Constructing antidictionaries in output-sensitive space

A word x that is absent from a word y is called minimal if all its proper factors occur in y. Given a collection of k words y₁, y₂,...,y_k over an alphabet Σ, we are asked to compute the set M(y ^1#...# y ^k ) ^ℓ of minimal absent words of length at most ℓ of word y=y₁#y₂#...#y_k, #∉Σ. In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. This computation generally requires Ω(n) space for n=|y| using any of the plenty available O(n)-time algorithms. This is because an Ω(n)-sized text index is constructed over y which can be impractical for large n. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when ||M(y ^1#...# y ^N ) ^ℓ || =o(n), for all N ϵ[1, k]. For instance, in the human genome, n ≈ 3 × 10 ⁹ but ||M (y ^1#...#yk ) ¹² || ≈ 10 ⁶. We consider a constant-sized alphabet for stating our results. We show that all M(y ₁ ) ^ℓ ,...,M(y _1#...# y _k ) ^ℓ can be computed in O(kn+Σ _N=1 ^k ||M(y ₁ #...#(y _N ) ^ℓ ||) total time using O(MaxIn+MaxOut) space, where MaxIn is the length of the longest word in y ₁ ,...,y _k and MaxOut=max{||M (y _1)#...# (y _N ) ^ℓ ||:N ϵ[1, k]. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution.