160 research outputs found
Prioritized Metric Structures and Embedding
Metric data structures (distance oracles, distance labeling schemes, routing
schemes) and low-distortion embeddings provide a powerful algorithmic
methodology, which has been successfully applied for approximation algorithms
\cite{llr}, online algorithms \cite{BBMN11}, distributed algorithms
\cite{KKMPT12} and for computing sparsifiers \cite{ST04}. However, this
methodology appears to have a limitation: the worst-case performance inherently
depends on the cardinality of the metric, and one could not specify in advance
which vertices/points should enjoy a better service (i.e., stretch/distortion,
label size/dimension) than that given by the worst-case guarantee.
In this paper we alleviate this limitation by devising a suit of {\em
prioritized} metric data structures and embeddings. We show that given a
priority ranking of the graph vertices (respectively,
metric points) one can devise a metric data structure (respectively, embedding)
in which the stretch (resp., distortion) incurred by any pair containing a
vertex will depend on the rank of the vertex. We also show that other
important parameters, such as the label size and (in some sense) the dimension,
may depend only on . In some of our metric data structures (resp.,
embeddings) we achieve both prioritized stretch (resp., distortion) and label
size (resp., dimension) {\em simultaneously}. The worst-case performance of our
metric data structures and embeddings is typically asymptotically no worse than
of their non-prioritized counterparts.Comment: To appear at STOC 201
Byzantine Agreement with Optimal Early Stopping, Optimal Resilience and Polynomial Complexity
We provide the first protocol that solves Byzantine agreement with optimal
early stopping ( rounds) and optimal resilience () using
polynomial message size and computation.
All previous approaches obtained sub-optimal results and used resolve rules
that looked only at the immediate children in the EIG (\emph{Exponential
Information Gathering}) tree. At the heart of our solution are new resolve
rules that look at multiple layers of the EIG tree.Comment: full version of STOC 2015 abstrac
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
Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion
This paper addresses the basic question of how well can a tree approximate
distances of a metric space or a graph. Given a graph, the problem of
constructing a spanning tree in a graph which strongly preserves distances in
the graph is a fundamental problem in network design. We present scaling
distortion embeddings where the distortion scales as a function of ,
with the guarantee that for each the distortion of a fraction
of all pairs is bounded accordingly. Such a bound implies, in
particular, that the \emph{average distortion} and -distortions are
small. Specifically, our embeddings have \emph{constant} average distortion and
-distortion. This follows from the following
results: we prove that any metric space embeds into an ultrametric with scaling
distortion . For the graph setting we prove that any
weighted graph contains a spanning tree with scaling distortion
. These bounds are tight even for embedding in arbitrary
trees.
For probabilistic embedding into spanning trees we prove a scaling distortion
of , which implies \emph{constant}
-distortion for every fixed .Comment: Extended abstrat apears in SODA 200
Lower Bounds on Implementing Robust and Resilient Mediators
We consider games that have (k,t)-robust equilibria when played with a
mediator, where an equilibrium is (k,t)-robust if it tolerates deviations by
coalitions of size up to k and deviations by up to players with unknown
utilities. We prove lower bounds that match upper bounds on the ability to
implement such mediators using cheap talk (that is, just allowing communication
among the players). The bounds depend on (a) the relationship between k, t, and
n, the total number of players in the system; (b) whether players know the
exact utilities of other players; (c) whether there are broadcast channels or
just point-to-point channels; (d) whether cryptography is available; and (e)
whether the game has a k+t$ players, guarantees that every player gets a
worse outcome than they do with the equilibrium strategy
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