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MIS on Trees Christoph Lenzen and Roger Wattenhofer

MIS on Trees Christoph Lenzen and Roger Wattenhofer. What is a Maximal Independet Set (MIS)?. inaugmentable set of non-adjacent nodes well-known symmetry breaking structure many algorithms build on a MIS. What is a Tree?. Let’s assume we all know. Talk Outline. in each phase:

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MIS on Trees Christoph Lenzen and Roger Wattenhofer

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  1. MIS on Trees Christoph Lenzen and Roger Wattenhofer

  2. What is a Maximal Independet Set (MIS)? inaugmentable set of non-adjacent nodes well-known symmetry breaking structure many algorithms build on a MIS

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

  4. Talk Outline

  5. 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 5 3 16 42 2

  6. ...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 ... ...

  7. ...and on Trees? • same analysis gives O(log n) • ...but let‘s have a closer look: • )v removed with probability • ¸ 1-(1-2ln ¢/¢)¢/2¼ 1-e-ln ¢ = 1-1/¢ survived until phase r with degree ¢> e(ln n ln ln n)1/2 v children that survived until phase r ¸¢/2 many with degree ·¢/(2ln ¢) Case 1

  8. ...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 v children that survived until phase r also true in phase r-1 ¸¢/2 many with degree ¸¢/(2ln ¢) Case 2

  9. ...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 v children that survived until phase r ... ...

  10. 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)

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

  12. 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)

  13. Thank you!

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