christoph lenzen, podc 2011. what is a maximal independet set (mis)? inaugmentable set of...
TRANSCRIPT
Christoph Lenzen, PODC 2011Christoph Lenzen, PODC 2011
MIS on TreesChristoph Lenzen and Roger Wattenhofer
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 talk
convincing motivation
impressive results
sketch key ideas
coherent conclusions
my talk
Well, let’s skip that...
We do it in O((ln n ln ln n)1/2) rounds!
give details
make 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 unlikely
or 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 many
with 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 many
with 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
Thank you!