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Optimal Determination of Source-destination Connectivity in Random Graphs

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. N nodes Each edge exists with probability p

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Optimal Determination of Source-destination Connectivity in Random Graphs

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  1. Optimal Determination of Source-destinationConnectivity 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. N nodes • Each edge exists with probability p • Proposed by Gilbert in 1959 • Called ER graph Random Graph: G(n,p) Model 2/19

  3. Goal: Determine whether S and D are connected or not • As quickly as possible • I.e., by testing the fewest expected number of edges Are S and D Connected? 3/19

  4. Determined S-D connectivity in 6 edges • By finding a path 4 2 3 6 1 5 edges tested 4/19

  5. Determined S-D disconnectivity in 10 edges • By finding a cut 7 8 3 6 5 4 1 2 edges tested 5/19

  6. S S D 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. 6/19

  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 edge between node 3 and CS • Based on counting edges (described in next slide) • Test the edge between node 2 and D CS D S CD 1 2 3 7/19

  8. Rule 1: • Test if direct edge exists between CS and CD. • Policy terminates if the direct 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 Mconnecting CS to C1. CD CS C1 The Optimal Policy: General Case C2 ……. M Cr 8/19

  9. Testing the direct edge at the first step is better than testing at the second step. S D S D S D S D Proof of Rule 1: Test If the Direct Edge Exists 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 9/19

  10. Proof of Rule 3 S D 1 2 … … r D S C1 C2 • Testing CS-C1 edge is better than testing CS-C2 edge. 10/19

  11. Proof of Rule 3 S D • Example: 1 1 3 1 2 2 Two policies: D S D S D S D S C1 C1 C1 C1 C2 C2 C2 C2 • Induction on the number of potential edges in the graph. 11/19

  12. Proof of Rule 3 • Stochastically couple edges under Agood and Abad. S D 1 2 Terminates earlier! S D 1 2 12/19

  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 D S C1 In the set M 13/19

  14. Proof of Rule 2 One step earlier! 14/19

  15. Phase Transition • 1000 nodes • Low values of p: 999 edges from S to establish disconnectedness • High values of p: 1 edge from S to D to establish connectedness • At Phase transition: Take a long time (around 15000 steps) to test • Algorithm is try to check both connectivity and disconnectivity • Our policy is optimal for all p! connected disconnected 15/19

  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 17/19

  17. Whether ER graphs are connected is very well studied topic. • Efficiently testing connectivity is not. • (Surprisingly) • We provide the optimal testing algorithm. • Optimal for all p and all N. Concluding Remarks 18/19

  18. Thank you ! 19/19

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