Efficient Enumeration of Non-Equivalent Squares in Partial Words with Few Holes

Abstract

A word of the form WW for some word $W\in \varSigma ^*$ is called a square, where $\varSigma$ is an alphabet. A partial word is a word possibly containing holes (also called don’t cares). The hole is a special symbol which matches (agrees with) any symbol from . 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. We denote by $psquares (T)$ the number of non-equivalent p-squares which are factors of a partial word T. Let $\mathrm {PSQUARES}_k(n)$ be the maximum value of $psquares (T)$ over all partial words of length n with at most k holes. We show asymptotically tight bounds:

$$ c_1\cdot \min (nk^2,\, n^2) \le \mathrm {PSQUARES}_k(n) \le c_2\cdot \min (nk^2,\, n^2) $$

for some constants $c_1,c_2>0$. We also present an algorithm that computes $psquares (T)$ in $\mathcal {O}(nk^3)$ time for a partial word T of length n with k holes. In particular, our algorithm runs in linear time for $k=\mathcal {O}(1)$ and its time complexity near-matches the maximum number of non-equivalent p-square factors in a partial word.

P. Charalampopoulos—Supported by the Graduate Teaching Scholarship scheme of the Department of Informatics at King’s College London.

T. Kociumaka—Supported by Polish budget funds for science in 2013–2017 as a research project under the ’Diamond Grant’ program.

J. Radoszewski, W. Rytter and T. Waleń—Supported by the Polish National Science Center, grant no. 2014/13/B/ST6/00770.