296 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
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}
On the number of pancake stacks requiring four flips to be sorted
Using existing classification results for the 7- and 8-cycles in the pancake
graph, we determine the number of permutations that require 4 pancake flips
(prefix reversals) to be sorted. A similar characterization of the 8-cycles in
the burnt pancake graph, due to the authors, is used to derive a formula for
the number of signed permutations requiring 4 (burnt) pancake flips to be
sorted. We furthermore provide an analogous characterization of the 9-cycles in
the burnt pancake graph. Finally we present numerical evidence that polynomial
formulae exist giving the number of signed permutations that require flips
to be sorted, with .Comment: We have finalized for the paper for publication in DMTCS, updated a
reference to its published version, moved the abstract to its proper
location, and added a thank you to the referees. The paper has 27 pages, 6
figures, and 2 table
Embedding Schemes for Interconnection Networks.
Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth
Lifeworld Analysis
We argue that the analysis of agent/environment interactions should be
extended to include the conventions and invariants maintained by agents
throughout their activity. We refer to this thicker notion of environment as a
lifeworld and present a partial set of formal tools for describing structures
of lifeworlds and the ways in which they computationally simplify activity. As
one specific example, we apply the tools to the analysis of the Toast system
and show how versions of the system with very different control structures in
fact implement a common control structure together with different conventions
for encoding task state in the positions or states of objects in the
environment.Comment: See http://www.jair.org/ for any accompanying file
An improved bound on the chromatic number of the Pancake graphs
In this paper an improved bound on the chromatic number of the Pancake graph
, is presented. The bound is obtained using a subadditivity
property of the chromatic number of the Pancake graph. We also investigate an
equitable coloring of . An equitable -coloring based on efficient
dominating sets is given and optimal equitable -colorings are considered for
small . It is conjectured that the chromatic number of coincides with
its equitable chromatic number for any
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