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