Download
graph sparsification by effective resistances n.
Skip this Video
Loading SlideShow in 5 Seconds..
Graph Sparsification by Effective Resistances PowerPoint Presentation
Download Presentation
Graph Sparsification by Effective Resistances

Graph Sparsification by Effective Resistances

339 Vues Download Presentation
Télécharger la présentation

Graph Sparsification by Effective Resistances

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  1. Graph Sparsification by Effective Resistances Daniel Spielman Nikhil Srivastava Yale

  2. Sparsification Approximate any graph G by a sparse graph H. • Nontrivial statement about G • H is faster to compute with than G G H

  3. Cut Sparsifiers [Benczur-Karger’96] H approximates G if for every cut S½V sum of weights of edges leaving S is preserved Can find H with O(nlogn/2) edges in time S S

  4. The Laplacian (quick review) Quadratic form Positive semidefinite Ker(LG)=span(1) if G is connected

  5. Cuts and the Quadratic Form For characteristic vector So BK says:

  6. A Stronger Notion For characteristic vector So BK says:

  7. Why?

  8. 1. All eigenvalues are preserved By Courant-Fischer, G and H have similar eigenvalues. For spectral purposes, G and H are equivalent.

  9. 1. All eigenvalues are preserved By Courant-Fischer, G and H have similar eigenvalues. For spectral purposes, G and H are equivalent. cf. matrix sparsifiers [AM01,FKV04,AHK05]

  10. 2. Linear System Solvers Conj. Gradient solves in ignore (time to mult. by A)

  11. 2. Preconditioning Find easy that approximates . Solve instead. Time to solve (mult.by )

  12. 2. Preconditioning Use B=LH ? Find easy that approximates . Solve instead. ? Time to solve (mult.by )

  13. 2. Preconditioning Spielman-Teng [STOC ’04] Nearly linear time. Find easy that approximates . Solve instead. Time to solve (mult.by )

  14. Examples

  15. Example: Sparsify Complete Graph by Ramanujan Expander G is complete on n vertices. H is d-regular Ramanujan graph.

  16. Example: Sparsify Complete Graph by Ramanujan Expander G is complete on n vertices. H is d-regular Ramanujan graph. So, is a good sparsifier for G. Each edge has weight (n/d)

  17. Example: Dumbell Kn Kn 1 d-regular Ramanujan, times n/d d-regular Ramanujan, times n/d 1

  18. G2 Kn Kn G1 G3 1 F2 d-regular Ramanujan, times n/d d-regular Ramanujan, times n/d 1 F1 F3 Example: Dumbell

  19. Example: Dumbell. Must include cut edge Kn Kn e Only this edge contributes to If

  20. Results

  21. Main Theorem Every G=(V,E,c) contains H=(V,F,d) with O(nlogn/2) edges such that:

  22. Main Theorem Every G=(V,E,c) contains H=(V,F,d) with O(nlogn/2) edges such that: Can find H in time by random sampling.

  23. Main Theorem Every G=(V,E,c) contains H=(V,F,d) with O(nlogn/2) edges such that: Can find H in time by random sampling. Improves [BK’96] Improves O(nlogc n) sparsifiers [ST’04]

  24. How? Electrical Flows.

  25. Effective Resistance Identify each edge of G with a unit resistor is resistance between endpoints of e 1 v u 1 1 a

  26. Effective Resistance Identify each edge of G with a unit resistor is resistance between endpoints of e 1 v u Resistance of path is 2 1 1 a

  27. Effective Resistance Identify each edge of G with a unit resistor is resistance between endpoints of e 1 v u Resistance from u to v is Resistance of path is 2 1 1 a

  28. Effective Resistance Identify each edge of G with a unit resistor is resistance between endpoints of e -1 +1 v u 1/3 -1/3 a 0

  29. Effective Resistance Identify each edge of G with a unit resistor is resistance between endpoints of e V +1 -1 = potential difference between endpoints when flow one unit from one endpoint to other

  30. Effective Resistance V +1 -1 [Chandra et al. STOC ’89]

  31. The Algorithm Sample edges of G with probability If chosen, include in H with weight Take q=O(nlogn/2) samples with replacement Divide all weights by q.

  32. An algebraic expression for Orient G arbitrarily.

  33. An algebraic expression for Orient G arbitrarily. Signed incidence matrix Bm£ n :

  34. An algebraic expression for Orient G arbitrarily. Signed incidence matrix Bm£ n : Write Laplacian as

  35. V +1 -1 An algebraic expression for

  36. An algebraic expression for Then

  37. An algebraic expression for Then

  38. An algebraic expression for Then Reduce thm. to statement about 

  39. Goal Want

  40. Sampling in 

  41. Reduction to  Lemma.

  42. New Goal Lemma.

  43. The Algorithm Sample edges of G with probability If chosen, include in H with weight Take q=O(nlogn/2) samples with replacement Divide all weights by q.

  44. The Algorithm Sample columns of with probability If chosen, include in with weight Take q=O(nlogn/2) samples with replacement Divide all weights by q.

  45. The Algorithm Sample columns of with probability If chosen, include in with weight Take q=O(nlogn/2) samples with replacement Divide all weights by q.

  46. The Algorithm Sample columns of with probability If chosen, include in with weight Take q=O(nlogn/2) samples with replacement Divide all weights by q.

  47. The Algorithm Sample columns of with probability If chosen, include in with weight Take q=O(nlogn/2) samples with replacement Divide all weights by q. cf. low-rank approx. [FKV04,RV07]

  48. A Concentration Result

  49. A Concentration Result So with prob. ½:

  50. A Concentration Result So with prob. ½: