New algorithms for binary jumbled pattern matching

Given a pattern P and a text T , both strings over a binary alphabet, the binary jumbled string matching problem consists in telling whether any permutation of P occurs in T . The indexed version of this problem, i.e., preprocessing a string to efficiently answer such permutation queries, is hard and has been studied in the last few years. Currently the best bounds for this problem are O(n/ log n) (with O(n) space and O(1) query time) [10] and O(r log r) (with O(|L|) space and O(log |L|) query time) [3], where r is the length of the run-length encoding of T and |L| = O(n) is the size of the index. In this paper we present new results for this problem. Our first result is an alternative construction of the index by Badkobeh et al. [3] that obtains a trade-off between the space and the time complexity. It has O(r log k + n/k) complexity to build the index, O(log k) query time, and uses O(n/k+ |L|) space, where k is a parameter. The second result is an O(n log w/w) algorithm (with O(n) space and O(1) query time), based on word-level parallelism where w is the word size in bits.