Sampling in Graphs node sparsifiers Alexandr Andoni Microsoft
![Small cut (node) sparsifiers [A-Gupta-Krauthgamer’ 14] •](https://slidetodoc.com/presentation_image_h/b619408bee23d28f45b65c00782cda2a/image-7.jpg "Small cut (node) sparsifiers [A-Gupta-Krauthgamer’ 14] •")
- Slides: 19
Sampling in Graphs: node sparsifiers Alexandr Andoni (Microsoft Research)
Graph compression Why smaller graphs? • use less storage space • faster algorithms • easier visualization
Sparsification of edges • Preserve some structure: e. g. , cuts • Also: distances, effective resistances, etc
Sparsification of nodes ? •
Node sparsifiers •
Results on cut sparsifiers Graph size Approximation Reference Comments [Moi 09, LM 10, CLLM 10, EGKRTCT 10, MM 10] [LM 10, CLLM 10, MM 10] [Chu 12, KW 12] [HKNR 98, KRTV 12] [KRTV 12, KR 13] bipartite* graphs [AGK’ 14] bipartite* graphs • Similar results for flow (node) sparsifier
Small cut (node) sparsifiers [A-Gupta-Krauthgamer’ 14] •
Main idea ? • Sampling edges doesn’t work here • Need to sample entire sub-structures of the graph
Sampling in Bipartite Graphs • Sample non-terminals, together with edges • reweight edges accordingly
Sampling in Bipartite Graphs • Sample non-terminals, together with edges • reweight edges accordingly • Uniform sampling doesn’t work
Non-uniform sampling •
Tool: Importance sampling •
Importance sampling •
Actual Sampling •
Checking importance sampling •
Flow (node) sparsifiers •
Remarks •
Graph compression via sampling •