A Randomized Distributed Algorithm for the Maximal Independent

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

Maximal Independent Set (MIS) - independent - maximal 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). PODC 2007 Beat Gfeller, Elias Vicari

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

Related Work ³ p Graphs: • In General ¢ time¡ randomized algorithm [Luby 85] log n= log n • an time lower bound [KMW 04] O no(1) • deterministic time algorithm [AGLP 89, PS 92] • an O(log n) ´ • In Growth-Bounded¤ Graphs: (log n) • Lower bound , holds even for ring networks (they are ¤ GBGs) [Linial 87, Naor 91] O(log ¢ log n) • deterministic time algorithm [Kuhn, Moscibroda, ¤ Nieberg, Wattenhofer, DISC 05] O(log n) • deterministic time algorithm with distance measuring [KMW 05] PODC 2007 Beat Gfeller, Elias Vicari

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 v r=2 |MIS| ≤ f(r) • Each node knows its neighbors and can distinguish them „Compute a MIS“ = each node knows whether it is in MIS PODC 2007 Beat Gfeller, Elias Vicari

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

Det. O(log Δ log*n)-time algorithm for GBGs • General idea [KMNW 05]: t=2 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: t iterations for a 2 t-ruling fast MISafter algorithm, this process should terminate quickly! w’’ v 1 2 3 w PODC 2007 4 5 6 w’ Beat Gfeller, Elias Vicari

Det. O(log Δ log*n)-time algorithm for GBGs • General idea [KMNW 05]: t=2 1. Compute a t-ruling independent set 2. expand this set into a MIS in O(t · log*n) time • [KMNW 05]: 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) PODC 2007 Beat Gfeller, Elias Vicari

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

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! PODC 2007 Beat Gfeller, Elias Vicari

Algorithm “Rand. Step” – view of a node u 1. 2. 3. 4. 5. 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 PODC 2007 Beat Gfeller, Elias Vicari

Analysis: ruling-property 1. neighbor v with dv>(du)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/(du)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: 2 t-ruling after t iterations v 1 2 3 w PODC 2007 4 5 6 w’’ w’ Beat Gfeller, Elias Vicari

[ Analysis: nodes outside S B 1. neighbor v with dv>(du)2 ? → u joins S 2. not in S: neighbor of u in S? → u joins B Thus, for each u not 2 in S or B: (du )1=2 (d ) · dnode · v u for all neighbors v of u PODC 2007 Beat Gfeller, Elias Vicari

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(d high degree → its neighbors also have 1=2 2: d ) (d ) · · u v u 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. PODC 2007 Beat Gfeller, Elias Vicari

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(d high degree → its neighbors also have 1=2 2: d ) (d ) · · u v u 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) > 2 du 7/8] if du ≥ 9 k 2 log 2 n PODC 2007 Beat Gfeller, Elias Vicari 1 nk

Analysis: Conclusion • W. h. p. , neither R nor G contains a node with degree > • • • 2Δ 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 2Δ 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 “Rand. Step”, the remaining set is O(loglog Δ)-ruling and has induced degree O(log 5 n) with probability 1 -O(1/nk), for any k > 3. PODC 2007 Beat Gfeller, Elias Vicari

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

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

Analysis: high-degreen nodes [detailed] • No high-degree node becomes green w. h. p. For node u in· G (d (i. e. )not 1=2 2 in S or B): (deach ) d · u v u 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 ³ ´ - all du-1 neighbors of u had probability ≥ 1/(du)1/2 to join R ¡ du - P[u joins G] = P[u joins G | u S, B] ≤ 1=2 · 1 ¡ du ¡ P[u and no neighbor of u joins R | u S, B] · e d 1=2 u : If du ≥ k 2 log 2 n, this is PODC 2007 · 1 : nk Beat Gfeller, Elias Vicari

Analysis: high-degreen nodes • High-degree nodes do not become green w. h. p. For node u in· G (d (i. e. )not 1=2 2 in S or B): (deach ) d · u v u for all neighbors v of u TODO: maybe omit u in G: altogether! just mention - u has no neighbor in S, B → each is a lemmaneighbor in red node candidate for R analysis. [ 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: 1 If du ≥ k 2 log 2 n, P[u joins G] ≤ nk. PODC 2007 Beat Gfeller, Elias Vicari

Analysis: high-degree red nodes • A high-degree red node reduces its degree a lot w. h. p. For node u in· R (d (i. e. )not 1=2 2 in S or B): (deach ) d · u v u 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: · e 13 d 7=8 · P[# neighbors of u that join R (+1) > 2 du 7/8] if du ≥ 9 k 2 log 2 n If du ≥ 9 k 4 log 4 n, P[any neighbor of u joins G] PODC 2007 Beat Gfeller, Elias Vicari · 1 nk¡ 1 1 nk

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

[ Analysis: nodes outside S B 1. neighbor v with dv>(du)2 ? → u joins S 2. not in S: neighbor of u in S? → u joins B Thus, for each u not 2 in S or B: (du )1=2 (d ) · dnode · v u for all neighbors v of u u PODC 2007 Beat Gfeller, Elias Vicari