Two-way string-matching algorithm

String matching algorithms

In computer science, the two-way string-matching algorithm is an string-searching algorithm, discovered by Maxime Crochemore and Dominique Perrin in 1991. It takes a pattern of size m, called a “needle”, preprocesses it in linear time O(m), producing information that can then be used to search for the needle in any “haystack” string, taking only linear time O(n) with n the haystack's length. The two-way algorithm can be viewed as a combination of the forward-going Knuth–Morris–Pratt algorithm (KMP) and the backward-running Boyer–Moore string-search algorithm (BM).Like those two, the 2-way algorithm preprocesses the pattern to find partially repeating periods and computes “shifts” based on them, indicating what offset to “jump” to in the haystack when a given character is encountered. Unlike BM and KMP, it uses only O(log m) additional space to store information about those partial repeats: the search pattern is split into two halves (its critical factorization), represented only by the position of that split. Being a number lesser than m, it can be represented in ⌈log₂ m⌉ bits. In most practical settings, this can be taken to be O(1), as the needle's size is limited by the size of addressable memory.The actual matching operation performs at most 2n − m comparisons. Breslauer later published two improved variants performing fewer comparisons, at the cost of storing additional data about the preprocessed needle: * The first one performs at most n + ⌊(n − m)/2⌋ comparisons, ⌈(n − m)/2⌉ fewer than the original. It must however store ⌈log m⌉ additional offsets in the needle, using O(log2 m) space. * The second adapts it to only store a constant number of such offsets, denoted c, but must perform n + ⌊(1⁄2 + ε) * (n − m)⌋ comparisons, with ε = 1⁄2(Fc+2 − 1)−1 = O(−c) going to zero exponentially quickly as c increases. The algorithm is considered fairly efficient in practice, being cache-friendly and using several operations that can be implemented in well-optimized subroutines. It is used by the C standard libraries glibc, newlib, and musl, to implement the memmem and strstr family of substring functions. As with most advanced string-search algorithms, the naïve implementation may be more efficient on small-enough instances; this is especially so if the needle isn't searched in multiple haystacks, which would amortize the preprocessing cost. (Wikipedia).

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Two-way string-matching algorithm

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