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

A word of the form WW for some word (Win 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 Open image in new window which matches (agrees with) any symbol from Open image in new window . 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: $$ c_1cdot min (nk^2,, n^2) le mathrm {PSQUARES}_k(n) le c_2cdot min (nk^2,, n^2) $$for some constants (c_1,c_2u003e0). 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.