21,578 research outputs found
Locating regions in a sequence under density constraints
Several biological problems require the identification of regions in a
sequence where some feature occurs within a target density range: examples
including the location of GC-rich regions, identification of CpG islands, and
sequence matching. Mathematically, this corresponds to searching a string of 0s
and 1s for a substring whose relative proportion of 1s lies between given lower
and upper bounds. We consider the algorithmic problem of locating the longest
such substring, as well as other related problems (such as finding the shortest
substring or a maximal set of disjoint substrings). For locating the longest
such substring, we develop an algorithm that runs in O(n) time, improving upon
the previous best-known O(n log n) result. For the related problems we develop
O(n log log n) algorithms, again improving upon the best-known O(n log n)
results. Practical testing verifies that our new algorithms enjoy significantly
smaller time and memory footprints, and can process sequences that are orders
of magnitude longer as a result.Comment: 17 pages, 8 figures; v2: minor revisions, additional explanations; to
appear in SIAM Journal on Computin
CD-independent subsets in meet-distributive lattices
A subset of a finite lattice is CD-independent if the meet of any two
incomparable elements of equals 0. In 2009, Cz\'edli, Hartmann and Schmidt
proved that any two maximal CD-independent subsets of a finite distributive
lattice have the same number of elements. In this paper, we prove that if
is a finite meet-distributive lattice, then the size of every CD-independent
subset of is at most the number of atoms of plus the length of . If,
in addition, there is no three-element antichain of meet-irreducible elements,
then we give a recursive description of maximal CD-independent subsets.
Finally, to give an application of CD-independent subsets, we give a new
approach to count islands on a rectangular board.Comment: 14 pages, 4 figure
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