complexity. More precisely, we present two variants of the algorithm, both using an O(n) size data structure, each of which can be constructed in O(n) time. The first solution is very simple and easy to implement and leads to an expected query time of O(n( σ log σ ) 1/2 log m √ m ), where m = i q i is the length of a string with Parikh vector q. The second uses wavelet trees and improves the expected runtime to O(n( σ log σ ) 1/2 1 √ m ), i.e., by a factor of log m. 357 Int. J. Found. Comput. Sci. 2012.23:357-374. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 02/20/15. For personal use only. 358 P. Burcsi et al.Notice that this is an overestimate, since line 7 is only executed if no occurrence was found after the current update of R (line 4). Standard algebraic manipulations using Jensen's inequality (see, e.g. [16]) yield J i=1 log(R i −R i−1 + m) ≤ J log n J + m . Therefore we obtain

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

“…The index provides only a yes/no answer for a query pattern; additional O(log n) time can be used to restore a witness occurrence [12]. The construction time was improved independently by Burcsi et al [6] (see also [7,8]) and Moosa and Rahman [20] to O n 2 log n , and then by Moosa and Rahman [21] to O n 2 (log n) 2 . All these results work in the word-RAM model.…”

Fast algorithms for Abelian periods in words and greatest common divisor queries

“…An example of such a word over five-letter alphabet was given by Pleasants [18] and afterwards the optimal example over four-letter alphabet was shown by Keränen [16]. Quite recently there have been several results on Abelian complexity in words [2,8,9, 10] and partial words [3,4] and on Abelian pattern matching [5,17]. Abelian periods were first defined and studied by Constantinescu and Ilie [6].…”

On the Lower Bound for the Maximum Consecutive Sub-Sums Problem

“…end if 11: end for 12: end for 13: return max It's not difficult to come up with simple Θ(n 2 ) algorithms, the Naive-MCSP method is an example of such an algorithm, and despite the interest generated by this problem (16,17,18,23,24,55,56) does not exists a better than O(n 2 / log 2 n) algorithm in the literature. Then, a natural question to ask is whether there exists a non-trivial lower bound for the MCSP.…”

“…For the Parikh vector pattern matching the best constructions for the binary sequences are as follows: Bursci et al (16,17,18) showed an O(n 2 / log n)-time algorithms based on the O(n 2 / log n) algorithm of Bremner et al (14,21) for computing a (min, +)-convolution, independently Moosa and Rahman (55) obtained the same result by a different use of (min, +)-convolution, then Mossa and Rahman (56) further improved it to O(n 2 / log 2 n) in the RAM model.…”

“…Given a sequence of n non-negative numbers, The MCSP asks for the computation of the maximum consecutive sums of lengths 1 through n. This problem has recently generated much interest (16,17,18,23,24,55,56). In particular, for a binary string s of size n, knowing for each = 1, 2, .…”