124 research outputs found
MinMax-Profiles: A Unifying View of Common Intervals, Nested Common Intervals and Conserved Intervals of K Permutations
Common intervals of K permutations over the same set of n elements were
firstly investigated by T. Uno and M.Yagiura (Algorithmica, 26:290:309, 2000),
who proposed an efficient algorithm to find common intervals when K=2. Several
particular classes of intervals have been defined since then, e.g. conserved
intervals and nested common intervals, with applications mainly in genome
comparison. Each such class, including common intervals, led to the development
of a specific algorithmic approach for K=2, and - except for nested common
intervals - for its extension to an arbitrary K.
In this paper, we propose a common and efficient algorithmic framework for
finding different types of common intervals in a set P of K permutations, with
arbitrary K. Our generic algorithm is based on a global representation of the
information stored in P, called the MinMax-profile of P, and an efficient data
structure, called an LR-stack, that we introduce here. We show that common
intervals (and their subclasses of irreducible common intervals and same-sign
common intervals), nested common intervals (and their subclass of maximal
nested common intervals) as well as conserved intervals (and their subclass of
irreducible conserved intervals) may be obtained by appropriately setting the
parameters of our algorithm in each case. All the resulting algorithms run in
O(Kn+N)-time and need O(n) additional space, where N is the number of
solutions. The algorithms for nested common intervals and maximal nested common
intervals are new for K>2, in the sense that no other algorithm has been given
so far to solve the problem with the same complexity, or better. The other
algorithms are as efficient as the best known algorithms.Comment: 25 pages, 2 figure
Pancake Flipping is Hard
Pancake Flipping is the problem of sorting a stack of pancakes of different
sizes (that is, a permutation), when the only allowed operation is to insert a
spatula anywhere in the stack and to flip the pancakes above it (that is, to
perform a prefix reversal). In the burnt variant, one side of each pancake is
marked as burnt, and it is required to finish with all pancakes having the
burnt side down. Computing the optimal scenario for any stack of pancakes and
determining the worst-case stack for any stack size have been challenges over
more than three decades. Beyond being an intriguing combinatorial problem in
itself, it also yields applications, e.g. in parallel computing and
computational biology. In this paper, we show that the Pancake Flipping
problem, in its original (unburnt) variant, is NP-hard, thus answering the
long-standing question of its computational complexity.Comment: Corrected reference
Easy identification of generalized common and conserved nested intervals
In this paper we explain how to easily compute gene clusters, formalized by
classical or generalized nested common or conserved intervals, between a set of
K genomes represented as K permutations. A b-nested common (resp. conserved)
interval I of size |I| is either an interval of size 1 or a common (resp.
conserved) interval that contains another b-nested common (resp. conserved)
interval of size at least |I|-b. When b=1, this corresponds to the classical
notion of nested interval. We exhibit two simple algorithms to output all
b-nested common or conserved intervals between K permutations in O(Kn+nocc)
time, where nocc is the total number of such intervals. We also explain how to
count all b-nested intervals in O(Kn) time. New properties of the family of
conserved intervals are proposed to do so
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