27,236 research outputs found
Faster generation of random spanning trees
In this paper, we set forth a new algorithm for generating approximately
uniformly random spanning trees in undirected graphs. We show how to sample
from a distribution that is within a multiplicative of uniform in
expected time \TO(m\sqrt{n}\log 1/\delta). This improves the sparse graph
case of the best previously known worst-case bound of , which has stood for twenty years.
To achieve this goal, we exploit the connection between random walks on
graphs and electrical networks, and we use this to introduce a new approach to
the problem that integrates discrete random walk-based techniques with
continuous linear algebraic methods. We believe that our use of electrical
networks and sparse linear system solvers in conjunction with random walks and
combinatorial partitioning techniques is a useful paradigm that will find
further applications in algorithmic graph theory
Faster generation of random spanning trees
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 39-40).In this thesis, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative (1+6) of uniform in expected time ... . This improves the sparse graph case of the best previously known worst-case bound of O(min{mn, n2. 376}), which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks to introduce a new approach to the problem that integrates discrete random walk-based techniques with continuous linear algebraic methods. We believe that our use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is a useful paradigm that will find further applications in algorithmic graph theory. This work was done in collaboration with Jonathan Kelner.by Aleksander Ma̧dry.S.M
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
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
SAT Modulo Monotonic Theories
We define the concept of a monotonic theory and show how to build efficient
SMT (SAT Modulo Theory) solvers, including effective theory propagation and
clause learning, for such theories. We present examples showing that monotonic
theories arise from many common problems, e.g., graph properties such as
reachability, shortest paths, connected components, minimum spanning tree, and
max-flow/min-cut, and then demonstrate our framework by building SMT solvers
for each of these theories. We apply these solvers to procedural content
generation problems, demonstrating major speed-ups over state-of-the-art
approaches based on SAT or Answer Set Programming, and easily solving several
instances that were previously impractical to solve
A nearly-mlogn time solver for SDD linear systems
We present an improved algorithm for solving symmetrically diagonally
dominant linear systems. On input of an symmetric diagonally
dominant matrix with non-zero entries and a vector such that
for some (unknown) vector , our algorithm computes a
vector such that
{ denotes the A-norm} in time
The solver utilizes in a standard way a `preconditioning' chain of
progressively sparser graphs. To claim the faster running time we make a
two-fold improvement in the algorithm for constructing the chain. The new chain
exploits previously unknown properties of the graph sparsification algorithm
given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning
properties. We also present an algorithm of independent interest that
constructs nearly-tight low-stretch spanning trees in time
, a factor of faster than the algorithm in
[Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the
construction time of the preconditioning chain.Comment: to appear in FOCS1
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