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 udnot 2 in S or B: · dnode · ( du ) 1=2 ( 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(high degree also have du ) 1=2 · dv → · its ( duneighbors )2: 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(high degree also have du ) 1=2 · dv → · its ( duneighbors )2: 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. Fordeach 2 in S or B): du )not · duvin·G ((i. e. ( u ) 1=2 node 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. Fordeach 2 in S or B): du )not · duvin·G ((i. e. ( u ) 1=2 node 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. Fordeach 2 in S or B): du )not · duvin·R ((i. e. ( u ) 1=2 node 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 udnot 2 in S or B: · dnode · ( du ) 1=2 ( v u) for all neighbors v of u u PODC 2007 Beat Gfeller, Elias Vicari
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