159 research outputs found
Optimal bounds for disjoint Hamilton cycles in star graphs
In interconnection network topologies, the n-dimensional star graph Stn has n! vertices
corresponding to permutations a (1) : : : a (n) of n symbols a1; : : : ; an and edges which
exchange the positions of the rst symbol a (1) with any one of the other symbols. The
star graph compares favorably with the familiar n-cube on degree, diameter and a number
of other parameters. A desirable property which has not been fully evaluated in star
graphs is the presence of multiple edge-disjoint Hamilton cycles which are important for
fault-tolerance. The only known method for producing multiple edge-disjoint Hamilton
cycles in Stn has been to label the edges in a certain way and then take images of a
known base 2-labelled Hamilton cycle under di erent automorphisms that map labels
consistently. However, optimal bounds for producing edge-disjoint Hamilton cycles in
this way, and whether Hamilton decompositions can be produced, are not known for
any Stn other than for the case of St5 which does provide a Hamilton decomposition.
In this paper we show that, for all n, not more than '(n)=2, where ' is Euler's totient
function, edge-disjoint Hamilton cycles can be produced by such automorphisms. Thus,
for non-prime n, a Hamilton decomposition cannot be produced. We show that the
'(n)=2 upper bound can be achieved for all even n. In particular, if n is a power of
2, Stn has a Hamilton decomposable spanning subgraph comprising more than half of
the edges of Stn. Our results produce a better than twofold improvement on the known
bounds for any kind of edge-disjoint Hamilton cycles in n-dimensional star graphs for
general n
Automorphisms generating disjoint Hamilton cycles in star graphs
In the first part of the thesis we define an automorphism φn for each star graph
Stn of degree n − 1, which yields permutations of labels for the edges of Stn
taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations
into permutation cycles, we are able to identify edge-disjoint Hamilton cycles
that are automorphic images of a known two-labelled Hamilton cycle H1 2(n)
in Stn. Our main result is an improvement from the existing lower bound of
bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number
of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the
improvement is from bn/8c to bn/5c. We extend this result to the cases when n
is the power of a prime other than 3 and 7.
The second part of the thesis studies ‘symmetric’ collections of edge-disjoint
Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under
general label-mapping automorphisms. We show that, for all even n, there exists
a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot
have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus,
Stn is not symmetrically Hamilton decomposable if n is not prime. We also give
cases of even n, in terms of Carmichael’s reduced totient function λ, for which
‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated
from H1 2(n) by a single automorphism, can and cannot attain the optimum
bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a
power of 2, then Stn has a spanning subgraph with more than half of the edges
of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains
an open problem as to whether the bϕ(n)/2c can be achieved for symmetric
collections, but we are able to show that, for certain odd n, a Ï•(n)/4 bound is
achievable and optimal for strongly symmetric collections.
The search for edge-disjoint Hamilton cycles in star graphs is important for the
design of interconnection network topologies in computer science. All our results
improve on the known bounds for numbers of any kind of edge-disjoint Hamilton
cycles in star graphs
Symmetry and optimality of disjoint Hamilton cycles in star graphs
Multiple edge-disjoint Hamilton cycles have been obtained in labelled
star graphs Stn of degree n-1, using number-theoretic means, as images
of a known base 2-labelled Hamilton cycle under label-mapping auto-
morphisms of Stn. However, no optimum bounds for producing such
edge-disjoint Hamilton cycles have been given, and no positive or nega-
tive results exist on whether Hamilton decompositions can be produced
by such constructions other than a positive result for St5. We show that
for all even n there exist such collections, here called symmetric collec-
tions, of φ(n)/2 edge-disjoint Hamilton cycles, where φ is Euler's totient
function, and that this bound cannot be improved for any even or odd n.
Thus, Stn is not symmetrically Hamilton decomposable if n is not prime.
Our method improves on the known bounds for numbers of any kind of
edge-disjoint Hamilton cycles in star graphs
Enumerating Subgraph Instances Using Map-Reduce
The theme of this paper is how to find all instances of a given "sample"
graph in a larger "data graph," using a single round of map-reduce. For the
simplest sample graph, the triangle, we improve upon the best known such
algorithm. We then examine the general case, considering both the communication
cost between mappers and reducers and the total computation cost at the
reducers. To minimize communication cost, we exploit the techniques of (Afrati
and Ullman, TKDE 2011)for computing multiway joins (evaluating conjunctive
queries) in a single map-reduce round. Several methods are shown for
translating sample graphs into a union of conjunctive queries with as few
queries as possible. We also address the matter of optimizing computation cost.
Many serial algorithms are shown to be "convertible," in the sense that it is
possible to partition the data graph, explore each partition in a separate
reducer, and have the total computation cost at the reducers be of the same
order as the computation cost of the serial algorithm.Comment: 37 page
Disjoint Hamilton cycles in transposition graphs
Most network topologies that have been studied have been subgraphs of transposition graphs.
Edge-disjoint Hamilton cycles are important in network topologies for improving fault-tolerance
and distribution of messaging traffic over the network. Not much was known about edge-disjoint
Hamilton cycles in general transposition graphs until recently Hung produced a construction
of 4 edge-disjoint Hamilton cycles in the 5-dimensional transposition graph and showed how
edge-disjoint Hamilton cycles in (n + 1)-dimensional transposition graphs can be constructed
inductively from edge-disjoint Hamilton cycles in n-dimensional transposition graphs. In the
same work it was conjectured that n-dimensional transposition graphs have n − 1 edge-disjoint
Hamilton cycles for all n greater than or equal to 5. In this paper we provide an edge-labelling
for transposition graphs and, by considering known Hamilton cycles in labelled star subgraphs
of transposition graphs, are able to provide an extra edge-disjoint Hamilton cycle at the inductive
step from dimension n to n + 1, and thereby prove the conjecture
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