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An optimal O(log log n) time concurrent-read concurrent-write parallel algorithm for detecting all squares in a string is presented. A tight lower bound shows that over general alphabets this is the fastest possible optimal algorithm. When p processors are available the bounds become \Theta(d n log n p e + log log d1+p=ne 2p). The algorithm uses an optimal parallel string-matching algorithm together with periodicity properties to locate the squares within the input string.

Crochemore and Perrin discovered an elegant linear-time constant-space string matching algorithm that makes at most 2n \Gamma m symbol comparison. This paper shows how to modify their algorithm to use fewer comparisons. Given any fixed ffl ? 0, the modified algorithm takes linear time, uses constant space and makes at most n+ b 1+ffl 2 (n \Gamma m)c comparisons. If O(log m) space is available, then the algorithm makes at most n + b 1 2 (n \Gamma m)c comparisons. The pattern preprocessing step also takes linear time and uses constant space. These are the first string matching algorithms that make fewer than 2n \Gamma m comparisons and use sub-linear space.

A new approach to time-space efficient string-matching is presented. The method is flexible, its implementation depends whether or not the alphabet is linearly ordered. The only known linear-time constant-space algorithm for string-matching over nonordered alphabets is the GalilSeiferas algorithm, see [8, 6] which is rather complicated. The zooming method gives probably the simplest string-matching algorithm working in constant space and linear time for nonordered alphabets. The novel feature of our algorithm is the application of the searching phase (which is usually simpler than preprocessing) in the preprocessing phase. The preprocessing has a recursive structure similar to selection in linear time, see [1]. For ordered alphabets the preprocessing part is much simpler, its basic component is a simple and well-known algorithm for finding the maximal suffix, see [7]. Hence we demonstrate a new application of this algorithm, see also [5]. The idea of the zooming method was applied in [...

A new string-matching algorithm working in constant space and linear time is presented. It is based on a powerful idea of sampling, originally introduced in parallel computations. The algorithm uses a sample S which consists of two positions inside the pattern P . First the positions of the sample S are tested against the corresponding positions of the text T , then a version of Knuth-Morris-Pratt algorithm is applied. This gives the simplest known string-matching algorithm which works in constant space and linear time and which does not use any linear order of the alphabet. A rened version of the algorithm gives the fastest (in the sense of number of comparisons) known algorithm for string-matching in constant space. It makes (1 + ")n +O( n m ) symbol comparisons. This improves substantially the result of [3], where a ( 3 2 + ")n comparisons constant space algorithm was designed. 1 Introduction Assume we are given two strings: a pattern P of length m and a text T of length n. The ...

In this paper we study the exact comparison complexity of the string prefix-matching problem in the deterministic sequential comparison model with equality tests. We derive almost tight lower and upper bounds on the number of symbol comparisons required in the worst case by on-line prefix-matching algorithms for any fixed pattern and variable text. Unlike previous results on the comparison complexity of string-matching and prefix-matching algorithms, our bounds are almost tight for any particular pattern. We also consider the special case where the pattern and the text are the same string. This problem, which we call the string self-prefix problem, is similar to the pattern preprocessing step of the Knuth-Morris-Pratt stringmatching algorithm that is used in several comparison efficient stringmatching and prefix-matching algorithms, including in our new algorithm. We obtain roughly tight lower and upper bounds on the number of symbol comparisons required in the worst case...

) Maxime Crochemore Universit`e de Marne-la-Vall`ee Leszek Gasieniec y Max-Planck Institut fur Informatik Wojciech Rytter z Warsaw University and University of Liverpool Abstract Given two strings: pattern P of length m and text T of length n. The stringmatching problem is to find all occurrences of the pattern P in the text T . We present a simple string-matching algorithms which works in average o(n) time with constant additional space for one-dimensional texts and two-dimensional arrays. This is the first attempt to the small-space string-matching problem in which sublinear time algorithms are delivered. More precisely we show that all occurrences of one- or two-dimensional patterns can be found in O( n r ) average time with constant memory, where r is the repetition size (size of the longest repeated subword) of P . Institut Gaspard Monge, Universit`e de Marne-la-Vall`ee, France (mac@univ-mlv.fr). y Max-Planck Institut fur Informatik, Im Stadtwald, D--6612...

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