Sampling in Graphs Alexandr Andoni Microsoft Research Graph

  • Slides: 32
Download presentation
Sampling in Graphs Alexandr Andoni (Microsoft Research)

Sampling in Graphs Alexandr Andoni (Microsoft Research)

Graph compression Why smaller graphs? • use less storage space • faster algorithms •

Graph compression Why smaller graphs? • use less storage space • faster algorithms • easier visualization

• Preserve some structure • Cuts • approximately • Other properties: • Distances,

• Preserve some structure • Cuts • approximately • Other properties: • Distances, (multi-commodity) flows, effective resistances…

Plan 1) Cut sparsifiers 2) More efficient cut sparsifiers 3) Node sparsifiers

Plan 1) Cut sparsifiers 2) More efficient cut sparsifiers 3) Node sparsifiers

Cut sparsifiers •

Cut sparsifiers •

Approach? [Karger’ 94, ’ 96]: •

Approach? [Karger’ 94, ’ 96]: •

Concentration •

Concentration •

Applying Chernoff bound •

Applying Chernoff bound •

Enough? •

Enough? •

Smaller size? •

Smaller size? •

Non-uniform sampling [Benczur-Karger’ 96] •

Non-uniform sampling [Benczur-Karger’ 96] •

Strong connectivity • Connectivity: 5 Strong conn. : 2

Strong connectivity • Connectivity: 5 Strong conn. : 2

Proof of theorem •

Proof of theorem •

ii) Cut values are approximated •

ii) Cut values are approximated •

Iterative sampling •

Iterative sampling •

Comments •

Comments •

BREAK

BREAK

Smaller relaxed cut sparsifiers [A-Krauthgamer-Woodruff’ 14]: •

Smaller relaxed cut sparsifiers [A-Krauthgamer-Woodruff’ 14]: •

Motivating example •

Motivating example •

Proof of theorem •

Proof of theorem •

i) Sketch description •

i) Sketch description •

ii) Sketch size • ? ? ?

ii) Sketch size • ? ? ?

iii) Estimation •

iii) Estimation •

Estimation illustration • dense components

Estimation illustration • dense components

iii) Correctness of estimation •

iii) Correctness of estimation •

Variance •

Variance •

Concluding remarks •

Concluding remarks •

Open questions •

Open questions •