Beat Gfeller, Elias Vicari ETH Zurich, Switzerland

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


Overview

• Related Work

• Model

• Our Algorithm and its Analysis

• Conclusion


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)

¢


• 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


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


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!


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


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


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!


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


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’’


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


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:



• 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


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.


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


Thank you!

Questions? Comments?


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):


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


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):


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.



• 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):


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



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..


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:


[

(du)1=2 · dv · (du)2

u

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Beat Gfeller, Elias Vicari ETH Zurich, Switzerland