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

A word of the form WW for some word $W\in Σ^*$ 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: $PSQUARES_k(n) = \Theta(\min(nk^2, n^2))$ and $USQUARES_k(n) = \Theta(nk)$. 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).