15 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
Another involution principle-free bijective proof of Stanley's hook-content formula
Another bijective proof of Stanley's hook-content formula for the generating
function for semistandard tableaux of a given shape is given that does not
involve the involution principle of Garsia and Milne. It is the result of a
merge of the modified jeu de taquin idea from the author's previous bijective
proof (``An involution principle-free bijective proof of Stanley's hook-content
formula", Discrete Math. Theoret. Computer Science, to appear) and the
Novelli-Pak-Stoyanovskii bijection (Discrete Math. Theoret. Computer Science 1
(1997), 53-67) for the hook formula for standard Young tableaux of a given
shape. This new algorithm can also be used as an algorithm for the random
generation of tableaux of a given shape with bounded entries. An appropriate
deformation of this algorithm gives an algorithm for the random generation of
plane partitions inside a given box.Comment: 23 pages, AmS-Te
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
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
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