k-Abelian Pattern Matching

Abstract

Two words are called k-abelian equivalent, if they share the same multiplicities for all factors of length at most k. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are k-abelian equivalent to some pattern P . Moreover, an optimal algorithm for finding the largest k for which two words are k-abelian equivalent is given. Solutions for various online versions of the k-abelian pattern matching problem are also proposed.? T.

Circular Sequence Comparison with q-grams

“…i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 17 1 8 13 19 4 15 11 20 0 7 18 2 9 5 16 12 3 14 We observe that when identifying the q-gram distance between two blocks, we can apply the idea in [13], with the only difference that we should also maintain a Parikh vector that stores the differences between the number of occurrences of q-grams in the current block of xx and y (in fact the new letters given by the ranks). Moreover, at the time of the construction of y , we also construct a Parikh vector P(y ), storing, for each letter of y , the number of its occurrences in y .…”

“…Thus, Step 3 cannot guarantee that i best , the local minimum obtained by shifting the window m/β positions to the right and left of j best , is minimal for all 0 ≤ i < m. In this section, we give a fast and exact algorithm, denoted by saCSC, to find i such that δ i = D β,q (x i , y) is minimal, based on the suffix array (see Section 2). We partially follow the idea from [13]. This work investigates the string matching problem in the setting of k-abelian equivalences: two strings are considered k-abelian equivalent for some positive integer k, if they have the same length and share the same factors of length at most k, including multiplicities.…”

“…In [13], the authors propose a linear-time algorithm to solve the string matching problem when looking at q-abelian equivalent strings: given a string x of length m, a string y of length n ≥ m, and a positive integer q < m, all factors of y that are q-abelian equivalent to x can be found in time and space O(m + n). The idea of the algorithm in [13] consists in constructing the suffix array of the string xy, and ranking sets of identical q-length prefixes of suffixes in the suffix array in the order of their appearance. Then it constructs new strings based on this ranking, and solves the problem as in the jumbled matching case [6], i.e.…”

k-Abelian pattern matching

“…✩ This work represents an extended version of a paper presented at the 18th International Conference on Developments in Language Theory, DLT 2014[5].…”

SAT and CP - Parallelisation and Applications

“…In k-abelian pattern matching, two words are considered equivalent if the subsequences or factors of length k occur in both words in the same multiplicity. Some algorithms for this problem, together with experimental results were presented in [91] and [92]. My contributition here was mostly the implementation of the developed algorithms and the experimental section.…”