19,728 research outputs found
An almost-linear time algorithm for uniform random spanning tree generation
We give an -time algorithm for generating a uniformly
random spanning tree in an undirected, weighted graph with max-to-min weight
ratio . We also give an -time algorithm for
generating a random spanning tree with total variation distance from
the true uniform distribution. Our second algorithm's runtime does not depend
on the edge weights. Our -time algorithm is the first
almost-linear time algorithm for the problem --- even on unweighted graphs ---
and is the first subquadratic time algorithm for sparse weighted graphs.
Our algorithms improve on the random walk-based approach given in
Kelner-M\k{a}dry and M\k{a}dry-Straszak-Tarnawski. We introduce a new way of
using Laplacian solvers to shortcut a random walk. In order to fully exploit
this shortcutting technique, we prove a number of new facts about electrical
flows in graphs. These facts seek to better understand sets of vertices that
are well-separated in the effective resistance metric in connection with Schur
complements, concentration phenomena for electrical flows after conditioning on
partial samples of a random spanning tree, and more
Vacancy localization in the square dimer model
We study the classical dimer model on a square lattice with a single vacancy
by developing a graph-theoretic classification of the set of all configurations
which extends the spanning tree formulation of close-packed dimers. With this
formalism, we can address the question of the possible motion of the vacancy
induced by dimer slidings. We find a probability 57/4-10Sqrt[2] for the vacancy
to be strictly jammed in an infinite system. More generally, the size
distribution of the domain accessible to the vacancy is characterized by a
power law decay with exponent 9/8. On a finite system, the probability that a
vacancy in the bulk can reach the boundary falls off as a power law of the
system size with exponent 1/4. The resultant weak localization of vacancies
still allows for unbounded diffusion, characterized by a diffusion exponent
that we relate to that of diffusion on spanning trees. We also implement
numerical simulations of the model with both free and periodic boundary
conditions.Comment: 35 pages, 24 figures. Improved version with one added figure (figure
9), a shift s->s+1 in the definition of the tree size, and minor correction
Sampling Random Spanning Trees Faster than Matrix Multiplication
We present an algorithm that, with high probability, generates a random
spanning tree from an edge-weighted undirected graph in
time (The notation hides
factors). The tree is sampled from a distribution
where the probability of each tree is proportional to the product of its edge
weights. This improves upon the previous best algorithm due to Colbourn et al.
that runs in matrix multiplication time, . For the special case of
unweighted graphs, this improves upon the best previously known running time of
for (Colbourn
et al. '96, Kelner-Madry '09, Madry et al. '15).
The effective resistance metric is essential to our algorithm, as in the work
of Madry et al., but we eschew determinant-based and random walk-based
techniques used by previous algorithms. Instead, our algorithm is based on
Gaussian elimination, and the fact that effective resistance is preserved in
the graph resulting from eliminating a subset of vertices (called a Schur
complement). As part of our algorithm, we show how to compute
-approximate effective resistances for a set of vertex pairs via
approximate Schur complements in time,
without using the Johnson-Lindenstrauss lemma which requires time. We
combine this approximation procedure with an error correction procedure for
handing edges where our estimate isn't sufficiently accurate
Optimal Path and Minimal Spanning Trees in Random Weighted Networks
We review results on the scaling of the optimal path length in random
networks with weighted links or nodes. In strong disorder we find that the
length of the optimal path increases dramatically compared to the known small
world result for the minimum distance. For Erd\H{o}s-R\'enyi (ER) and scale
free networks (SF), with parameter (), we find that the
small-world nature is destroyed. We also find numerically that for weak
disorder the length of the optimal path scales logaritmically with the size of
the networks studied. We also review the transition between the strong and weak
disorder regimes in the scaling properties of the length of the optimal path
for ER and SF networks and for a general distribution of weights, and suggest
that for any distribution of weigths, the distribution of optimal path lengths
has a universal form which is controlled by the scaling parameter
where plays the role of the disorder strength, and
is the length of the optimal path in strong disorder. The
relation for is derived analytically and supported by numerical
simulations. We then study the minimum spanning tree (MST) and show that it is
composed of percolation clusters, which we regard as "super-nodes", connected
by a scale-free tree. We furthermore show that the MST can be partitioned into
two distinct components. One component the {\it superhighways}, for which the
nodes with high centrality dominate, corresponds to the largest cluster at the
percolation threshold which is a subset of the MST. In the other component,
{\it roads}, low centrality nodes dominate. We demonstrate the significance
identifying the superhighways by showing that one can improve significantly the
global transport by improving a very small fraction of the network.Comment: review, accepted at IJB
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
Graph Sparsification by Edge-Connectivity and Random Spanning Trees
We present new approaches to constructing graph sparsifiers --- weighted
subgraphs for which every cut has the same value as the original graph, up to a
factor of . Our first approach independently samples each
edge with probability inversely proportional to the edge-connectivity
between and . The fact that this approach produces a sparsifier resolves
a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan
and Panigrahi also resolves this question. Our second approach constructs a
sparsifier by forming the union of several uniformly random spanning trees.
Both of our approaches produce sparsifiers with
edges. Our proofs are based on extensions of Karger's contraction algorithm,
which may be of independent interest
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