65,703 research outputs found
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
An optimal quantum algorithm for the oracle identification problem
In the oracle identification problem, we are given oracle access to an
unknown N-bit string x promised to belong to a known set C of size M and our
task is to identify x. We present a quantum algorithm for the problem that is
optimal in its dependence on N and M. Our algorithm considerably simplifies and
improves the previous best algorithm due to Ambainis et al. Our algorithm also
has applications in quantum learning theory, where it improves the complexity
of exact learning with membership queries, resolving a conjecture of Hunziker
et al.
The algorithm is based on ideas from classical learning theory and a new
composition theorem for solutions of the filtered -norm semidefinite
program, which characterizes quantum query complexity. Our composition theorem
is quite general and allows us to compose quantum algorithms with
input-dependent query complexities without incurring a logarithmic overhead for
error reduction. As an application of the composition theorem, we remove all
log factors from the best known quantum algorithm for Boolean matrix
multiplication.Comment: 16 pages; v2: minor change
A graph-theory method for pattern identification in geographical epidemiology - a preliminary application to deprivation and mortality
Background: Graph theoretical methods are extensively used in the field of computational chemistry to search datasets of compounds to see if they contain particular molecular substructures or patterns. We describe a preliminary application of a graph theoretical method, developed in computational chemistry, to geographical epidemiology in relation to testing a prior hypothesis. We tested the methodology on the hypothesis that if a socioeconomically deprived neighbourhood is situated in a wider deprived area, then that neighbourhood would experience greater adverse effects on mortality compared with a similarly deprived neighbourhood which is situated in a wider area with generally less deprivation.
Methods: We used the Trent Region Health Authority area for this study, which contained 10,665 census enumeration districts (CED). Graphs are mathematical representations of objects and their relationships and within the context of this study, nodes represented CEDs and edges were determined by whether or not CEDs were neighbours (shared a common boundary). The overall area in this study was represented by one large graph comprising all CEDs in the region, along with their adjacency information. We used mortality data from 1988-1998, CED level population estimates and the Townsend Material Deprivation Index as an indicator of neighbourhood level deprivation. We defined deprived CEDs as those in the top 20% most deprived in the Region. We then set out to classify these deprived CEDs into seven groups defined by increasing deprivation levels in the neighbouring CEDs. 506 (24.2%) of the deprived CEDs had five adjacent CEDs and we limited pattern development and searching to these CEDs. We developed seven query patterns and used the RASCAL (Rapid Similarity Calculator) program to carry out the search for each of the query patterns. This program used a maximum common subgraph isomorphism method which was modified to handle geographical data.
Results: Of the 506 deprived CEDs, 10 were not identified as belonging to any of the seven groups because they were adjacent to a CED with a missing deprivation category quintile, and none fell within query Group 1 (a deprived CED for which all five adjacent CEDs were affluent). Only four CEDs fell within Group 2, which was defined as having four affluent adjacent CEDs and one non-affluent adjacent CED. The numbers of CEDs in Groups 3-7 were 17, 214, 95, 81 and 85 respectively. Age and sex adjusted mortality rate ratios showed a non-significant trend towards increasing mortality risk across Groups (Chi-square = 3.26, df = 1, p = 0.07).
Conclusion: Graph theoretical methods developed in computational chemistry may be a useful addition to the current GIS based methods available for geographical epidemiology but further developmental work is required. An important requirement will be the development of methods for specifying multiple complex search patterns. Further work is also required to examine the utility of using distance, as opposed to adjacency, to describe edges in graphs, and to examine methods for pattern specification when the nodes have multiple attributes attached to them
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
An Efficient Local Search for Partial Latin Square Extension Problem
A partial Latin square (PLS) is a partial assignment of n symbols to an nxn
grid such that, in each row and in each column, each symbol appears at most
once. The partial Latin square extension problem is an NP-hard problem that
asks for a largest extension of a given PLS. In this paper we propose an
efficient local search for this problem. We focus on the local search such that
the neighborhood is defined by (p,q)-swap, i.e., removing exactly p symbols and
then assigning symbols to at most q empty cells. For p in {1,2,3}, our
neighborhood search algorithm finds an improved solution or concludes that no
such solution exists in O(n^{p+1}) time. We also propose a novel swap
operation, Trellis-swap, which is a generalization of (1,q)-swap and
(2,q)-swap. Our Trellis-neighborhood search algorithm takes O(n^{3.5}) time to
do the same thing. Using these neighborhood search algorithms, we design a
prototype iterated local search algorithm and show its effectiveness in
comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX
and LocalSolver.Comment: 17 pages, 2 figure
Machine learning stochastic design models.
Due to the fluid nature of the early stages of the design process, it is difficult to obtain deterministic product design evaluations. This is primarily due to the flexibility of the design at this stage, namely that there can be multiple interpretations of a single design concept. However, it is important for designers to understand how these design concepts are likely to fulfil the original specification, thus enabling the designer to select or bias towards solutions with favourable outcomes. One approach is to create a stochastic model of the design domain. This paper tackles the issues of using a product database to induce a Bayesian model that represents the relationships between the design parameters and characteristics. A greedy learning algorithm is presented and illustrated using a simple case study
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