) Alberto Apostolico Purdue University and Universit`a di Padova Dany Breslauer yyz Columbia University Zvi Galil z Columbia University and Tel-Aviv University Summary of results Optimal concurrent-read concurrent-write parallel algorithms for two problems are presented: ffl Finding all the periods of a string. The period of a string can be computed by previous efficient parallel algorithms only if it is shorter than half of the length of the string. Our new algorithm computes all the periods in optimal O(log log n) time, even if they are longer. The algorithm can be used to compute all initial palindromes of a string within the same bounds. ffl Testing if a string is square-free. We present an optimal O(log log n) time algorithm for testing if a string is square-free, improving the previous bound of O(log n) given by Apostolico [1] and Crochemore and Rytter [12]. We show matching lower bounds for the optimal parallel algorithms that solve the problems above on a general alphab...

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.

A string w covers another string z if every symbol of z is within some occurrence of w in z. A string is called superprimitive if it is covered only by itself, and quasiperiodic if it is covered by some shorter string. This paper presents an O(log log n) time n log n log log n -processor CRCW-PRAM algorithm that tests if a string is superprimitive. The algorithm is the fastest possible with this number of processors over a general alphabet. 1 Introduction Quasiperiodicity, as defined by Apostolico and Ehrenfeucht [3], is an avoidable regularity of strings that is strongly related to other regularities such as periods and squares [12]. Apostolico, Farach and Iliopoulos [4] and Breslauer [7] gave linear-time sequential algorithms that tests if a string is superprimitive. Apostolico and Ehrenfeucht [3] presented an algorithm that finds all maximal quasiperiodic substrings of a string. This paper presents a parallel algorithm that tests if a string of length n is superprimitive i...

Problems involving strings arise in many areas of computer science and have numerous practical applications. We consider several problems from a theoretical perspective and provide efficient algorithms and lower bounds for these problems in sequential and parallel models of computation. In the sequential setting, we present new algorithms for the string matching problem improving the previous bounds on the number of comparisons performed by such algorithms. In parallel computation, we present tight algorithms and lower bounds for the string matching problem, for finding the periods of a string, for detecting squares and for finding initial palindromes.

An O(log log m) time n log m log log m -processor CRCW-PRAM algorithm for the string prefix-matching problem over a general alphabet is presented. The algorithm can also be used to compute the KMP failure function in O(log log m) time on m log m log log m processors. These results improve on the running time of the best previous algorithm for both problems, which was O(log m), while preserving the same number of operations. 1 Introduction String matching is the problem of finding all occurrences of a short pattern string P[1::m] in a longer text string T [1::n]. The classical sequential algorithm of Knuth, Morris and Pratt [12] solves the string matching problem in time that is linear in the length of the input strings. The Knuth-Morris-Pratt [12] string matching algorithm can be easily generalized to find the longest pattern prefix that starts at each text position within the same time bound. We refer to this problem as string prefix-matching. In parallel, the string matching p...

This paper presents two efficient concurrent-read concurrent-write parallel algorithms that find all palindromes in a given string: 1. An O(log n) time, n-processor algorithm over general alphabets. In case of constant size alphabets the algorithm requires only n= log n processors, and thus achieves an optimal-speedup. 2. An O(log log n) time, n log n= log log n-processor algorithm over general alphabets. This is the fastest possible time with the number of processors used. These new results improve on the known parallel palindrome detection algorithms by using smaller auxiliary space and either by making fewer operations or by achieving a faster running time. 1 Introduction Palindromes are symmetric strings that read the same forward and backward. Palindromes have been studied for centuries as word puzzles and more recently have found several important uses in formal languages and computability theory. Formally, a non-empty string w is a palindrome if w = w R , where w R denotes...

We provide general transformations of lower bounds in Valiant's parallel-comparison-decision-tree model to lower bounds in the priority concurrent-read concurrent-write parallel-random-access-machine model. The proofs rely on standard Ramsey-theoretic arguments that simplify the structure of the computation by restricting the input domain. The transformation of comparison model lower bounds, which are usually easier to obtain, to the parallel-random-access-machine, unifies some known lower bounds and gives new lower bounds for several problems.

String Pattern Matching concerns itself with algorithmic and combinatorial issues related to matching and searching on linearly arranged sequences of symbols, arguably the simplest possible discrete structures. As unprecedented volumes of sequence data are amassed, disseminated and shared at an increasing pace, effective access to, and manipulation of such data depend crucially on the efficiency with which strings are structured, compressed, transmitted, stored, searched and retrieved. This paper samples from this perspective, and with the authors' own bias, a rich arsenal of ideas and techniques developed in more than three decades of history.

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...

Abstract. We study the problem of finding, in a given word, all maximal gapped palindromes verifying two types of constraints, that we call longarmed and length-constrained palindromes. For both classes, we propose algorithms that run in time O(n + S), where S is the number of output palindromes. Both algorithms can be extended to compute biological gapped palindromes within the same time bound. 1