8,473 research outputs found
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
Low Diameter Graph Decompositions by Approximate Distance Computation
In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of low diameter graph decomposition with small edge cutting probabilities inherently rely on the subtractive form of the triangle inequality, which fails to hold under distance approximation.
The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 2013), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the CONGEST, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 1996) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only logaritmically many times, which is of interest for capacitated problems and simulating CONGEST algorithms on the tree into which the graph is embedded
Parallel Graph Decompositions Using Random Shifts
We show an improved parallel algorithm for decomposing an undirected
unweighted graph into small diameter pieces with a small fraction of the edges
in between. These decompositions form critical subroutines in a number of graph
algorithms. Our algorithm builds upon the shifted shortest path approach
introduced in [Blelloch, Gupta, Koutis, Miller, Peng, Tangwongsan, SPAA 2011].
By combining various stages of the previous algorithm, we obtain a
significantly simpler algorithm with the same asymptotic guarantees as the best
sequential algorithm
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