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 y
₁
, y
₂
,...,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=y
₁
#y
₂
#...#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.