Exponential Speed-Up

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Exponential Speed-Up of Local Algorithms using
Non-Local Communication
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distributed systems with "non-local" communication
abstracting communication
"classical" local model
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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?
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Model

• Task
• graph problem on G (V,E)
• maximum degree of G polylogarithmic
• nodes know neighbors in G

• Communication
• V n nodes that may exchange messages directly
• bandwidth and memory limitation of n² (² constant)

MIS, MDS, max. matching, coloring, etc.
large degrees implied small diameter anyway
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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
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Results

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

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Examples
problem running 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)
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Thank You!Questions Comments?
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