1 / 24

On Local Fixing

On Local Fixing. Michael König Roger Wattenhofer. ETH Zurich – Distributed Computing – www.disco.ethz.ch. Motivation. Motivation. Motivation by Example. Motivation by Example. Motivation by Example. Motivation by Example. Motivation by Example. Motivation by Example.

mayes
Download Presentation

On Local Fixing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. On Local Fixing Michael König Roger Wattenhofer ETH Zurich – Distributed Computing – www.disco.ethz.ch

  2. Motivation

  3. Motivation

  4. Motivation by Example

  5. Motivation by Example

  6. Motivation by Example

  7. Motivation by Example

  8. Motivation by Example

  9. Motivation by Example

  10. Motivation by Example

  11. Motivation by Example

  12. Model • Neighborhood model • Unbounded message sizes • Synchronous rounds t = 2 t = 1 t = 0

  13. Model (cont.) • We say a problem can be fixed locally if any solution can be fixed within O(1) rounds after a graph change. 2 (-e) Edge Deletion (w → w') Weight Change (+e) Edge Insertion 1 (-v1) (-v*) (+v1) (+v*) Node Deletion Node Insertion

  14. Computation vs. Fixing • Computing solutions is very well-studied • Different “complexity classes” have been defined: 4 3 3 5 5 4 3 Γ1-Count “local” Maximal Matching “polylog” Spanning Tree “global” • Previous work on local fixing • 1995: Maximal Independent Sets (MIS), by Kutten and Peleg • 2007: O(1)-Maximum Weighted Matchings, by Lotker, Patt-Shamir and Rosén

  15. Problems • LBound UBound +e -e w → w' +v1 -v1 +v* -v* Γ1-Count Ω(1) O(1) o(n)-MDS Ω(1) O(1) MIS Ω(√log(n)) O(log(n)) O(1)-MWM Ω(√log(n)) O(log(n)) MM Ω(√log(n)) O(log(n)) 2-MVC Ω(√log(n)) O(log(n)) Γ log(n)-Count Ω(log(n)) O(log(n)) ST Ω(D) O(D) MST Ω(D) O(D) SPT Ω(D) O(D) Flow Ω(D) O(D) Leader Ω(D) O(D) Count Ω(D) O(D)

  16. Problems • LBound UBound +e -e w → w' +v1 -v1 +v* -v* Γ1-Count Ω(1) O(1) o(n)-MDS Ω(1) O(1) MIS Ω(√log(n)) O(log(n)) O(1)-MWM Ω(√log(n)) O(log(n)) MM Ω(√log(n)) O(log(n)) 2-MVC Ω(√log(n)) O(log(n)) Γ log(n)-Count Ω(log(n)) O(log(n)) ST Ω(D) O(D) MST Ω(D) O(D) SPT Ω(D) O(D) Flow Ω(D) O(D) Leader Ω(D) O(D) Count Ω(D) O(D) • ü ü ü ü ü • ü ü ü ü ü û û ûûûû û ûûûû û • ü ü ü ü ü • ü ü ü ü ü ü • ü ü ü ü ü • ü ü ü ü ü • ü ü ü ü ü û ûûûû û û û û û û û û û û û û û û û û û û û û û û û û û û û ûûûû û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û û • ü ü ü ü ü • ü ü ü ü ü ü • ü ü ü ü ü ü • ü ü ü ü ü ü • ü ü ü ü ü • ü ü ü ü ü

  17. Problems • LBound UBound +e -e w → w' +v1 -v1 +v* -v* Γ1-Count Ω(1) O(1) o(n)-MDS Ω(1) O(1) MIS Ω(√log(n)) O(log(n)) O(1)-MWM Ω(√log(n)) O(log(n)) MM Ω(√log(n)) O(log(n)) 2-MVC Ω(√log(n)) O(log(n)) Γ log(n)-Count Ω(log(n)) O(log(n)) ST Ω(D) O(D) MST Ω(D) O(D) SPT Ω(D) O(D) Flow Ω(D) O(D) Leader Ω(D) O(D) Count Ω(D) O(D) ü û üûüû û üûüûü üûüûü û û ü û û ü û ü ü û ü û ü û û ü û ü û ü ü û ü û ü û üûüûü û û ü û û ü û ü ü û üü û ü û û ü û û ü û ü ü û ü û ü û û ü û ü û ü

  18. Results • LBound UBound +e -e w → w' +v1 -v1 +v* -v* Γ1-Count Ω(1) O(1) o(n)-MDS Ω(1) O(1) MIS Ω(√log(n)) O(log(n)) O(1)-MWM Ω(√log(n)) O(log(n)) MM Ω(√log(n)) O(log(n)) 2-MVC Ω(√log(n)) O(log(n)) Γ log(n)-Count Ω(log(n)) O(log(n)) ST Ω(D) O(D) MST Ω(D) O(D) SPT Ω(D) O(D) Flow Ω(D) O(D) Leader Ω(D) O(D) Count Ω(D) O(D) üüüüüü û û û û û û üüüüüü üüüüüüü üüüüüü üüüüüü û û û û û û üûüüü û û û û üü û û û û û üü û û û û û üü û û üüüüüü üü û û û û

  19. Maximal Independent Set (MIS) ? ? ? ? ?

  20. “Last Will”: during settling

  21. “Last Will”: in action ! !

  22. o(n)-Minimum Dominating Set • We know algorithms which compute a o(n)-MDS in constant time[Kuhn et al., 2005] • But we can construct o(n)-MDS solutions which cannot be fixed locally! Recipe for a k-MDS which cannot be fixed within c steps: x = ⌊(k - 1)(c + 1)⌋, y = 3c + 1 V = (a1, a2, …, ax, b1, b2, …, by) E = {(ai, b1) | 1 ≤ i ≤ x} ∪ {(bi, bi+1) | 1 ≤ i ≤ y} a2 a2 a1 a1 For k = 2.8, c = 2, x = 5 and y = 7: a3 a3 b1 b1 b2 b2 b3 b3 b4 b4 b5 b5 b6 b6 b7 b7 a4 a4 a5 a5

  23. Conclusion • Traditional distributed complexity classes don’t tell the whole story. • A set of orthogonal “fixing complexity” classes may be interesting!

  24. Thanks!Questions & Comments?

More Related