Efficient Enumeration of Distinct Factors Using Package Representations

We investigate properties and applications of a new compact representation of string factors: families of packages. In a string T , each package (i, `, k) represents the factors of T of length ` that start in the interval [i, i + k]. A family F of packages represents the set Factors(F) defined as the union of the sets of factors represented by individual packages in F . We show how to efficiently enumerate Factors(F) and showcase that this is a generic tool for enumerating important classes of factors of T , such as powers and antipowers. Our approach is conceptually simpler than problem-specific methods and provides a unifying framework for such problems, which we hope can be further exploited. We also consider a special case of the problem in which every occurrence of every factor represented by F is captured by some package in F . For both applications mentioned above, we construct an efficient package representation that satisfies this property. We develop efficient algorithms that, given a family F of m packages in a string of length n, report all distinct factors represented by these packages in O(n log n+m logn+ |Factors(F)|) time for the general case and in the optimal O(n + m + |Factors(F)|) time for the special case. We can also compute |Factors(F)| in O(n log n + m logn) time in the general case and in O(n+m) time in the special case. In particular, we improve over the state-of-the-art O(nk log k logn)-time algorithm for computing the number of distinct k-antipower factors, by providing an algorithm that runs in O(nk) time, and we obtain an alternative linear-time algorithm to enumerate distinct squares. ? Partially supported by ERC grant TOTAL under the EU’s Horizon 2020 Research and Innovation Programme (agreement no. 677651). ?? Supported by ISF grants no. 1278/16 and 1926/19, by a BSF grant no. 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064). ? ? ? Supported by the Polish National Science Center, grant no. 2018/31/D/ST6/03991.