937 research outputs found
Sufficient Conditions for Efficient Indexing Under Different Matchings
The most important task derived from the massive digital data accumulation in the world, is efficient access to this data, hence the importance of indexing. In the last decade, many different types of matching relations were defined, each requiring an efficient indexing scheme. Cole and Hariharan in a ground breaking paper [Cole and Hariharan, SIAM J. Comput., 33(1):26-42, 2003], formulate sufficient conditions for building an efficient indexing for quasi-suffix collections, collections that behave as suffixes. It was shown that known matchings, including parameterized, 2-D array and order preserving matchings, fit their indexing settings. In this paper, we formulate more basic sufficient conditions based on the order relation derived from the matching relation itself, our conditions are more general than the previously known conditions
Exact Algorithm for Sampling the 2D Ising Spin Glass
A sampling algorithm is presented that generates spin glass configurations of
the 2D Edwards-Anderson Ising spin glass at finite temperature, with
probabilities proportional to their Boltzmann weights. Such an algorithm
overcomes the slow dynamics of direct simulation and can be used to study
long-range correlation functions and coarse-grained dynamics. The algorithm
uses a correspondence between spin configurations on a regular lattice and
dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson,
Proc. 8th Symp. Discrete Algorithms 258, (1997)] for sampling dimer coverings
on a planar lattice is adapted to generate samplings for the dimer problem
corresponding to both planar and toroidal spin glass samples. This algorithm is
recursive: it computes probabilities for spins along a "separator" that divides
the sample in half. Given the spins on the separator, sample configurations for
the two separated halves are generated by further division and assignment. The
algorithm is simplified by using Pfaffian elimination, rather than Gaussian
elimination, for sampling dimer configurations. For n spins and given floating
point precision, the algorithm has an asymptotic run-time of O(n^{3/2}); it is
found that the required precision scales as inverse temperature and grows only
slowly with system size. Sample applications and benchmarking results are
presented for samples of size up to n=128^2, with fixed and periodic boundary
conditions.Comment: 18 pages, 10 figures, 1 table; minor clarification
Toward Entity-Aware Search
As the Web has evolved into a data-rich repository, with the standard "page view," current search engines are becoming increasingly inadequate for a wide range of query tasks. While we often search for various data "entities" (e.g., phone number, paper PDF, date), today's engines only take us indirectly to pages. In my Ph.D. study, we focus on a novel type of Web search that is aware of data entities inside pages, a significant departure from traditional document retrieval. We study the various essential aspects of supporting entity-aware Web search. To begin with, we tackle the core challenge of ranking entities, by distilling its underlying conceptual model Impression Model and developing a probabilistic ranking framework, EntityRank, that is able to seamlessly integrate both local and global information in ranking. We also report a prototype system built to show the initial promise of the proposal. Then, we aim at distilling and abstracting the essential computation requirements of entity search. From the dual views of reasoning--entity as input and entity as output, we propose a dual-inversion framework, with two indexing and partition schemes, towards efficient and scalable query processing. Further, to recognize more entity instances, we study the problem of entity synonym discovery through mining query log data. The results we obtained so far have shown clear promise of entity-aware search, in its usefulness, effectiveness, efficiency and scalability
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Embedding bounded degree spanning trees in random graphs
We prove that if a tree has vertices and maximum degree at most
, then a copy of can almost surely be found in the random graph
.Comment: 14 page
Efficient Exact Inference in Planar Ising Models
We give polynomial-time algorithms for the exact computation of lowest-energy
(ground) states, worst margin violators, log partition functions, and marginal
edge probabilities in certain binary undirected graphical models. Our approach
provides an interesting alternative to the well-known graph cut paradigm in
that it does not impose any submodularity constraints; instead we require
planarity to establish a correspondence with perfect matchings (dimer
coverings) in an expanded dual graph. We implement a unified framework while
delegating complex but well-understood subproblems (planar embedding,
maximum-weight perfect matching) to established algorithms for which efficient
implementations are freely available. Unlike graph cut methods, we can perform
penalized maximum-likelihood as well as maximum-margin parameter estimation in
the associated conditional random fields (CRFs), and employ marginal posterior
probabilities as well as maximum a posteriori (MAP) states for prediction.
Maximum-margin CRF parameter estimation on image denoising and segmentation
problems shows our approach to be efficient and effective. A C++ implementation
is available from http://nic.schraudolph.org/isinf/Comment: Fixed a number of bugs in v1; added 10 pages of additional figures,
explanations, proofs, and experiment
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