1 / 23

Beat Gfeller, Elias Vicari ETH Zurich, Switzerland

A Randomized Distributed Algorithm for the Maximal Independent Set Problem in Growth-Bounded Graphs. Beat Gfeller, Elias Vicari ETH Zurich, Switzerland. PODC 2007. Maximal Independent Set ( MIS ). In general: captures some aspects of distributed symmetry-breaking

ebliss
Télécharger la présentation

Beat Gfeller, Elias Vicari ETH Zurich, Switzerland

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. A Randomized Distributed Algorithmfor the Maximal Independent Set Problem in Growth-Bounded Graphs Beat Gfeller, Elias Vicari ETH Zurich, Switzerland PODC 2007

  2. Maximal Independent Set (MIS) In general: • captures some aspects of distributed symmetry-breaking • important building block for many distributed algorithms In growth-bounded graphs (wireless networks): • (1+)-approximation MDS and MCDS in O(TMIS) time. • O(1) degree, O(1) stretch spanner in O(TMIS). • independent • maximal

  3. Overview • Related Work • Model • Our Algorithm and its Analysis • Conclusion

  4. ³ ´ ( ) ¡ ¢ 1 ¤ ¤ ¤ ( ( ( ( ) ) ) ) p l l l l l O O ­ O O ¢ o = l l l ­ n o o o o g g g g n n n o g n o g n o g o g n Related Work • In General Graphs: • an time randomized algorithm [Luby85] • an time lower bound [KMW04] • deterministic time algorithm [AGLP89, PS92] • In Growth-Bounded Graphs: • Lower bound , holds even for ring networks (they are GBGs) [Linial87, Naor91] • deterministic time algorithm [Kuhn, Moscibroda, Nieberg, Wattenhofer, DISC 05] • deterministic time algorithm with distance measuring [KMW05]

  5. The Model • Synchronous message passing, synchronous wake-up • Message size O(log n) bits • No node/transmission failures, no collisions • Network modelled as a Growth-Bounded Graph • Each node knows its neighbors and can distinguish them v r = 2 |MIS| ≤ f(r) „Compute a MIS“ = each node knows whether it is in MIS

  6. µ A crucial concept: t-ruling set t-ruling set RV: every node has a node in R within distance t t = 2 independent t-ruling set

  7. Det. O(log Δ log*n)-time algorithm for GBGs • General idea [KMNW05]: • Compute a t-ruling independent set • expand this set into a MIS in O(t ·log*n) time • Structure of step 1: Repeat: compute a 2-ruling set R on G. G’ = G[R]. Until: R is an independent set. By induction: 2t-ruling after t iterations t = 2 for a fast MIS algorithm, this process should terminate quickly! w’’ 6 4 5 3 1 2 v w’ w

  8. Det. O(log Δ log*n)-time algorithm for GBGs • General idea [KMNW05]: • Compute a t-ruling independent set • expand this set into a MIS in O(t ·log*n) time • [KMNW05]: step 1 in O(log Δ · log*n) time, t = O(log Δ), deterministic → MIS in O(log Δ · log*n) • [This work]: step 1 in O(loglog n· log*n) time, t = O(loglog n), randomized → MIS in O(loglog n· log*n) t = 2

  9. Our Randomized Ruling Set – Algorithm • Compute O(loglog Δ)-ruling set with induced degree O(log5 n) in O(loglog Δ · log*n) time using randomization • Make this set independent, but still O(loglog n)-ruling using the det. O(log Δ log*n) time algorithm “Interleaving” the two algorithms: → knowledge of n not required

  10. The Main Ideas • Repeatedly choose a 2-ruling subset which induces a “low” degree. • Reduce the degree from d to dc for some c < 1 → O(loglog Δ) steps (logarithm decreases geometrically) • In a d-regular graph, each node should stay with probability 1/d(1-c) → expected degree dc, 2-ruling with high probability • In general graph? → first, remove nodes with much smaller or larger degree!

  11. [ [ Algorithm “RandStep” – view of a node u •  neighbor v with dv>(du)2 ? → u joins S (“small”) • not in S:  neighbor of u in S? → u joins B (“big”) • not in S or B: u joins R with probability 1/(du)1/4 (“red”) • not in S,B,R, no neighbor in S,B,R→ u joins G (“green”) • G’=G[S R G] dv=2 du=5 dw=2 dq=2

  12. [ [ w’’ 6 4 5 3 1 2 v w’ w Analysis: ruling-property •  neighbor v with dv>(du)2 ? → u joins S (“small”) • not in S:  neighbor of u in S? → u joins B (“big”) • not in S or B: u joins R with probability 1/(du)1/4 (“red”) • not in S,B,R, no neighbor in S,B,R→ u joins G (“green”) • G’=G[S R G] By construction: 2-ruling after one iteration By induction: 2t-ruling after t iterations

  13. [ = 1 2 2 ( ) ( ) d d d · · u v u Analysis: nodes outside SB •  neighbor v with dv>(du)2 ? → u joins S • not in S:  neighbor of u in S? → u joins B Thus, for each node u not in S or B: for all neighbors v of u

  14. = 1 2 2 ( ) ( ) d d d · · u v u : Analysis: high-degree red nodes • A high-degree red node u reduces its degree a lot w.h.p. - Neighbors of red nodes: in R or G (never in S) - red node u has high degree → its neighbors also have high degree: Green neighbors: Lemma:High-degree nodes do not become green w.h.p. → high-degree red node has no green neighbors w.h.p.

