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 $(x_1,x_2,\ldots,x_n)$ 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 $x_j$ will depend on the rank $j$ 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 $j$. 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 ($\min\{f+2,t+1\}$ rounds) and optimal resilience ($n>3t$) 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 $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, 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 $\tilde O(n^{2+3/4})$ 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 $\epsilon$, with the guarantee that for each $\epsilon$ the distortion of a fraction $1-\epsilon$ of all pairs is bounded accordingly. Such a bound implies, in particular, that the \emph{average distortion} and $\ell_q$-distortions are small. Specifically, our embeddings have \emph{constant} average distortion and $O(\sqrt{\log n})$ $\ell_2$-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/\epsilon})$. For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/\epsilon})$. These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/\epsilon))$, which implies \emph{constant} $\ell_q$-distortion for every fixed $q<\infty$.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 $t$ 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 $k+t)-punishment strategy; that is, a strategy that, if used by all but at most$k+t$ players, guarantees that every player gets a worse outcome than they do with the equilibrium strategy