A substring w[i.j] in w is called a repetition of period p if w[k] = w[k + p] for any i ≤ k ≤ j - p. Especially, a maximal repetition, which cannot be extended neither to left nor to right, is called a run. The ratio of the length of the run to its period, i.e. [Formula: see text], is called an exponent. The sum of exponents of runs in a string is of interest. The maximal value of the sum is still unknown, and the current upper bound is 2.9n given by Crochemore and Ilie, where n is the length of a string. In this paper we show a closed formula which exactly expresses the average value of it for any n and any alphabet size, and the limit of this value per unit length as n approaches infinity. For binary strings, the limit value is approximately 1.13103. We also show the average number of squares in a string of length n and its limit value.
We address the problems of detecting and counting various forms of regularities in a string represented as a straight-line program (SLP) which is essentially a context free grammar in the Chomsky normal form. Given an SLP of size n that represents a string s of length N,
our algorithm computes all runs and squares in s inh is the height of the derivation tree of the SLP. We also show an algorithm to compute all gapped-palindromes in O (n 3 h + gnh log N) time and O (n 2 ) space, where g is the length of the gap. As one of the main components of the above solution, we propose a new technique called approximate doubling which seems to be a useful tool for a wide range of algorithms on SLPs. Indeed, we show that the technique can be used to compute the periods and covers of the string in O (n 2 h) time and O (nh(n + log 2 N)) time, respectively.
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