1. Luoyi Fu, Xinbing Wang, P. R. Kumar
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

2. 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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3. 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
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4. 2 3 6 4 5 1 Determined S-D connectivity in 6 number of edges
By finding a path 2 3 6 4 5 1
edges tested
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5. Determined S-D disconnectivity in 10 number of edges
By finding a cut 10 2 9 7 3 6 4 8 5 1
edges tested
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6. 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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7. 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 CS S D CD 1 2 3
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8. The Optimal Policy: General Case
Rule 1:
Test if edge exists between CS and CD.
Policy terminates if the edge exists.
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.
Rule 3:
Sharpen Rule 2 by specifying which particular edge in M should be tested.
Test any edges in M connecting CS to C1. CS CD C1 C2
……. M Cr
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9. 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
terminate
Terminate one step earlier! S D S D S D
Same
probability
Induction on the number of edges tested before the direct edge is tested
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10. Proof of Rule 3 S D 1 2 … … r
Testing CS-C1 edge is better than testing CS-C2 edge. S D C1 C2
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11. Proof of Rule 3 S D Take the graph on the right as example. 1 1 3 1 2
Two policies: S D S D S D S D C1 C1 C1 C1 C2 C2 C2 C2
Induction on the number of potential edges in the graph.
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12. Proof of Rule 3 Stochastically couple edges under Agood and Abad. S D1
Terminates earlier! 2 S D 1 2
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13. Proof of Rule 2 Testing CS -C1 edge is better than testing C1-CD edge.
Stochastic coupling argument
Induction on the number of potential edges in the graph S D C1
In the set M
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14. Proof of Rule 2
One step earlier!
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15. Phase Transition 1000 nodes P~0: 999 edges from S P~1: 1 edge to D
Phase transition: take a long time (around steps) to test
Our policy is optimal for all p!
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16. 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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17. 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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18. Thank you !
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