Christoph 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 2011 What is a Tree? Let’s assume we all know...

Christoph 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 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)

Christoph Lenzen, PODC 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 r with degree ¢ > e (ln n ln ln n) 1/2... v

Christoph Lenzen, PODC 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 r with degree ¢ > e (ln n ln ln n) 1/2 children that survived until phase r Case 1 ¸ ¢ /2 many with degree · ¢ /(2ln ¢ ) v

Christoph Lenzen, PODC 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 r with degree ¢ > e (ln n ln ln n) 1/2 children that survived until phase r Case 2 ¸ ¢ /2 many with degree ¸ ¢ /(2ln ¢ ) also true in phase r-1 v

Christoph Lenzen, PODC 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 r with degree ¢ > e (ln n ln ln n) 1/2 children that survived until phase r... v

Christoph 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. probably O((ln n) 1/2 )

Christoph 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 neighbors Cole and Vishkin, Inf. & Control’86 Linial, SIAM J. on Comp.‘92 Schneider and Wattenhofer, PODC’08 Naor, SIAM J. on Disc. Math.‘91 Barenboim and Elkin, Dist. Comp.‘09 e.g. Barenboim and Elkin, PODC‘10

Christoph 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 2011