An Efficient Algorithm to Test Square-Freeness of Strings Compressed by Balanced Straight Line Programs

In this paper we study the problem of deciding whether a given compressed string contains a square. A string x is called a square if x = zz and z = uk implies k = 1 and u = z. A string w is said to be square-free if no substrings of w are squares. However, very little is known for testing square-freeness of a given compressed string. In this paper, we give an O(max(n2,n log2 N))-time O(n2)-space solution to test squarefreeness of a given compressed string, where n and N are the size of a given compressed string and the corresponding decompressed string, respectively. Our input strings are compressed by balanced straight line program (BSLP). We remark that BSLP has exponential compression, that is, N = O(2n). Hence no decompress-then-test approaches can be better than our method in the worst case.