1,415 research outputs found
Directed Hamiltonicity and Out-Branchings via Generalized Laplacians
We are motivated by a tantalizing open question in exact algorithms: can we
detect whether an -vertex directed graph has a Hamiltonian cycle in time
significantly less than ? We present new randomized algorithms that
improve upon several previous works:
1. We show that for any constant and prime we can count the
Hamiltonian cycles modulo in
expected time less than for a constant that depends only on and
. Such an algorithm was previously known only for the case of counting
modulo two [Bj\"orklund and Husfeldt, FOCS 2013].
2. We show that we can detect a Hamiltonian cycle in
time and polynomial space, where is the size of the maximum
independent set in . In particular, this yields an time
algorithm for bipartite directed graphs, which is faster than the
exponential-space algorithm in [Cygan et al., STOC 2013].
Our algorithms are based on the algebraic combinatorics of "incidence
assignments" that we can capture through evaluation of determinants of
Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed
graphs. In addition to the novel algorithms for directed Hamiltonicity, we use
the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting
out-branchings. Specifically, we give an -time randomized algorithm
for detecting out-branchings with at least internal vertices, improving
upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015].
We also present an algebraic algorithm for the directed -Leaf problem, based
on a non-standard monomial detection problem
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
Elastic Registration of Geodesic Vascular Graphs
Vascular graphs can embed a number of high-level features, from morphological
parameters, to functional biomarkers, and represent an invaluable tool for
longitudinal and cross-sectional clinical inference. This, however, is only
feasible when graphs are co-registered together, allowing coherent multiple
comparisons. The robust registration of vascular topologies stands therefore as
key enabling technology for group-wise analyses. In this work, we present an
end-to-end vascular graph registration approach, that aligns networks with
non-linear geometries and topological deformations, by introducing a novel
overconnected geodesic vascular graph formulation, and without enforcing any
anatomical prior constraint. The 3D elastic graph registration is then
performed with state-of-the-art graph matching methods used in computer vision.
Promising results of vascular matching are found using graphs from synthetic
and real angiographies. Observations and future designs are discussed towards
potential clinical applications
Listing Subgraphs by Cartesian Decomposition
We investigate a decomposition technique for listing problems in graphs and set systems. It is based on the Cartesian product of some iterators, which list the solutions of simpler problems. Our ideas applies to several problems, and we illustrate one of them in depth, namely, listing all minimum spanning trees of a weighted graph G. Here iterators over the spanning trees for unweighted graphs can be obtained by a suitable modification of the listing algorithm by [Shioura et al., SICOMP 1997], and the decomposition of G is obtained by suitably partitioning its edges according to their weights. By combining these iterators in a Cartesian product scheme that employs Gray coding, we give the first algorithm which lists all minimum spanning trees of G in constant delay, where the delay is the time elapsed between any two consecutive outputs. Our solution requires polynomial preprocessing time and uses polynomial space
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