Maximal Independent Set
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Transcript of Maximal Independent Set
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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
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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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Applications in Distributed Systems (2)
•A MIS is always a Dominating Set (DS) of the graph (the converse in not true), namely every node in G must be at distance at most 1 from at least one node in the MIS
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
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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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Remove and neighbors )( 1vN1v
2G
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Pick a node and add it to I2v
2v
Phase 2:
2G
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2v
Remove and neighbors )( 2vN2v
2G
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Remove and neighbors )( 2vN2v
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, 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 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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remove and neighbors 1I )( 1IN
2G
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Phase 2:
Find any independent set 2I
And insert to :2I I 2III
On new graph
2G
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remove and neighbors 2I )( 2IN
2G
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remove and neighbors 2I )( 2IN
3G
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Phase 3:
Find any independent set 3I
And insert to :3I I 3III
On new graph
3G
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remove and neighbors 3I )( 3IN
3G
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remove and neighbors 3I )( 3IN
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 subgraph removed at the end of a phase, the smaller the residual graph, and then 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: is at least the probability that a node elects itself and no neighbor elects itself, i.e.:
1 2 y
p
p1
p1p1
yp1p
z
No neighbor should elect itself
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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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) at most:
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?