Here we introduce two new notions of approximate matching with application in computer assisted music analysis. We present algorithms for each notion of approximation: for approximate string matching and for computing approximate squares.

In this paper we give an overview of four algorithms that we have developed for pattern matching, pattern discovery and data compression in multidimensional datasets. We show that these algorithms can fruitfully be used for processing musical data. In particular, we show that our algorithms can discover instances of perceptually signifrant musica 1 repetition that cannot be found using previous approaches. We also describe results that suggest the possibility of using our datacompression algorithm for modelling expert motivic-thematic music analysis.

A repetition in a word is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in, that is those for which any extended subword of has a bigger period. The set of such repetitions represents in a compact way all repetitions in.We first prove a combinatorial result asserting that the sum of exponents of all maximal repetitions of a word of length is bounded by a linear function in. This implies, in particular, that there is only a linear number of maximal repetitions in a word. This allows us to construct a linear-time algorithm for finding all maximal repetitions. Some consequences and applications of these results are discussed, as well as related works. 1.

A tandem repeat (or square) is a string ffff, where ff is a nonempty string. We present an O(jSj)-time algorithm that operates on the suffix tree T (S) for a string S, finding and marking the endpoint in T (S) of every tandem repeat that occurs in S. This decorated suffix tree implicitly represents all occurrences of tandem repeats in S, and can be used to efficiently solve many questions concerning tandem repeats and tandem arrays in S. This improves and generalizes several prior efforts to efficiently capture large subsets of tandem repeats.

A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first study maximal repetitions in Fibonacci words -- we count their exact number, and estimate the sum of their exponents. These quantities turn out to be linearly-bounded in the length of the word. We then prove that the maximal number of maximal repetitions in general words (on arbitrary alphabet) of length n is linearly-bounded in n, and we mention some applications and consequences of this result.

The problem of computing tandem repetitions with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK log K + S) algorithm is proposed (S the size of the output). This improves, in particular, the bound obtained in [LS93]. Finally, other possible definions are briefly analyzed.

A pair in a string is the occurrence of the same substring twice. A pair is maximal if the two occurrences of the substring cannot be extended to the left and right without making them different. The gap of a pair is the number of characters between the two occurrences of the substring. In this paper we present methods for finding all maximal pairs under various constraints on the gap. In a string of length n we can find all maximal pairs with gap in an upper and lower bounded interval in time O(n log n + z) where z is the number of reported pairs. If the upper bound is removed the time reduces to O(n+z). Since a tandem repeat is a pair where the gap is zero, our methods can be seen as a generalization of finding tandem repeats. The running time of our methods equals the running time of well known methods for finding tandem repeats.

. All our words (strings) are over a fixed alphabet. A square is a subword of the form uu = u 2 , where u is a nonempty word. Two squares are distinct if they are of different shape, not just translates of each other. A word u is primitive if u cannot be written in the form u = v j for some j 2. A square u 2 with u primitive is primitive rooted. Let M (n) denote the maximum number of distinct squares, P (n) the number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduce M (n) ! 2n for all n ? 0, and P (n) = n \Gamma o(n) for infinitely many n. 1. Introduction We consider words (strings, sequences) over a fixed alphabet throughout. A square is a pair of identical adjacent subwords, such as 1011010110 = 10110 2 over the binary alphabet. Two squares are distinct if they are of different shape, not just translates of each other. Denote by M (n...

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 (finite) Fibonacci string F n is defined as follows: F 0 = b, F 1 = a; for every integer n 2, F n = F n\Gamma1 F n\Gamma2 . For n 1, the length of F n is denoted by f n = jF n j. The infinite Fibonacci string F is the string which contains every F n , n 1, as a prefix. Apart from their general theoretical importance, Fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the "Abelian squares" in a given string. In this paper we provide a characterization of all the squares in F , hence in every prefix F n ; this characterization naturally gives rise to a \Theta(f n ) algorithm which specifies all the squares of F n in an appropriate encoding. This encoding is made possible by the fact that the squares of F n occur consecutively, in "runs", the number of which is \Theta(f n ). By contrast, the known general algorithms for the computation of the repetitions in an arbitrary string require \Theta(f n log f n ) time (and pro...