Constructing Antidictionaries of Long Texts in Output-Sensitive Space - Ayad, Lorraine A.K.; Badkobeh, Golnaz; Fici, Gabriele; Héliou, Alice; Pissis, Solon P. (Theory of computing systems, 2021)
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 y 1 , … , 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 the collection { y 1 , … , y k }. The set M { y 1 , … , y k } ℓ contains all the words x such that x is absent from all the words of the collection while there exist i , j , such that the maximal proper suffix of x is a factor of y i and the maximal proper prefix of x is a factor of y j . 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. Indeed, the set M y ℓ of minimal absent words of a word y is equal to M { y 1 , … , y k } ℓ for any decomposition of y into a collection of words y 1 , … , y k such that there is an overlap of length at least ℓ − 1 between any two consecutive words in the collection. 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 ∈ [1, k ], where ∥ S ∥ denotes the sum of the lengths of words in set S . For instance, in the human genome, n ≈ 3 × 10 9 but ∥ M { y 1 , … , y k } 12 ∥ ≈ 1 0 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 ( k n + ∑ 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 ∈ [ 1 , k ] } . Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution.