Transcript MobiHoc09

Optimal Determination of Source-destination
Connectivity in Random Graphs
Luoyi Fu, Xinbing Wang, P. R. Kumar
Dept. of Electronic Engineering
Shanghai Jiao Tong University
Dept. of Electrical & Computer Engineering
Texas A&M University
Random Graph: G(n,p) Model




N nodes
Each edge exists with probability p
Proposed by Gilbert in 1959
It can also be called ER graph
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Are S and D Connected?
 Goal: Determine whether S and D are connected or not
 As quickly as possible
 i.e., by testing the fewest expected number of edges
S
D
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 Determined S-D connectivity in 6 number of edges
 By finding a path
edges tested
S
D
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 Determined S-D disconnectivity in 10 number of edges
 By finding a cut
edges tested
S
D
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S
D
S
D
 Sometimes, S and D may be connected.
 Sometimes, S and D may be disconnected.
 Termination time may be random.
 We want to determine whether S and D are connected or not
 By either finding a Path or a Cut
 By testing the fewest number of edges
 Quickest discovery of an S-D route has not been studied before.
 Finding a shortest path is not the goal here.
 Finding the shortest path is a well studied problem.
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The Optimal Policy: A Five-node Example
 Test the direct edge between S and D
 Test a potential edge between S and a randomly
chosen node
 Contract S and the node into a component if an
edge exists between them
 Test the direct edge between CS and D
 2 potential edges between nodes and D
 3 potential edges between nodes and CS
 Test an edge between D and a randomly chosen
node
 2 potential edges between node 2 and CS
 1 potential edges between node 3 and CS
 Test the edge between node 2 and D
 Similar rules in general
S
CS
D
CD
1
2
3
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The Optimal Policy: General Case
 Rule 1:
 Test if edge exists between CS and CD.
 Policy terminates if the edge exists.
n1
C1
n2
C2
…….
 Rule 2:
 List all the paths connecting CS to CD with the
minimum number of potential edges.
 Not CS-C1-C2-CD
 But CS-C1-CD
 Find Set M that contains the minimum potential
Cut between CS and CD.
M
 Rule 3:
n1
 Sharpen Rule 2 by specifying which particular
edge in M should be tested.
 Test any edges in M connecting CS to C1.
CD
CS
nr
Cr
 max n1, n2 ,
, nr 
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Proof of Rule 1: Test If the Direct Edge Exists
Testing the direct edge at the first step is better than testing
at the second step.
S
D
S
D
S
D
S
D
terminate
S
D
S
D
S
D
S
D
S
D
S
D
terminate
Terminate
one step
earlier!
Agood
Same
probability
Abad
Induction on the number of edges tested before the direct edge is
tested
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Proof of Rule 3
S
1
n1
2
n2
…
…
M
D
r
nr n1  max n1, n2 ,
, nr 
Testing CS-C1 edge is better than testing CS-C2 edge.
D
S
kS 1
C1
k1D
kS 2
C2
k2 D
k1D  k2 D
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Proof of Rule 3
Take the graph on the right as example.
Two policies:
Agood
D
S
1
3
1
2
Abad
D
S
D
S
D
S
C1
C1
C1
C1
C2
C2
C2
C2
Induction on the number of potential edges in
the graph.
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Proof of Rule 3
 Stochastically couple edges under Agood and Abad.
Agood
S
D

Agood

1



2
Terminates earlier!
Abad
S


D
Abad
1 

2

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Proof of Rule 2
Testing CS -C1 edge is better than testing C1-CD edge.
D
S
k11
C1
k12
In the set M
Stochastic coupling argument
Induction on the number of potential edges in the graph
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Proof of Rule 2
Agood
Abad
One step earlier!
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Phase Transition
1000 nodes
P~0: 999 edges from S
P~1: 1 edge to D
Phase transition: take a long time (around 15000 steps) to test
Our policy is optimal for all p!
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Extension to Slightly More General Graphs
 Series graphs
 Parallel graphs
 SP graphs
 PS graphs
 Series of parallel of series (SPS) graphs
 Parallel of series of parallel (PSP) graphs
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Concluding Remarks
 Whether ER are connected graphs is very well studied topic.
 Quickly testing connectivity is not.
 (Surprisingly)
 We provide the optimal testing algorithm.
 Optimal for all p.
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Thank you !
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