Abstract
Finding an approximate period in a given string S of length n is defined as follows. Let S′ be a periodic string closest to S under some distance metric, find the smallest period of S′. This period is called an approximate period of S under the given metric. Let the distance between the input string S and a closest periodic string under the Hamming distance S′ be k. We develop algorithms that construct an approximate period of S under the Hamming distance in time O(nkloglogn) and under the swap distance in time O(n2). Finally, we show an O(nlogn) algorithm for finite alphabets, and an O(nlog3n) algorithm for infinite alphabets, that approximate the minimum number of mismatches between the input string and a closest periodic string under the Hamming distance.
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