beat gfeller, elias vicari eth zurich, switzerland
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
Portland, Oregon, 13 August, 2007
A Randomized Distributed Algorithmfor the Maximal Independent Set Problem in
Growth-Bounded GraphsBeat Gfeller, Elias Vicari
ETH Zurich, Switzerland
PODC 2007
2PODC 2007 Beat Gfeller, Elias Vicari
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
3PODC 2007 Beat Gfeller, Elias Vicari
Overview
• Related Work
• Model
• Our Algorithm and its Analysis
• Conclusion
4PODC 2007 Beat Gfeller, Elias Vicari
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(logn)
O(log¢ log¤n)
O(log¤n)
(log¤n)
³ p
logn=loglogn´
O¡no(1)
¢
5PODC 2007 Beat Gfeller, Elias Vicari
• 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
6PODC 2007 Beat Gfeller, Elias Vicari
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
7PODC 2007 Beat Gfeller, Elias Vicari
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
1v 2 3 4 56
w w’
w’’
t = 2
for a fast MIS algorithm, this process should terminate quickly!
for a fast MIS algorithm, this process should terminate quickly!
8PODC 2007 Beat Gfeller, Elias Vicari
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
9PODC 2007 Beat Gfeller, Elias Vicari
Our Randomized Ruling Set – Algorithm
1. Compute O(loglog Δ)-ruling set with induced degree O(log5 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
10PODC 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!
11PODC 2007 Beat Gfeller, Elias Vicari
Algorithm “RandStep” – view of a node u
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]
dv=2
du=5
[ [
dw=2
dq=2
12PODC 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: 2t-ruling after t iterations
[ [
1v 2 3 4 56
w w’
w’’
13PODC 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 node u not in S or B:
for all neighbors v of u
[
(du)1=2 · dv · (du)2
14PODC 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 → 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.
(du)1=2 · dv · (du)2:
15PODC 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 → 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
(du)1=2 · dv · (du)2:
· 1nk
16PODC 2007 Beat Gfeller, Elias Vicari
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.
17PODC 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
18PODC 2007 Beat Gfeller, Elias Vicari
Thank you!
Questions? Comments?
19PODC 2007 Beat Gfeller, Elias Vicari
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· e¡ d
1=2u :
·³1¡ d¡ 1=2u
´du
· 1nk :
(du)1=2 · dv · (du)2
20PODC 2007 Beat Gfeller, Elias Vicari
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] ≤ .
(du)1=2 · dv · (du)2
1nk
TODO: maybe omit altogether! just mention lemma in red node analysis.
21PODC 2007 Beat Gfeller, Elias Vicari
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] · 1nk ¡ 1
· e¡ 13d7=8 · 1
nk
(du)1=2 · dv · (du)2
22PODC 2007 Beat Gfeller, Elias Vicari
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..
23PODC 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 node u not in S or B:
for all neighbors v of u
[
(du)1=2 · dv · (du)2
u