ETH Zurich – Distributed Computing Group Roger Wattenhofer 1ETH Zurich – Distributed Computing – www.disco.ethz.ch

Christoph LenzenRoger Wattenhofer

Exponential Speed-Upof Local Algorithms

using Non-Local Communication

Christoph Lenzen@PODC 2010

"classical"local model

distributed systemswith "non-local"communication

abstracting

communication

Christoph Lenzen@PODC 2010

Local vs. Non-Local Communication

local non-local

communication range

neighbors anyone

limiting factor locality bandwidth

strength highly concurrent algorithms known

global information

exchange

Can we combine it?

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 n² (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

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

Christoph Lenzen@PODC 2010

Results

Theorem:

Given that inputs are of size at most n²/2,any local algorithm terminating within r 2 O(log n) roundscan be simulated within O(log r) rounds.

Christoph Lenzen@PODC 2010

Examples

problem running timemaximal

independent setO(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)

ETH Zurich – Distributed Computing Group Roger Wattenhofer 8ETH Zurich – Distributed Computing – www.disco.ethz.ch

Christoph LenzenRoger Wattenhofer

Thank You!Questions & Comments?