1 maximal independent set. 2 independent set (is): in a graph g=(v,e), |v|=n, |e|=m, any set of...

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Maximal Independent Set

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Independent Set (IS):

In a graph G=(V,E), |V|=n, |E|=m, 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

Remark: finding a maximum IS is an NP-hard problem, while a MIS can be found in polynomial time!

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Applications in DS: network topology

•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 interferences, a conflict graph is built, and a MIS on that defines a clustering of the nodes enabling efficient routing

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Applications in DS: network monitoring

• A MIS is always a Dominating Set (DS) of the graph (the converse in not true, unless the DS is independent), namely every node in G is at distance at most 1 from at least one node in the MIS (otherwise the MIS could be enlarged, against the assumption of maximality!)

In a network graph G consisting of nodes representing processors, a MIS defines a set of processors which can monitor the correct functioning of all the nodes in G (in such an application, one should find a minimum MIS)

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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

12

2v


2G

13


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 a MIS of I G

G

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Worst case graph (for number of phases):

n nodes, n-1 phases

Running time of the algorithm: Θ(m)

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, instead of a single node, we now select any independent set (this selection should be seen as a black box at this stage)

The underlying idea is that this approach will be useful for a distributed algorithm

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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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1. The algorithm is correct, since independence and maximality follow by construction

2. The number of phases depends on the choice of the independent set in each phase: The larger the subgraph removed at the end of a phase, the smaller the residual graph, and then the faster the algorithm. Then, how do we choose such a set, so that independence is guaranteed and the convergence is fast?

Analysis

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Example:If is MIS, 1 phase is needed

1I

Example:If each contains one node, phases may be needed

kI)(n

(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 remove many nodes from the current residual graph•Works with synchronous, anonymous, uniform models

Remark: It is randomized in a Las Vegas sense, i.e., it uses randomization only to reduce the expected running time, but always terminates with a correct result

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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 are elected simultaneously (nodes can check it out by testing their neighborhood)

Problematic nodeskG

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All the problematic nodes step back to the unelected status, and proceed to the next phase. 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 ProbabilityLet A and B denote two events in a probability space;

let

1. A (i.e., not A) be the event that A does not occur;2. AՈB be the event that both A and B occur;3. AUB be the event that A or (non-exclusive) B occurs.

Then, we have that:1. P(A)=1-P(A);2. P(AՈB)=P(A)P(B) (if A and B are independent)3. P(AUB)=P(A)+P(B)-P(AՈB) (if A and B are

mutually exclusive, then P(AՈB)=0, and P(AUB)=P(A)+P(B)).

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Fundamental inequalities

tn

t ent

nt

e

11

21n

nt ||

10 p

1k

k

kp

p

11

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Probability for a node z of success in a phase: P(success z) = P((z enters Ik) OR (z enters N(Ik))

i.e., is at least the probability that it enters Ik, i.e., it elects itself AND no neighbor elects itself, i.e. (notice that events are independent):

1 2 y

p

p1

p1p1

z

No neighbor should elect itself

elects itself

P(success z) ≥ p(1-p)y (recall that p=1/d)

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Probability of success for a node in a phase:

2ed1

d1

1ed1

d1

1d1

p)p(1p1pd

dy

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

Node z does not disappear at the end of phase k with probability at most

2ed1

1

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after phases

Definition: Bad event for node :

ned ln4

node did not disappear

This happens with probability P(ANDk=1,..,4ed lnn (z does not disappear at the end of phase

k))

i.e., at most:

2ln2

ln22ln411

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12

11

needed n

nedned

z

z

(first (right) ineq. with t =-1 and n =2ed)

Independent events

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after phases

Bad event for G:

ned ln4

at least one node did not disappear

This happens with probability (notice that events are not mutually exclusive): 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

(i.e., with 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 it is in Ik, and notifies neighbors;

3. In round 3, each node receiving notifications from elected neighbors, realizes to be in N(Ik),

notifies its neighbors about that, and stops. total # of rounds: )log( ndO

(w.h.p.)

Time complexity

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Homework

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

1 maximal independent set. 2 independent set (is): in a graph g=(v,e), |v|=n, |e|=m, any set of...

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