Optimal Determination of Source-Optimal Determination of Source-destinationdestination

Connectivity in Random Graphs Connectivity in Random Graphs

Luoyi Fu, Xinbing Wang, P. R. Kumar Dept. of Electronic EngineeringShanghai Jiao Tong University

Dept. of Electrical & Computer EngineeringTexas A&M University

Random Graph: G(n,p) ModelRandom Graph: G(n,p) Model N nodesN nodes Each edge exists with probability pEach edge exists with probability p Proposed by Gilbert in 1959Proposed by Gilbert in 1959 It can also be called ER graphIt can also be called ER graph

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Are S and D Connected?Are S and D Connected? Goal: Determine whether S and D are connected or notGoal: Determine whether S and D are connected or not

As quickly as possibleAs quickly as possible

i.e., by testing the fewest expected number of edgesi.e., by testing the fewest expected number of edges

S

D

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Determined S-D connectivity in 6 number of edgesDetermined S-D connectivity in 6 number of edges

By finding a pathBy finding a path

S

D

edges testededges tested

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Determined S-D disconnectivity in 10 number of edgesDetermined S-D disconnectivity in 10 number of edges

By finding a cutBy finding a cut

S

D

edges testededges tested

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SS DD SS DD

Sometimes, S and D may be connected.Sometimes, S and D may be connected. Sometimes, S and D may be disconnected.Sometimes, S and D may be disconnected.

Termination time may be random.Termination time may be random.

We want to determine whether S and D are connected or not We want to determine whether S and D are connected or not By either finding a Path or a CutBy either finding a Path or a Cut By testing the fewest number of edgesBy testing the fewest number of edges

Quickest discovery of an S-D route has not been studied before.Quickest discovery of an S-D route has not been studied before.

Finding a shortest path is not the goal here.Finding a shortest path is not the goal here. Finding the shortest path is a well studied problem.Finding the shortest path is a well studied problem.

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Test the direct edge between S and DTest the direct edge between S and D

Test a potential edge between S and a randomly Test a potential edge between S and a randomly chosen nodechosen node Contract S and the node into a component if an Contract S and the node into a component if an

edge exists between themedge exists between them Test the direct edge between CTest the direct edge between CSS and D and D

2 potential edges between nodes and D2 potential edges between nodes and D 3 potential edges between nodes and C3 potential edges between nodes and CSS

Test an edge between D and a randomly chosen Test an edge between D and a randomly chosen nodenode 2 potential edges between node 2 and CS2 potential edges between node 2 and CS 1 potential edges between node 3 and CS1 potential edges between node 3 and CS Test the edge between node 2 and DTest the edge between node 2 and D

Similar rules in generalSimilar rules in general

The Optimal Policy: A Five-node ExampleThe Optimal Policy: A Five-node Example

11

33

22

SS DDCCSS

CCDD

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The Optimal Policy: General CaseThe Optimal Policy: General Case

……

……

..

Rule 1Rule 1: : Test if edge exists between CTest if edge exists between CSS and C and CD.D.

Policy terminates if the edge exists.Policy terminates if the edge exists.

Rule 2: Rule 2: List all the paths connecting CList all the paths connecting CSS to C to CDD with the with the

minimum number of potential edges.minimum number of potential edges. Not CNot CSS-C-C11-C-C22-C-CDD

But CBut CSS-C-C11-C-CDD

Find Set M that contains the minimum potential Find Set M that contains the minimum potential Cut between CCut between CSS and C and CD.D.

Rule 3: Rule 3: Sharpen Rule 2 by specifying which particular Sharpen Rule 2 by specifying which particular

edge in M should be tested.edge in M should be tested. Test any edges in MTest any edges in M connecting Cconnecting CSS to C to C11..

CCSS CCDD

CC11

CC22

CCrrMM

1n

2n

rn

1 1 2max , , , rn n n n

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Proof of Rule 1: Test If the Direct Edge ExistsProof of Rule 1: Test If the Direct Edge Exists

Testing the direct edge at the first step is better than testing Testing the direct edge at the first step is better than testing at the second step.at the second step.

badAgoodA

SS DD SS DD

SS DD SS DD

SS DD SS DD

SS DD SS DD

SS DD

SS DD

terminateterminate

Same Same probabilityprobability

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Induction on the number of edges tested before the direct edge is Induction on the number of edges tested before the direct edge is testedtested

terminateterminate

Terminate Terminate one step one step earlier!earlier!

Proof of Rule 3Proof of Rule 3

Testing CTesting CSS-C-C11 edge is better than testing C edge is better than testing CSS-C-C22 edge. edge.

SS DD

CC11

CC22

1Dk

2Dk

1Sk

2Sk1 2D Dk k

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SS DD

11

rr

22

…

1n

2n

rn

…

1 1 2max , , , rn n n n

M

Proof of Rule 3Proof of Rule 3

Take the graph on the right as example.Take the graph on the right as example.

Two policies:Two policies:

badAgoodA

SS DD

CC11

SS DD

CC11

SS DD

CC11

SS DD

CC11

CC22 CC22

11

11 22

33

CC22 CC22

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Induction on the number of potential edges in Induction on the number of potential edges in the graph.the graph.

Proof of Rule 3Proof of Rule 3

Stochastically couple edges under AStochastically couple edges under Agoodgood and A and Abadbad..

goodA

badA

SS DD

11

22

SS DD

11

22

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goodA

badA

Terminates earlier!Terminates earlier!

Testing CS -C1 edge is better than testing C1-CD edge.

Stochastic coupling argument

Induction on the number of potential edges in the graph

Proof of Rule 2Proof of Rule 2

DD

CC11 12k11k

In the set MIn the set M

SS

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Proof of Rule 2Proof of Rule 2

goodA

badA

One step earlier!One step earlier!

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Phase TransitionPhase Transition

1000 nodes1000 nodes

P~0: 999 edges from SP~0: 999 edges from S

P~1: 1 edge to DP~1: 1 edge to D

Phase transition: take a long time (around 15000 steps) to testPhase transition: take a long time (around 15000 steps) to test

Our policy is optimal for all p!Our policy is optimal for all p!

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Extension to Slightly More General GraphsExtension to Slightly More General Graphs

Series graphsSeries graphs

Parallel graphsParallel graphs

SP graphsSP graphs

PS graphsPS graphs

Series of parallel of series (SPS) graphsSeries of parallel of series (SPS) graphs

Parallel of series of parallel (PSP) graphsParallel of series of parallel (PSP) graphs

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Concluding RemarksConcluding Remarks

Whether ER are connected graphs is very well studied topic.Whether ER are connected graphs is very well studied topic.

Quickly testing connectivity is not.Quickly testing connectivity is not. (Surprisingly)(Surprisingly)

We provide the optimal testing algorithm.We provide the optimal testing algorithm.

Optimal for all p.Optimal for all p.

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Thank you !Thank you !

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