  15. = 1 2 2 ( ) ( ) d d d · · 1 · u v u : k n Analysis: high-degree red nodes • A high-degree red node u reduces its degree a lot w.h.p. - Neighbors of red nodes: in R or G (never in S) - red node u has high degree → its neighbors also have high degree: Red neighbors: → neighbors of u join R with probability 1/(dv)1/4 ≤ 1/(du)1/8 → E[# neighbors of u that join R (+1)] ≤ du · (du)-1/8 = (du)7/8 Chernoff-Bound: P[# neighbors of u that join R (+1) > 2du7/8] if du ≥ 9k2log2 n

  16. Analysis: Conclusion • W.h.p., neither R nor G contains a node with degree > 2Δ7/8 as long as Δ> c·log5n • S contains only nodes with degree ≤ Δ1/2 • W.h.p., the degree decreases in each iteration from Δ to 2Δ7/8, as long as Δ > c·log5n. • W.h.p., after O(loglog Δ) iterations Δ < c·log5n. Theorem: In any graph, after O(loglog Δ) iterations of Algorithm “RandStep”, the remaining set is O(loglog Δ)-ruling and has induced degree O(log5n) with probability 1-O(1/nk), for any k > 3.

  17. Conclusion Summary: • Randomized MIS-computation in GBGs vs. in general graphs: O(loglog n log* n) vs. O(log n) • Randomized MIS computation in GBGs can be done almost as fast as with distance information in UDGs/UBGs. Open problems: • Is O(loglog n log*n) tight? Or is O(log*n) achievable? • Still open: polylog-time deterministic MIS algorithm in general graphs

  18. Thank you! Questions? Comments?

  19. 1 d · ³ ´ u = = k 1 2 1 2 = ¡ : 1 2 2 d d ¡ ( ) ( ) 1 · n d d d ¡ · · · e u u u v u : Analysis: high-degree green nodes [detailed] • No high-degree node becomes green w.h.p. For each node u in G (i.e. not in S or B): for all neighbors v of u Recall: 3. not in S or B→ u joins R with probability 1/(du)1/4 u in G: - u has no neighbor in S,B→ each neighbor is a candidate for R - all du-1 neighbors of u had probability ≥ 1/(du)1/2 to join R - P[u joins G] = P[u joins G | u S,B] ≤ P[u and no neighbor of u joins R | u S,B] If du≥ k2 log2 n, this is

  20. 1 k = 1 2 2 ( ) ( ) n d d d · · u v u Analysis: high-degree green nodes • High-degree nodes do not become green w.h.p. For each node u in G (i.e. not in S or B): for all neighbors v of u u in G: - u has no neighbor in S,B→ each neighbor is a candidate for R [ 3.not in S or B: u joins R with probability 1/(du)1/4 ] - all du-1 neighbors of u had probability ≥ 1/(du)1/2 to join R Lemma: If du≥ k2 log2 n, P[u joins G] ≤ . TODO: maybe omit altogether! just mention lemma in red node analysis.

  21. = 7 8 1 1 d 1 ¡ · · · 3 e k ¡ 1 = k 1 2 2 ( ) ( ) n d d d n · · u v u Analysis: high-degree red nodes • A high-degree red node reduces its degree a lot w.h.p. For each node u in R (i.e. not in S or B): for all neighbors v of u Recall: 3. not in S or B→ u joins R with probability 1/(du)1/4 → neighbors of u join R with probability at most 1/(du)1/8 → E[# neighbors of u that join R (+1)] ≤ du · (du)-1/8 = (du)7/8 Chernoff-Bound: P[# neighbors of u that join R (+1) > 2du7/8] if du ≥ 9k2log2 n If du ≥ 9k4log4 n, P[any neighbor of u joins G]

  22. Analysis: high-degree red nodes neighbors of red nodes: red or green (never small) if a red node has high degree, its neighbors also have high degree (although possibly smaller) we show: high-degree nodes are very unlikely to become green -> w.h.p. a high-degree red node has no green neighbors. what about the number of red neighbors? well, they all become red with probability at most … so expected number.. chernoff..

  23. [ = 1 2 2 ( ) ( ) d d d · · u v u Analysis: nodes outside SB •  neighbor v with dv>(du)2 ? → u joins S • not in S:  neighbor of u in S? → u joins B Thus, for each node u not in S or B: for all neighbors v of u u

More Related