108 research outputs found
Notes on the connectivity of Cayley coset digraphs
Hamidoune's connectivity results for hierarchical Cayley digraphs are
extended to Cayley coset digraphs and thus to arbitrary vertex transitive
digraphs. It is shown that if a Cayley coset digraph can be hierarchically
decomposed in a certain way, then it is optimally vertex connected. The results
are obtained by extending the methods used by Hamidoune. They are used to show
that cycle-prefix graphs are optimally vertex connected. This implies that
cycle-prefix graphs have good fault tolerance properties.Comment: 15 page
Invertible families of sets of bounded degree
Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then
hypergraph H is invertible iff there exists a permutation pi of V such that for
all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility
critical if H is not invertible but every hypergraph obtained by removing an
edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x
belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of
edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =<
(d-1) {2d-1 choose d} + 1. The proof of this result leads to the following
covering problem on graphs: Let G be a graph. A family H is subset of (2^{V(G)}
is an edge cover of G iff for every edge e of G, there is an E belongs to
H(edge set) which includes e. H(edge set) is a minimal edge cover of G iff for
H' subset of H, H' is not an edge cover of G. Let b(d) (c(d)) be the maximum
cardinality of a minimal edge cover H(edge set) of a complete bipartite graph
(complete graph) where H(edge set) has degree d. Theorem: c(d)=< i(d)=<b(d)=<
c(d+1) and 3. 2^{d-1} - 2 =< b(d)=< (d-1) {2d-1choose d} +1. The proof of this
result uses Sperner theory. The bounds b(d) also arise as bounds on the maximum
number of elements in the union of minimal covers of families of sets.Comment: 9 page
Lower bounds for identifying subset members with subset queries
An instance of a group testing problem is a set of objects \cO and an
unknown subset of \cO. The task is to determine by using queries of
the type ``does intersect '', where is a subset of \cO. This
problem occurs in areas such as fault detection, multiaccess communications,
optimal search, blood testing and chromosome mapping. Consider the two stage
algorithm for solving a group testing problem. In the first stage a
predetermined set of queries are asked in parallel and in the second stage,
is determined by testing individual objects. Let n=\cardof{\cO}. Suppose that
is generated by independently adding each x\in \cO to with
probability . Let () be the number of queries asked in the
first (second) stage of this algorithm. We show that if
, then \Exp(q_2) = n^{1-o(1)}, while there
exist algorithms with and \Exp(q_2) =
o(1). The proof involves a relaxation technique which can be used with
arbitrary distributions. The best previously known bound is q_1+\Exp(q_2) =
\Omega(p\log(n)). For general group testing algorithms, our results imply that
if the average number of queries over the course of ()
independent experiments is , then with high probability
non-singleton subsets are queried. This
settles a conjecture of Bill Bruno and David Torney and has important
consequences for the use of group testing in screening DNA libraries and other
applications where it is more cost effective to use non-adaptive algorithms
and/or too expensive to prepare a subset for its first test.Comment: 9 page
Concatenated Quantum Codes
One of the main problems for the future of practical quantum computing is to
stabilize the computation against unwanted interactions with the environment
and imperfections in the applied operations. Existing proposals for quantum
memories and quantum channels require gates with asymptotically zero error to
store or transmit an input quantum state for arbitrarily long times or
distances with fixed error. In this report a method is given which has the
property that to store or transmit a qubit with maximum error
requires gates with error at most and storage or channel elements
with error at most , independent of how long we wish to store the
state or how far we wish to transmit it. The method relies on using
concatenated quantum codes with hierarchically implemented recovery operations.
The overhead of the method is polynomial in the time of storage or the distance
of the transmission. Rigorous and heuristic lower bounds for the constant
are given.Comment: 16 pages in PostScirpt, the paper is also avalaible at
http://qso.lanl.gov/qc
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