1. ETH Zurich – Distributed Computing Group Roger Wattenhofer 1ETH Zurich – Distributed Computing – www.disco.ethz.ch Christoph Lenzen Roger Wattenhofer Exponential Speed-Up of Local Algorithms using Non-Local Communication

2. Christoph Lenzen@PODC 2010 "classical" local model distributed systems with "non-local" communication abstracting communication

3. Christoph Lenzen@PODC 2010 Local vs. Non-Local Communication localnon-local communication range neighborsanyone limiting factorlocalitybandwidth strengthhighly concurrent algorithms known global information exchange Can we combine it?

4. Christoph Lenzen@PODC 2010 MIS, MDS, max. matching, coloring, etc. Model Communication: |V| = n nodes that may exchange messages directly bandwidth and memory limitation of n ² ( ² constant) Task: graph problem on G = (V,E) maximum degree ¢ of G polylogarithmic nodes know neighbors in G large degrees implied small diameter anyway

5. Christoph Lenzen@PODC 2010 Idea collect entire (multi-hop) neighorhoods in G distance to which graph is known grows exponentially simulate multiple rounds of local algorithm in one v

6. Christoph Lenzen@PODC 2010 Results Theorem: Given that inputs are of size at most n ² / 2, any local algorithm terminating within r 2 O(log n) rounds can be simulated within O(log r) rounds.

7. Christoph Lenzen@PODC 2010 Examples problemrunning time maximal independent set O(log log n) w.h.p. O(log ¢ )-approx. of min. dominating set O(log log ¢ ) µ O(log log log n) (2+ ± )-approx. of max. weighted matching O(log log n) 2-approx. of min. vertex cover O(log log n)

8. ETH Zurich – Distributed Computing Group Roger Wattenhofer 8ETH Zurich – Distributed Computing – www.disco.ethz.ch Christoph Lenzen Roger Wattenhofer Thank You! Questions & Comments? TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AAA A A