29 research outputs found
Near-Optimal Distributed Maximum Flow
We present a near-optimal distributed algorithm for -approximation of single-commodity maximum flow in undirected weighted networks that runs in communication rounds in the \Congest model. Here, and denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of , and it nearly matches the round complexity lower bound. The development of the algorithm contains two results of independent interest: (i) A -round distributed construction of a spanning tree of average stretch . (ii) A -round distributed construction of an -congestion approximator consisting of the cuts induced by virtual trees. The distributed representation of the cut approximator allows for evaluation in rounds. All our algorithms make use of randomization and succeed with high probability
Lessons from the Congested Clique Applied to MapReduce
The main results of this paper are (I) a simulation algorithm which, under
quite general constraints, transforms algorithms running on the Congested
Clique into algorithms running in the MapReduce model, and (II) a distributed
-coloring algorithm running on the Congested Clique which has an
expected running time of (i) rounds, if ;
and (ii) rounds otherwise. Applying the simulation theorem to
the Congested-Clique -coloring algorithm yields an -round
-coloring algorithm in the MapReduce model.
Our simulation algorithm illustrates a natural correspondence between
per-node bandwidth in the Congested Clique model and memory per machine in the
MapReduce model. In the Congested Clique (and more generally, any network in
the model), the major impediment to constructing fast
algorithms is the restriction on message sizes. Similarly, in the
MapReduce model, the combined restrictions on memory per machine and total
system memory have a dominant effect on algorithm design. In showing a fairly
general simulation algorithm, we highlight the similarities and differences
between these models.Comment: 15 page
High Entropy Random Selection Protocols
In this paper, we construct protocols for two parties that do not trust each other,
to generate random variables with high Shannon entropy.
We improve known bounds for the trade off between the number of rounds, length of communication and the entropy of the outcome
High Entropy Random Selection Protocols
We study the two party problem of randomly selecting a common string among all the strings of length n. We want the protocol to have the property that the output distribution has high Shannon entropy or high min entropy, even when one of the two parties is dishonest and deviates from the protocol. We develop protocols that achieve high, close to n, Shannon entropy and simultaneously min entropy close to n/2. In the literature the randomness guarantee is usually expressed in terms of “resilience”. The notion of Shannon entropy is not directly comparable to that of resilience, but we establish a connection between the two that allows us to compare our protocols with the existing ones. We construct an explicit protocol that yields Shannon entropy n- O(1) and has O(log ∗n) rounds, improving over the protocol of Goldreich et al. (SIAM J Comput 27: 506–544, 1998) that also achieves this entropy but needs O(n) rounds. Both these protocols need O(n2) bits of communication. Next we reduce the number of rounds and the length of communication in our protocols. We show the existence, non-explicitly, of a protocol that has 6 rounds, O(n) bits of communication and yields Shannon entropy n- O(log n) and min entropy n/ 2 - O(log n). Our protocol achieves the same Shannon entropy bound as, also non-explicit, protocol of Gradwohl et al. (in: Dwork (ed) Advances in Cryptology—CRYPTO ‘06, 409–426, Technical Report , 2006), however achieves much higher min entropy: n/ 2 - O(log n) versus O(log n). Finally we exhibit a very simple 3-round explicit “geometric” protocol with communication length O(n). We connect the security parameter of this protocol with the well studied Kakey
With Great Speed Come Small Buffers: Space-Bandwidth Tradeoffs for Routing
We consider the Adversarial Queuing Theory (AQT) model, where packet arrivals are subject to a maximum average rate and burstiness . In this model, we analyze the size of buffers required to avoid overflows in the basic case of a path. Our main results characterize the space required by the average rate and the number of distinct destinations: we show that space suffice, where is the number of distinct destinations and ; and we show that space is necessary. For directed trees, we describe an algorithm whose buffer space requirement is at most where is the maximum number of destinations on any root-leaf path