Portland, Oregon, 13 August, 2007 A Randomized Distributed Algorithm for the Maximal Independent Set Problem in Growth-Bounded Graphs Beat Gfeller, Elias Vicari ETH Zurich, Switzerland PODC 2007

Beat Gfeller, Elias Vicari 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(T MIS ) time. O(1) degree, O(1) stretch spanner in O(T MIS ). - independent - maximal

PODC 2007 Beat Gfeller, Elias Vicari 3 Overview Related Work Model Our Algorithm and its Analysis Conclusion

PODC 2007 Beat Gfeller, Elias Vicari 4 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] O ( l ogn ) O ( l og ¢ l og ¤ n ) O ( l og ¤ n ) ­ ( l og ¤ n ) ­ ³ p l ogn = l og l ogn ´ O ¡ n o ( 1 ) ¢

PODC 2007 Beat Gfeller, Elias Vicari 5 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 The Model „Compute a MIS“ = each node knows whether it is in MIS r = 2 |MIS| ≤ f(r) v

PODC 2007 Beat Gfeller, Elias Vicari 6 A crucial concept: t-ruling set t-ruling set R V: every node has a node in R within distance t t = 2 µ independent t-ruling set

PODC 2007 Beat Gfeller, Elias Vicari 7 Det. O(log Δ log*n)-time algorithm for GBGs General idea [KMNW05]: 1. Compute a t-ruling independent set 2. 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 1 v w w’ w’’ t = 2 for a fast MIS algorithm, this process should terminate quickly!

PODC 2007 Beat Gfeller, Elias Vicari 8 Det. O(log Δ log*n)-time algorithm for GBGs General idea [KMNW05]: 1. Compute a t-ruling independent set 2. 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

PODC 2007 Beat Gfeller, Elias Vicari 9 Our Randomized Ruling Set – Algorithm 1. Compute O(loglog Δ )-ruling set with induced degree O(log 5 n) in O(loglog Δ · log*n) time using randomization 2. 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

PODC 2007 Beat Gfeller, Elias Vicari 10 The Main Ideas Repeatedly choose a 2-ruling subset which induces a “low” degree. Reduce the degree from d to d c 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 d c, 2-ruling with high probability In general graph? → first, remove nodes with much smaller or larger degree!

PODC 2007 Beat Gfeller, Elias Vicari 11 Algorithm “RandStep” – view of a node u 1.  neighbor v with d v >(d u ) 2 ? → u joins S (“small”) 2. not in S:  neighbor of u in S? → u joins B (“big”) 3. not in S or B: u joins R with probability 1/(d u ) 1/4 (“red”) 4. not in S,B,R, no neighbor in S,B,R → u joins G (“green”) 5. G’ = G[S R G] d v =2 d u =5 [ [ d w =2 d q =2

PODC 2007 Beat Gfeller, Elias Vicari 12 Analysis: ruling-property 1.  neighbor v with d v >(d u ) 2 ? → u joins S (“small”) 2. not in S:  neighbor of u in S? → u joins B (“big”) 3. not in S or B: u joins R with probability 1/(d u ) 1/4 (“red”) 4. not in S,B,R, no neighbor in S,B,R → u joins G (“green”) 5. G’ = G[S R G] By construction: 2-ruling after one iteration By induction: 2t-ruling after t iterations [ [ 1 v w w’ w’’

PODC 2007 Beat Gfeller, Elias Vicari 13 Analysis: nodes outside S B 1.  neighbor v with d v >(d u ) 2 ? → u joins S 2. 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 [ ( d u ) 1 = 2 · d v · ( d u ) 2

PODC 2007 Beat Gfeller, Elias Vicari 14 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. ( d u ) 1 = 2 · d v · ( d u ) 2 :

PODC 2007 Beat Gfeller, Elias Vicari 15 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/(d v ) 1/4 ≤ 1/(d u ) 1/8 → E[# neighbors of u that join R (+1)] ≤ d u · (d u ) -1/8 = (d u ) 7/8 Chernoff-Bound: P[# neighbors of u that join R (+1) > 2d u 7/8 ] if d u ≥ 9k 2 log 2 n ( d u ) 1 = 2 · d v · ( d u ) 2 : · 1 n k

PODC 2007 Beat Gfeller, Elias Vicari 16 Analysis: Conclusion W.h.p., neither R nor G contains a node with degree > 2 Δ 7/8 as long as Δ > c·log 5 n S contains only nodes with degree ≤ Δ 1/2 W.h.p., the degree decreases in each iteration from Δ to 2 Δ 7/8, as long as Δ > c·log 5 n. W.h.p., after O(loglog Δ ) iterations Δ < c·log 5 n. 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(log 5 n) with probability 1-O(1/n k ), for any k > 3.

PODC 2007 Beat Gfeller, Elias Vicari 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

PODC 2007 Beat Gfeller, Elias Vicari 18 Thank you! Questions? Comments?

PODC 2007 Beat Gfeller, Elias Vicari 19 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/(d u ) 1/4 u in G: - u has no neighbor in S,B → each neighbor is a candidate for R - all d u -1 neighbors of u had probability ≥ 1/(d u ) 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 d u ≥ k 2 log 2 n, this is · e ¡ d 1 = 2 u : · ³ 1 ¡ d ¡ 1 = 2 u ´ d u · 1 n k : ( d u ) 1 = 2 · d v · ( d u ) 2

PODC 2007 Beat Gfeller, Elias Vicari 20 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/(d u ) 1/4 ] - all d u -1 neighbors of u had probability ≥ 1/(d u ) 1/2 to join R Lemma: If d u ≥ k 2 log 2 n, P[u joins G] ≤. ( d u ) 1 = 2 · d v · ( d u ) 2 1 n k TODO: maybe omit altogether! just mention lemma in red node analysis.

PODC 2007 Beat Gfeller, Elias Vicari 21 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/(d u ) 1/4 → neighbors of u join R with probability at most 1/(d u ) 1/8 → E[# neighbors of u that join R (+1)] ≤ d u · (d u ) -1/8 = (d u ) 7/8 Chernoff-Bound: P[# neighbors of u that join R (+1) > 2d u 7/8 ] if d u ≥ 9k 2 log 2 n If d u ≥ 9k 4 log 4 n, P[any neighbor of u joins G] · 1 n k ¡ 1 · e ¡ 1 3 d 7 = 8 · 1 n k ( d u ) 1 = 2 · d v · ( d u ) 2

PODC 2007 Beat Gfeller, Elias Vicari 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..

PODC 2007 Beat Gfeller, Elias Vicari 23 Analysis: nodes outside S B 1.  neighbor v with d v >(d u ) 2 ? → u joins S 2. 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 [ ( d u ) 1 = 2 · d v · ( d u ) 2 u