1. Maximum Independent Set Minty's Algorithm as corrected by Nakamura et.al. parallelized

2. Why should you care? Maximum independent sets allow network scheduling algorithms to work well [1] Because its cool! NP hard but solvable in this case Other applications in: Computer vision Information theory Map labeling Pattern Recognition

3. So how does it work?[2][3] On a high level 'find augmenting paths' Have an ind set (black nodes) increase in size +1 'Clean' graph of 'useless' nodes Partition the white nodes next to black nodes Create an 'Edmond's graph' Fix graph as prescribed by Nakamura Run trees and blossoms from Edmond's

4. What is expensive serial? n=10x; p=50n=10x; p=10x Total Time n=10x; p=10x

5. How can you parallelize that? Embarrassingly parallel Issue is all data must be on all processors Parallelize graph cleaning Partition graph on 'black nodes' Run graph cleaning with minimal communication

6. How can you parallelize that? Trivially Parallel Find maximal ind set, superset of black nodes Every node must have the list of black nodes and the whole graph Improves more than the RBS does RBS % time Trivially Parallel max improvement

7. Future work Find efficient ways of removing claws from conflict graphs in ad-hoc wireless sensor networks Determine if the wireless network itself could be leveraged for efficiency (unrelated) Parallelize encoding/decoding for network coding

8. Citations [1] Scheduling for Network Coded Multicast: A Conflict Graph Formulation, IEEE, Danail Traskov, Michael Heindlmaier, Muriel M ́ dard, Ralf Koetter, and Desmond S. Lun [2] Minty's Algorithm http://www.sciencedirect.com/science/article/pii/009589568090074X [3] Edmond's http://ci.nii.ac.jp/naid/110001183942/

9. Questions? Thanks to Jeff and Dr.Edelman #Graphs by slide

10. More Mint! Key Idea: how can we make finding augmenting paths easier? Consider white nodes:

11. Even More Mint! So we have this way of generating paths from black to black nodes, we no longer need to consider white nodes except for the parity Now we can make black nodes lines where each end node represents a parity Now we just want to find an augmenting path of EDGES in this new graph...Edmond's!