Efficient enumeration of non-equivalent squares in partial words with few holes

A word of the form WW for some word W∈ Σ ^∗ is called a square. A partial word is a word possibly containing holes (also called don’t cares). The hole is a special symbol ◊∉ Σ which matches any symbol from Σ∪ { ◊}. A p-square is a partial word matching at least one square WW without holes. Two p-squares are called equivalent if they match the same set of squares. A p-square is called here unambiguous if it matches exactly one square WW without holes. Such p-squares are natural counterparts of classical squares. Let PSQUARES _k (n) and USQUARES _k (n) be the maximum number of non-equivalent p-squares and non-equivalent unambiguous p-squares in T over all partial words T of length n with at most k holes. We show asymptotically tight bounds: PSQUARESk(n)=Θ(min(nk2,n2)),USQUARESk(n)=Θ(nk).We present an algorithm that reports all non-equivalent p-squares in O(nk ³ ) time for a partial word of length n with k holes, for an integer alphabet. In particular, it runs in linear time for k= O(1) and its time complexity near-matches the asymptotic bound for PSQUARES _k (n). We also show an O(n) -time algorithm that reports all non-equivalent p-squares of a given length. The paper is a full and improved version of Charalampopoulos et al. (in Cao Y, Chen Y (eds) Proceedings of the 23rd international conference on computing and combinatorics, COCOON 2017; Springer, 2017).