3,962 research outputs found
Cycles in the burnt pancake graphs
The pancake graph is the Cayley graph of the symmetric group on
elements generated by prefix reversals. has been shown to have
properties that makes it a useful network scheme for parallel processors. For
example, it is -regular, vertex-transitive, and one can embed cycles in
it of length with . The burnt pancake graph ,
which is the Cayley graph of the group of signed permutations using
prefix reversals as generators, has similar properties. Indeed, is
-regular and vertex-transitive. In this paper, we show that has every
cycle of length with . The proof given is a
constructive one that utilizes the recursive structure of . We also
present a complete characterization of all the -cycles in for , which are the smallest cycles embeddable in , by presenting their
canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42
pages, 9 figures, 20 pages of appendice
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
Fault-tolerance embedding of rings and arrays in star and pancake graphs
The star and pancake graphs are useful interconnection networks for connecting processors in a parallel and distributed computing environment. The star network has been widely studied and is shown to possess attactive features like sublogarithmic diameter, node and edge symmetry and high resilience. The star/pancake interconnection graphs, {dollar}S\sb{n}/P\sb{n}{dollar} of dimension n have n! nodes connected by {dollar}{(n-1).n!\over2}{dollar} edges. Due to their large number of nodes and interconnections, they are prone to failure of one or more nodes/edges; In this thesis, we present methods to embed Hamiltonian paths (H-path) and Hamiltonian cycles (H-cycle) in a star graph {dollar}S\sb{n}{dollar} and pancake graph {dollar}P\sb{n}{dollar} in a faulty environment. Such embeddings are important for solving computational problems, formulated for array and ring topologies, on star and pancake graphs. The models considered include single-processor failure, double-processor failure, and multiple-processor failures. All the models are applied to an H-cycle which is formed by visiting all the ({dollar}{n!\over4!})\ S\sb4/P\sb4{dollar}s in an {dollar}S\sb{n}/P\sb{n}{dollar} in a particular order. Each {dollar}S\sb4/P\sb4{dollar} has an entry node where the cycle/path enters that particular {dollar}S\sb4/P\sb4{dollar} and an exit node where the path leaves it. Distributed algorithms for embedding hamiltonian cycle in the presence of multiple faults, are also presented for both {dollar}S\sb{n}{dollar} and {dollar}P\sb{n}{dollar}
Continuum mechanics at nanoscale. A tool to study trees' watering and recovery
The cohesion-tension theory expounds the crude sap ascent thanks to the
negative pressure generated by evaporation of water from leaves. Nevertheless,
trees pose multiple challenges and seem to live in unphysical conditions: the
negative pressure increases cavitation; it is possible to obtain a water
equilibrium between connected parts where one is at a positive pressure and the
other one is at negative pressure; no theory is able to satisfactorily account
for the refilling of vessels after embolism events. A theoretical form of our
paper in the Journal of Theoretical Biology is proposed together with new
results: a continuum mechanics model of the disjoining pressure concept refers
to the Derjaguin School of physical chemistry. A comparison between liquid
behaviour both in tight-filled microtubes and in liquid thin-films is offered
when the pressure is negative in liquid bulks and is positive in liquid
thin-films and vapour bulks. In embolized xylem microtubes, when the air-vapour
pocket pressure is greater than the air-vapour bulk pressure, a refilling flow
occurs between the air-vapour domains to empty the air-vapour pockets although
the liquid-bulk pressure remains negative. The model has a limit of validity
taking the maximal size of trees into account. These results drop inkling that
the disjoining pressure is an efficient tool to study biological liquids in
contact with substrates at a nanoscale range.Comment: The paper is a review and overlap of my different papers about the
watering of trees as a mathematical development of my paper in The Journal of
Theoretical Biology. These results are presented together with new
researches: transfer of liquid water and vapour between xylem microtubes, an
explanation of ultrasounds generated in the watering network considered as
sound pipes, numerical calculations of flows in thin liquid films and of
Poiseuille flows in xylem microtubes, an estimation of the velocity for the
ascent of crude sap and of the recovery time of trees during the spring
perio
Theoretical and experimental study of AC loss in HTS single pancake coils
The electromagnetic properties of a pancake coil in AC regime as a function
of the number of turns is studied theoretically and experimentally.
Specifically, the AC loss, the coil critical current and the voltage signal are
discussed. The coils are made of Bi2Sr2Ca2Cu3O10/Ag (BiSCCO) tape, although the
main qualitative results are also applicable to other kinds of superconducting
tapes, such as coated conductors. The AC loss and the voltage signal are
electrically measured using different pick up coils with the help of a
transformer. One of them avoids dealing with the huge coil inductance. Besides,
the critical current of the coils is experimentally determined by conventional
DC measurements. Furthermore, the critical current, the AC loss and the voltage
signal are simulated, showing a good agreement with the experiments. For all
simulations, the field dependent critical current density inferred from DC
measurements on a short tape sample is taken into account.Comment: 22 pages, 15 figures; contents extended (sections 3.2 and 4); one new
figure (figure 5) and two figures replaced (figures 3 and 8); typos
corrected; title change
Average number of flips in pancake sorting
We are given a stack of pancakes of different sizes and the only allowed
operation is to take several pancakes from top and flip them. The unburnt
version requires the pancakes to be sorted by their sizes at the end, while in
the burnt version they additionally need to be oriented burnt-side down. We
present an algorithm with the average number of flips, needed to sort a stack
of n burnt pancakes, equal to 7n/4+O(1) and a randomized algorithm for the
unburnt version with at most 17n/12+O(1) flips on average.
In addition, we show that in the burnt version, the average number of flips
of any algorithm is at least n+\Omega(n/log n) and conjecture that some
algorithm can reach n+\Theta(n/log n).
We also slightly increase the lower bound on g(n), the minimum number of
flips needed to sort the worst stack of n burnt pancakes. This bound, together
with the upper bound found by Heydari and Sudborough in 1997, gives the exact
number of flips to sort the previously conjectured worst stack -I_n for n=3 mod
4 and n>=15. Finally we present exact values of f(n) up to n=19 and of g(n) up
to n=17 and disprove a conjecture of Cohen and Blum by showing that the burnt
stack -I_{15} is not the worst one for n=15.Comment: 21 pages, new computational results for unburnt pancakes (up to n=19
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