Covering Problems for Partial Words and for Indeterminate Strings

We consider the problem of computing a shortest solid cover of an indeterminate string. An indeterminate string may contain non-solid symbols, each of which specifies a subset of the alphabet that could be present at the corresponding position. We also consider covering partial words, which are a special case of indeterminate strings where each non-solid symbol is a dont care symbol. We prove that indeterminate string covering problem and partial word covering problem are NP-complete for binary alphabet and show that both problems are fixed-parameter tractable with respect to $k$, the number of non-solid symbols. For the indeterminate string covering problem we obtain a $2^{O(k log k)} + n k^{O(1)}$-time algorithm. For the partial word covering problem we obtain a $2^{O(sqrt{k}log k)} + nk^{O(1)}$-time algorithm. We prove that, unless the Exponential Time Hypothesis is false, no $2^{o(sqrt{k})} n^{O(1)}$-time solution exists for either problem, which shows that our algorithm for this case is close to optimal. We also present an algorithm for both problems which is feasible in practice.

A Note on the Longest Common Compatible Prefix Problem for Partial Words

For a partial word $w$ the longest common compatible prefix of two positions $i,j$, denoted $lccp(i,j)$, is the largest $k$ such that $w[i,i+k-1]uparrow w[j,j+k-1]$, where $uparrow$ is the compatibility relation of partial words (it is not an equivalence relation). The LCCP problem is to preprocess a partial word in such a way that any query $lccp(i,j)$ about this word can be answered in $O(1)$ time. It is a natural generalization of the longest common prefix (LCP) problem for regular words, for which an $O(n)$ preprocessing time and $O(1)$ query time solution exists. Recently an efficient algorithm for this problem has been given by F. Blanchet-Sadri and J. Lazarow (LATA 2013). The preprocessing time was $O(nh+n)$, where $h$ is the number of holes in $w$. The algorithm was designed for partial words over a constant alphabet and was quite involved. We present a simple solution to this problem with slightly better runtime that works for any linearly-sortable alphabet. Our preprocessing is in time $O(nmu+n)$, where $mu$ is the number of blocks of holes in $w$. Our algorithm uses ideas from alignment algorithms and dynamic programming.