Approximate periodicity

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(nklog⁡log⁡n) and under the swap distance in time O(n2). Finally, we show an O(nlog⁡n) algorithm for finite alphabets, and an O(nlog3⁡n) algorithm for infinite alphabets, that approximate the minimum number of mismatches between the input string and a closest periodic string under the Hamming distance.