45 research outputs found
Reachability and Shortest Paths in the Broadcast CONGEST Model
In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter D of the underlying network is constant.
We show that for the case where D = 1 there is, quite surprisingly, a very simple algorithm that solves the reachability problem in 1(!) round. In contrast, for networks with D = 2, we show that any distributed algorithm (possibly randomized) for this problem requires Omega(sqrt{n/ log{n}}) rounds. Our results therefore completely resolve (up to a small polylog factor) the complexity of the single-source reachability problem for a wide range of diameters.
Furthermore, we show that when D = 1, it is even possible to get an almost 3 - approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just 2 rounds. We also prove a stronger lower bound of Omega(sqrt{n}) for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is 2. As far as we know this is the first lower bound that achieves Omega(sqrt{n}) for this problem
Robust Fault Tolerant uncapacitated facility location
In the uncapacitated facility location problem, given a graph, a set of
demands and opening costs, it is required to find a set of facilities R, so as
to minimize the sum of the cost of opening the facilities in R and the cost of
assigning all node demands to open facilities. This paper concerns the robust
fault-tolerant version of the uncapacitated facility location problem (RFTFL).
In this problem, one or more facilities might fail, and each demand should be
supplied by the closest open facility that did not fail. It is required to find
a set of facilities R, so as to minimize the sum of the cost of opening the
facilities in R and the cost of assigning all node demands to open facilities
that did not fail, after the failure of up to \alpha facilities. We present a
polynomial time algorithm that yields a 6.5-approximation for this problem with
at most one failure and a 1.5 + 7.5\alpha-approximation for the problem with at
most \alpha > 1 failures. We also show that the RFTFL problem is NP-hard even
on trees, and even in the case of a single failure
Improved Distance Oracles and Spanners for Vertex-Labeled Graphs
Consider an undirected weighted graph G=(V,E) with |V|=n and |E|=m, where
each vertex v is assigned a label from a set L of \ell labels. We show how to
construct a compact distance oracle that can answer queries of the form: "what
is the distance from v to the closest lambda-labeled node" for a given node v
in V and label lambda in L.
This problem was introduced by Hermelin, Levy, Weimann and Yuster [ICALP
2011] where they present several results for this problem. In the first result,
they show how to construct a vertex-label distance oracle of expected size
O(kn^{1+1/k}) with stretch (4k - 5) and query time O(k). In a second result,
they show how to reduce the size of the data structure to O(kn \ell^{1/k}) at
the expense of a huge stretch, the stretch of this construction grows
exponentially in k, (2^k-1). In the third result they present a dynamic
vertex-label distance oracle that is capable of handling label changes in a
sub-linear time. The stretch of this construction is also exponential in k, (2
3^{k-1}+1).
We manage to significantly improve the stretch of their constructions,
reducing the dependence on k from exponential to polynomial (4k-5), without
requiring any tradeoff regarding any of the other variables.
In addition, we introduce the notion of vertex-label spanners: subgraphs that
preserve distances between every node v and label lambda. We present an
efficient construction for vertex-label spanners with stretch-size tradeoff
close to optimal
Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees
The average distance from a node to all other nodes in a graph, or from a
query point in a metric space to a set of points, is a fundamental quantity in
data analysis. The inverse of the average distance, known as the (classic)
closeness centrality of a node, is a popular importance measure in the study of
social networks. We develop novel structural insights on the sparsifiability of
the distance relation via weighted sampling. Based on that, we present highly
practical algorithms with strong statistical guarantees for fundamental
problems. We show that the average distance (and hence the centrality) for all
nodes in a graph can be estimated using single-source
distance computations. For a set of points in a metric space, we show
that after preprocessing which uses distance computations we can compute
a weighted sample of size such that the average
distance from any query point to can be estimated from the distances
from to . Finally, we show that for a set of points in a metric
space, we can estimate the average pairwise distance using
distance computations. The estimate is based on a weighted sample of
pairs of points, which is computed using distance
computations. Our estimates are unbiased with normalized mean square error
(NRMSE) of at most . Increasing the sample size by a
factor ensures that the probability that the relative error exceeds
is polynomially small.Comment: 21 pages, will appear in the Proceedings of RANDOM 201
Fully dynamic all-pairs shortest paths with worst-case update-time revisited
We revisit the classic problem of dynamically maintaining shortest paths
between all pairs of nodes of a directed weighted graph. The allowed updates
are insertions and deletions of nodes and their incident edges. We give
worst-case guarantees on the time needed to process a single update (in
contrast to related results, the update time is not amortized over a sequence
of updates).
Our main result is a simple randomized algorithm that for any parameter
has a worst-case update time of and answers
distance queries correctly with probability , against an adaptive
online adversary if the graph contains no negative cycle. The best
deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time
of and assumes non-negative weights. This is the first
improvement for this problem for more than a decade. Conceptually, our
algorithm shows that randomization along with a more direct approach can
provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201
Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem
In the Single Source Replacement Paths (SSRP) problem we are given a graph , and a shortest paths tree rooted at a node , and
the goal is to output for every node and for every edge in
the length of the shortest path from to avoiding .
We present an time randomized combinatorial
algorithm for unweighted directed graphs. Previously such a bound was known in
the directed case only for the seemingly easier problem of replacement path
where both the source and the target nodes are fixed.
Our new upper bound for this problem matches the existing conditional
combinatorial lower bounds. Hence, (assuming these conditional lower bounds)
our result is essentially optimal and completes the picture of the SSRP problem
in the combinatorial setting.
Our algorithm extends to the case of small, rational edge weights. We
strengthen the existing conditional lower bounds in this case by showing that
any time (combinatorial or algebraic) algorithm for some
fixed yields a truly subcubic algorithm for the weighted All
Pairs Shortest Paths problem (previously such a bound was known only for the
combinatorial setting).Comment: 38 pages, 9 figures, to appear in the Proceedings of the 47th
International Colloquium on Automata, Languages and Programming (ICALP