Maximal Independent Set

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Maximal Independent Set

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Independent Set (IS):

In a graph, any set of nodes that are not adjacent

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Maximal Independent Set (MIS):

An independent set that is nosubset of any other independent set

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Maximum Independent Set:

A MIS of maximum size

A graph G… …a MIS of G… …a MIS of max size

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Applications in Distributed Systems

•In a network graph consisting of nodes representing processors, a MIS defines a set of processors which can operate in parallel without interference

•For instance, in wireless ad hoc networks, to avoid interference, a conflict graph is built, and a MIS on that defines a clustering of the nodes enabling efficient routing

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A Sequential Greedy algorithm

Suppose that will hold the final MISI

Initially I

G

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Pick a node and add it to I1v

1v

Phase 1:

1GG

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Remove and neighbors )( 1vN1v

1G

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2G

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Pick a node and add it to I2v

2v

Phase 2:

2G

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2v


2G

12


3G

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Repeat until all nodes are removed

Phases 3,4,5,…:

3G

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Repeat until all nodes are removed

No remaining nodes

Phases 3,4,5,…,x:

1xG

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At the end, set will be an MIS of I G

G

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Worst case graph (for number of phases):

n nodes

Running time of the algorithm: Θ(|E|)

Number of phases of the algorithm: O(n)

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Homework

Can you see a distributed version of the algorithm just given?

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A General Algorithm For Computing MIS

Same as the sequential greedy algorithm,but at each phase we may select any independent set (instead of a single node)

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Suppose that will hold the final MISI

Initially I

Example:

G

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Find any independent set 1I

Phase 1:

And insert to :1I I 1III

1GG

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1I )( 1INremove and neighbors

1G

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remove and neighbors 1I )( 1IN

1G

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2G

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Phase 2:



On new graph

2G

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2G

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3G

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Phase 3:



On new graph

3G

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3G

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No nodes are left

4G

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Final MIS I

G

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The number of phases depends on the choice of independent set in each phase:

The larger the independent set at eachphase, the faster the algorithm

Observation:

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Example:If is MIS, 1 phase is needed

1I

Example:If each contains one node, phases are needed

kI)(nO

(sequential greedy algorithm)

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A Randomized Sync. Distributed Algorithm

Follows the general MIS algorithm paradigm, by choosing randomly at each phase the independent set, in such a way that it is expected to include many nodes of the remaining graph

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Let be the maximum node degree in the whole graph

d

1 2 d

Suppose that d is known to all the nodes (this may require a pre-processing)

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Elected nodes are candidates forindependent set

Each node elects itself with probability

At each phase :k

kI

dp

1

1 2 d

kGz

z

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However, it is possible that neighbor nodes may be elected simultaneously

Problematic nodeskG

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All the problematic nodes must be un-elected. The remaining elected nodes formindependent set kI

kGkI

kIkI

kI

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Success for a node in phase : disappears at end of phase(enters or )

Analysis:

kGz

kI

1 2 y

No neighbor elects itself

z

z

k)( kIN

k

A good scenariothat guaranteessuccess

elects itself

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Basics of Probability

E: finite universe of events; let A and B denote two events in E; then:

1. A B is the event that A or (non-exclusive) B occurs;

2. A B is the event that both A and B occur.

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Probability of success in a phase:

1 2 y

p

p1

p1p1

dy pppp )1(1

z

No neighbor should elect itself

At least

elects itself

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Fundamental inequalities

tn

t ent

nt

e

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21n

nt ||

10 p

1k

k

kp

p

11

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Probability of success in phase:

ed

ded

dd

ppppd

dy

2

1

11

1

11

1

)1(1

At least

For 2d

First (left) ineq. with t =-1

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Therefore, node disappears at the end of phase with probability at least

1 2 y

z

z

ed21k

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Expected number of phases until nodedisappears:

at most ed2phase in success of yprobabilit

1 phases

z

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after phases

Definition: Bad event for node :

ned ln4

node did not disappear

This happens with probability (first (right) ineq. with t =-1 and n =2ed):

2ln2

ln22ln411

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12

11

needed n

nedned

z

z

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after phases

Bad event for G:

ned ln4

at least one node did not disappear

This happens with probability: P(ORxG(bad event for x)) ≤

nnnx

Gx

11) for event P(bad 2

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within phases

Good event for G:

ned ln4

all nodes disappear

This happens with probability:

n1

-1G] for event bad of ty[probabili1

(high probability)

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Total number of phases:

)log(ln4 ndOned

# rounds for each phase: 31. In round 1, each node tries to elect itself and

notifies neighbors;2. In round 2, each node receives notifications

from neighbors, decide whether is in Ik, and notifies neighbors;

3. In round 3, each node receiving notifications

from elected neighbors, realizes to be in N(Ik).

total # of rounds: )log( ndO

(with high probability)

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Homework

Can you provide a good bound on the number of messages?

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Maximal Independent Set