3 research outputs found
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^ℓ_y_1#...#y_k 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^ℓ_y_1#...#y_N||=o(n), for all N∈[1,k]. For instance, in the human genome, n ≈ 3× 10^9 but ||M^12_y_1#...#y_k|| ≈ 10^6. We consider a constant-sized alphabet for stating our results. We show that all M^ℓ_y_1,...,M^ℓ_y_1#...#y_k can be computed in O(kn+∑^k_N=1||M^ℓ_y_1#...#y_N||) total time using O(MaxIn+MaxOut) space, where MaxIn is the length of the longest word in {y_1,...,y_k} and MaxOut={||M^ℓ_y_1#...#y_N||:N∈[1,k]}. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution
Substring Complexity in Sublinear Space
Shannon's entropy is a definitive lower bound for statistical compression.
Unfortunately, no such clear measure exists for the compressibility of
repetitive strings. Thus, ad-hoc measures are employed to estimate the
repetitiveness of strings, e.g., the size of the Lempel-Ziv parse or the
number of equal-letter runs of the Burrows-Wheeler transform. A more recent
one is the size of a smallest string attractor. Unfortunately, Kempa
and Prezza [STOC 2018] showed that computing is NP-hard. Kociumaka et
al. [LATIN 2020] considered a new measure that is based on the function
counting the cardinalities of the sets of substrings of each length of ,
also known as the substring complexity. This new measure is defined as and lower bounds all the measures previously
considered. In particular, always holds and can be
computed in time using working space. Kociumaka et
al. showed that if is given, one can construct an -sized representation of supporting efficient direct
access and efficient pattern matching queries on . Given that for highly
compressible strings, is significantly smaller than , it is natural
to pose the following question: Can we compute efficiently using
sublinear working space?
It is straightforward to show that any algorithm computing using
space requires time through a reduction
from the element distinctness problem [Yao, SIAM J. Comput. 1994]. We present
the following results: an -time and
-space algorithm to compute , for any ; and
an -time and -space algorithm to
compute , for any
Internal Shortest Absent Word Queries in Constant Time and Linear Space
International audienceGiven a string T of length n over an alphabet Σ ⊂ {1, 2,. .. , n O(1) } of size σ, we are to preprocess T so that given a range [i, j], we can return a representation of a shortest string over Σ that is absent in the fragment T [i] • • • T [j] of T. We present an O(n)-space data structure that answers such queries in constant time and can be constructed in O(n log σ n) time