Results

**1 - 3**of**3**### Near-Optimal Distributed Tree Embedding

"... Tree embeddings are a powerful tool in the area of graph approximation algorithms. Roughly speaking, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms n-node metrics into (p ..."

Abstract
- Add to MetaCart

Tree embeddings are a powerful tool in the area of graph approximation algorithms. Roughly speaking, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms n-node metrics into (probability distributions over) trees, while stretching each pairwise distance by at most anO(log n) factor in expectation. This O(log n) stretch is optimal. Khan et al. [PODC’08] present a distributed algorithm that implements FRT in O(SPD log n) rounds, where SPD is the shortest-path-diameter of the weighted graph, and they explain how to use this embedding for various distributed approximation problems. Note that SPD can be as large as Θ(n), even in graphs where the hop-diameter D is a constant. Khan et al. noted that it would be interesting to improve this complexity. We show that this is indeed possible. More precisely, we present a distributed algorithm that constructs a tree embedding that is essentially as good as FRT in Õ(min{n0.5+ε,SPD} + D) rounds, for any constant ε> 0. A lower bound of Ω̃(min{n0.5,SPD}+D) rounds follows from Das Sarma et al. [STOC’11], rendering our round complexity near-optimal.

### Can Quantum Communication Speed Up Distributed Computation?

"... The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main re-sult is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, ..."

Abstract
- Add to MetaCart

The focus of this paper is on quantum distributed computation, where we investigate whether quantum communication can help in speeding up distributed network algorithms. Our main re-sult is that for certain fundamental network problems such as minimum spanning tree, minimum cut, and shortest paths, quantum communication does not help in substantially speeding up dis-tributed algorithms for these problems compared to the classical setting. In order to obtain this result, we extend the technique of Das Sarma et al. [SICOMP 2012] to obtain a uniform approach to prove non-trivial lower bounds for quantum distributed algo-rithms for several graph optimization (both exact and approximate versions) as well as verifica-tion problems, some of which are new even in the classical setting, e.g. tight randomized lower bounds for Hamiltonian cycle and spanning tree verification, answering an open problem of Das Sarma et al., and a lower bound in terms of the weight aspect ratio, matching the upper bounds of Elkin [STOC 2004]. Our approach introduces the Server model and Quantum Simulation Theorem which together provide a connection between distributed algorithms and communica-tion complexity. The Server model is the standard two-party communication complexity model