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

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 2/18

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 3/18

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 4/18

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 5/18

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. 6/18

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 7/18

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 8/18

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 9/18

Proof of Rule 3 S D 1 2 … … r Testing CS-C1 edge is better than testing CS-C2 edge. S D C1 C2 10/18

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. 11/18

Proof of Rule 3 Stochastically couple edges under Agood and Abad. S D 1 Terminates earlier! 2 S D 1 2 12/18

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 13/18

Proof of Rule 2 One step earlier! 14/18

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! 15/18

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 16/18

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. 17/18

Thank you ! 18/18