Sampling in Graphs Alexandr Andoni (Microsoft
Description: Sampling in Graphs Alexandr Andoni (Microsoft Research) Graph compression Why smaller graphs? use less storage space faster algorithms easier visualization Preserve some structure Cuts approximately Other properties: Distances,
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slide1. Sampling in Graphs Alexandr Andoni
(Microsoft Research)<br>
slide2. Graph compression Why smaller graphs?
use less storage space
faster algorithms
easier visualization<br>
slide3. Preserve some structure
Cuts
approximately
Other properties:
Distances, (multi-commodity) flows, effective resistances…<br>
slide4. Plan 1) Cut sparsifiers
2) More efficient cut sparsifiers
3) Node sparsifiers<br>
slide5. Cut sparsifiers<br>
slide6. Approach? [Karger’94,’96]:<br>
slide9. Concentration<br>
slide10. Applying Chernoff bound<br>
slide11. Enough?<br>
slide12. Smaller size?<br>
slide13. Non-uniform sampling [Benczur-Karger’96]<br>
slide14. Strong connectivity Connectivity: 5
Strong conn.: 2<br>
slide15. Proof of theorem<br>
slide16. ii) Cut values are approximated<br>
slide17. Iterative sampling<br>
slide18. Comments<br>
slide19. BREAK<br>
slide21. Smaller relaxed cut sparsifiers [A-Krauthgamer-Woodruff’14]:<br>
slide22. Motivating example<br>
slide23. Proof of theorem<br>
slide24. i) Sketch description<br>
slide25. ii) Sketch size ???<br>
slide26. iii) Estimation<br>
slide27. Estimation illustration dense components<br>
slide28. iii) Correctness of estimation<br>
slide30. Variance<br>
slide31. Dense component estimate<br>
slide32. Concluding remarks<br>
slide33. Open questions<br>
(Microsoft Research)<br>
slide2. Graph compression Why smaller graphs?
use less storage space
faster algorithms
easier visualization<br>
slide3. Preserve some structure
Cuts
approximately
Other properties:
Distances, (multi-commodity) flows, effective resistances…<br>
slide4. Plan 1) Cut sparsifiers
2) More efficient cut sparsifiers
3) Node sparsifiers<br>
slide5. Cut sparsifiers<br>
slide6. Approach? [Karger’94,’96]:<br>
slide9. Concentration<br>
slide10. Applying Chernoff bound<br>
slide11. Enough?<br>
slide12. Smaller size?<br>
slide13. Non-uniform sampling [Benczur-Karger’96]<br>
slide14. Strong connectivity Connectivity: 5
Strong conn.: 2<br>
slide15. Proof of theorem<br>
slide16. ii) Cut values are approximated<br>
slide17. Iterative sampling<br>
slide18. Comments<br>
slide19. BREAK<br>
slide21. Smaller relaxed cut sparsifiers [A-Krauthgamer-Woodruff’14]:<br>
slide22. Motivating example<br>
slide23. Proof of theorem<br>
slide24. i) Sketch description<br>
slide25. ii) Sketch size ???<br>
slide26. iii) Estimation<br>
slide27. Estimation illustration dense components<br>
slide28. iii) Correctness of estimation<br>
slide30. Variance<br>
slide31. Dense component estimate<br>
slide32. Concluding remarks<br>
slide33. Open questions<br>