18 research outputs found
Optimal Oblivious Routing in Polynomial Time
A recent seminal result of Räcke is that for any network there is an oblivious routing algorithm with a polylog competitive ratio with respect to congestion. Unfortunately, Räcke's construction is not polynomial time. We give a polynomial time construction that guarantee's Räcke's bounds, and more generally gives the true optimal ratio for any network
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
Set covering with our eyes closed
Given a universe of elements and a weighted collection of subsets of , the universal set cover problem is to a priori map each element to a set containing such that any set is covered by S(X)=\cup_{u\in XS(u). The aim is to find a mapping such that the cost of is as close as possible to the optimal set cover cost for . (Such problems are also called oblivious or a priori optimization problems.) Unfortunately, for every universal mapping, the cost of can be times larger than optimal if the set is adversarially chosen. In this paper we study the performance on average, when is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (nonmetric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multicut and disc-covering problems and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors
Finding effective support-tree preconditioners
In 1995, Gremban, Miller, and Zagha introduced support-tree preconditioners and a parallel algorithm called support-tree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the off-diagonal entries are non-positive, and the row and column sums are zero. A Laplacian A with 2m non-zeros can be interpreted as an undirected positively-weighted graph G with n vertices and m edges, where there is an edge between two nodes i and j with weight c((i, j)) = −Ai,j = −Aj,i if Ai,j = Aj,i < 0. Gremban et al. showed experimentally that STCG performs well on several classes of graphs commonly used in scientific computations. In his thesis, Gremban also proved upper bounds on the number of iterations required for STCG to converge for certain classes of graphs. In this paper, we present an algorithm for finding a preconditioner for an arbitrary graph G = (V, E) with n nodes, m edges, and a weight function c> 0 on the edges, where w.l.o.g., mine∈E c(e) = 1. Equipped with this preconditioner, STCG requires O(log 4 n · � ∆/α) iterations, where α = min U⊂V,|U|≤|V |/2 c(U, V \U)/|U | is the minimum edge expansion of the graph, and ∆ = maxv∈V c(v) is the maximum incident weight on any vertex. Each iteration requires O(m) work and can be implemented in O(log n) steps in parallel, using only O(m) space. Our results generalize to matrices that are symmetric and diagonally-dominant (SDD).