# On Approximate Jumbled Pattern Matching in Strings

Burcsi P, Cicalese F, Fici G, Lipták Z (2012)
Theory of Computing Systems 50(1): 35-51.

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Journal Article | Published | English

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Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of a Parikh vector q in the text s requires finding a substring t of s with p(t) = q. This can be viewed as the task of finding a jumbled (permuted) version of a query pattern, hence the term Jumbled Pattern Matching. We present several algorithms for the approximate version of the problem: Given a string s and two Parikh vectors u, v (the query bounds), find all maximal occurrences in s of some Parikh vector q such that u <= q <= v. This definition encompasses several natural versions of approximate Parikh vector search. We present an algorithm solving this problem in sub-linear expected time using a wavelet tree of s, which can be computed in time O(n) in a preprocessing phase. We then discuss a Scrabble-like variation of the problem, in which a weight function on the letters of s is given and one has to find all occurrences in s of a substring t with maximum weight having Parikh vector p(t) <= v. For the case of a binary alphabet, we present an algorithm which solves the decision version of the Approximate Jumbled Pattern Matching problem in constant time, by indexing the string in subquadratic time.
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Burcsi P, Cicalese F, Fici G, Lipták Z. On Approximate Jumbled Pattern Matching in Strings. Theory of Computing Systems. 2012;50(1):35-51.
Burcsi, P., Cicalese, F., Fici, G., & Lipták, Z. (2012). On Approximate Jumbled Pattern Matching in Strings. Theory of Computing Systems, 50(1), 35-51.
Burcsi, P., Cicalese, F., Fici, G., and Lipták, Z. (2012). On Approximate Jumbled Pattern Matching in Strings. Theory of Computing Systems 50, 35-51.
Burcsi, P., et al., 2012. On Approximate Jumbled Pattern Matching in Strings. Theory of Computing Systems, 50(1), p 35-51.
P. Burcsi, et al., “On Approximate Jumbled Pattern Matching in Strings”, Theory of Computing Systems, vol. 50, 2012, pp. 35-51.
Burcsi, P., Cicalese, F., Fici, G., Lipták, Z.: On Approximate Jumbled Pattern Matching in Strings. Theory of Computing Systems. 50, 35-51 (2012).
Burcsi, Peter, Cicalese, Ferdinando, Fici, Gabriele, and Lipták, Zsuzsanna. “On Approximate Jumbled Pattern Matching in Strings”. Theory of Computing Systems 50.1 (2012): 35-51.
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