Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
What is a Maximal Independet Set (MIS)?
• inaugmentable set of non-adjacent nodes• well-known symmetry breaking structure• many algorithms build on a MIS
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
What is a Tree?
Let’s assume we all know...
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
Talk Outline
good talkconvincing motivationimpressive resultssketch key ideascoherent conclusions
my talkWell, let’s skip that...We do it in O((ln n ln ln n)1/2) rounds!give detailsmake up for the bad talk
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
• in each phase:– draw uniformly random “ID”– if own ID is larger than all neighbors’ IDs ) join & terminate– if neighbor joined independent set ) do not join & terminate
• removes const. fraction of edges with const. probability) running time O(log n) w.h.p.
An Algorithm for General Graphs (Luby, STOC’85)
12
2
3
5
16
42
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
...and on Trees?
• same analysis gives O(log n)• ...but let‘s have a closer look:
• show that either this event is unlikelyor subtree of v contains >n nodes
survived until phase rwith degree ¢ > e(ln n ln ln n)1/2
...
...
v
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
...and on Trees?
• same analysis gives O(log n)• ...but let‘s have a closer look:
) v removed with probability¸ 1-(1-2ln ¢/¢)¢/2 ¼ 1-e-ln ¢ = 1-1/¢
survived until phase rwith degree ¢ > e(ln n ln ln n)1/2
children that surviveduntil phase r
Case 1¸ ¢/2 manywith degree · ¢/(2ln ¢)
v
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
...and on Trees?
• same analysis gives O(log n)• ...but let‘s have a closer look:
) each of them removed in phase r-1 with prob. ¸ 1-2ln ¢/¢or has ¢/(4ln ¢) high-degree children in phase r-1
survived until phase rwith degree ¢ > e(ln n ln ln n)1/2
children that surviveduntil phase r
Case 2¸ ¢/2 manywith degree ¸ ¢/(2ln ¢)
also true inphase r-1
v
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
...and on Trees?
• same analysis gives O(log n)• ...but let‘s have a closer look:
• recursion, r ¸ (ln n)1/2, and a small miracle...) v is removed in phase r with probability ¸ 1-O(1/¢)
survived until phase rwith degree ¢ > e(ln n ln ln n)1/2
children that surviveduntil phase r
...
...
v
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
Getting a Fast Uniform Algorithm
• (very) roughly speaking, we argue as follows:– degrees · e(ln n ln ln n)1/2 after O((ln n)1/2) rounds– degrees fall exponentially till O((ln n)1/2)– coloring techniques + eleminating leaves deal with small
degrees– guess (ln n ln ln n)1/2 and loop, increasing guess exponentially
) termination within O((ln n ln ln n)1/2) rounds w.h.p.
probablyO((ln n)1/2)
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
Trees - Why Should we Care?
• previous sublogarithmic MIS algorithms require small independent sets in considered neighborhood:– Cole-Vishkin type algorithms (£(log* n), directed trees, rings,
UDG‘s, etc.)– forest decomposition (£(log n/log log n), bounded arboricity)– “general coloring”-based algorithms (£(¢), small degrees)
• our proof utilizes independence of neighborsCole and Vishkin,Inf. & Control’86
Linial, SIAM J. on Comp.‘92
Schneider and Wattenhofer,
PODC’08Naor, SIAM J. on
Disc. Math.‘91
Barenboim and Elkin,Dist. Comp.‘09
e.g. Barenboim and Elkin,PODC‘10
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
Some Speculation
• bounded arboricity = “everywhere sparse” ) little dependencies
) generalization possible?
• combination with techniques relying on dependence) hope for sublogarithmic solution on general graphs?
• take home message:Don‘t give up on matching the ((ln n)1/2) lower bound!
Kuhn et al., PODC’04(recently improved)
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
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