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Graph Sparsification by Effective Resistances

Graph Sparsification by Effective Resistances. Daniel Spielman Nikhil Srivastava Yale. Sparsification. Approximate any graph G by a sparse graph H . Nontrivial statement about G H is faster to compute with than G. G. H. Cut Sparsifiers [Benczur-Karger’96]. H approximates G if

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Graph Sparsification by Effective Resistances

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  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:

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