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.

The so called "four Russians technique" is often used to speed up algorithms by encoding several data items in a single memory cell. Given a sequence of n symbols over a constant size alphabet, one can encode the sequence into O(n=) memory cells in O(log ) time using n= log processors. This paper presents an efficient CRCW-PRAM string-matching algorithm for coded texts that takes O(log log(m=)) time 1 making only O(n=) operations, an improvement by a factor of = O(logn) on the number of operations used in previous algorithms. Using this stringmatching algorithm one can test if a string is square-free and find all palindromes in a string in O(log log n) time using n= log log n processors. 1 Introduction In the string-matching problem one is searching for occurrences of a pattern string P[1::m] in a text string T [1::n]. There exist several O(n + m) time sequential string-matching algorithms that are used in a large variety of applications. Galil [23] published the first efficient...

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