Faster pattern matching with character classes using prime number encoding

Abstract

In pattern matching with character classes the goal is to find all occurrences of a pattern of length m in a text of length n, where each pattern position consists of an allowed set of characters from a finite alphabet Σ. We present an FFT-based algorithm that uses a novel prime-numbers encoding scheme, which is log n/ log m times faster than the fastest extant approaches, which are based on boolean convolutions. In particular, if m |Σ| = n O (1) , our algorithm runs in time O (n log m), matching the complexity of the fastest techniques for wildcard matching, a special case of our problem. A major advantage of our algorithm is that it allows a tradeoff between the running time and the RAM word size. Our algorithm also speeds up solutions to approximate matching with character classes problems-namely, matching with k mismatches and Hamming distance, as well as to the subset matching problem.

Pattern matching with wildcards using words of shorter length

“…The convolution, or cross-correlation, of two O (n log m). According to [8], we assume that σ 2m, since each 2m-long piece of the text contains at most 2m symbols.…”

“…For c k , we have c k ≡ 1 (mod ρ k ) and c k ≡ 0 (mod ρ k ), k = k. By the CRT, code(S) = ρ k ∈S c k . In [8], this method is used to encode the character classes where each character is associated with a distinct prime number.…”

“…Very recently, Linhart and Shamir [8] introduced the prime number encoding. This approach solved a series of matching problems, including: matching with character classes, approximate matching with character classes, and the subset matching problem.…”

“…Based on the prime number encoding [8], we present an integer encoding scheme using less than 2 lg σ prime numbers, and each code word uses at most 2 lg σ lg(5m) bits. Based on the encoding, the problem of matching with wildcards can be solved by one convolution in O (n log m) time.…”

Fast Convolutions and Their Applications in Approximate String Matching

“…Bit-parallelism can be used to simulate a nondeterministic automaton; the best result along these lines [10] yields O(nm/w) time. A prime-number encoding scheme was proposed in [11] for k-mismatches with character classes, to achieve O(nσ) time if, for example, both log 2 m = O(log n) and log m = O(σ) hold. Table 1.…”