Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families

Transcription

1 Graph Sparsification III: Ramanujan Graphs, Lifts, and Interlacing Families Nikhil Srivastava Microsoft Research India Simons Institute, August 27, 2014

2 The Last Two Lectures Lecture 1. Every weighted undirected G has a weighted subgraph H with O edges n log n ε 2 which satisfies L G L H 1 + ε L G random sampling

3 The Last Two Lectures Lecture 1. Every weighted undirected G has a weighted subgraph H with O edges n log n ε 2 which satisfies L G L H 1 + ε L G Lecture 2. Improved this to 4 n ε 2. greedy

4 The Last Two Lectures Lecture 1. Every weighted undirected G has a weighted subgraph H with O edges n log n ε 2 which satisfies L G L H 1 + ε L G Lecture 2. Improved this to 4 n ε 2. Suboptimal for K n in two ways: weights, and 2n/ε 2.

5 Good Sparsifiers of K n G=K n H = random d-regular x (n/d) E G = O(n 2 ) E H = O(dn) L Kn (1 ε) L H L Kn 1 + ε

6 Good Sparsifiers of K n G=K n H = random d-regular x (n/d) E G = O(n 2 ) E H = O(dn) n(1 ε) L H n 1 + ε

7 Regular Unweighted Sparsifiers of K n G=K n Rescale weights back to 1 H = random d-regular E G = O(n 2 ) E H = dn/2 d(1 ε) L H d 1 + ε

8 Regular Unweighted Sparsifiers of K n G=K n H = random d-regular E G = O(n 2 ) E H = dn/2 [Friedman 08] d 2 d 1 L H d + 2 d 1 +o(1)

9 Regular Unweighted Sparsifiers of K n G=K n H = random d-regular will try to match this E G = O(n 2 ) E H = dn/2 [Friedman 08] d 2 d 1 L H d + 2 d 1 +o(1)

10 Why do we care so much about K n? Unweighted d-regular approximations of K n are called expanders. They behave like random graphs: the right # edges across cuts fast mixing of random walks Prototypical pseudorandom object. Many uses in CS and math (Routing, Coding, Complexity )

11 Switch to Adjacency Matrix Let G be a graph and A be its adjacency matrix b c a d e eigenvalues λ 1 λ 2 λ n L = di A

12 Switch to Adjacency Matrix Let G be a graph and A be its adjacency matrix b c a d e eigenvalues λ 1 λ 2 λ n If d-regular, then A1 = d1 so If bipartite then eigs are symmetric about zero so λ 1 = d λ n = d trivial

13 Spectral Expanders Definition: G is a good expander if all non-trivial eigenvalues are small 0 [ ] -d d

14 Spectral Expanders Definition: G is a good expander if all non-trivial eigenvalues are small [ ] -d 0 d e.g. K d and K d,d have all nontrivial eigs 0.

15 Spectral Expanders Definition: G is a good expander if all non-trivial eigenvalues are small [ ] -d 0 d e.g. K d and K d,d have all nontrivial eigs 0. Challenge: construct infinite families.

16 Spectral Expanders Definition: G is a good expander if all non-trivial eigenvalues are small [ ] -d 0 d e.g. K d and K d,d have all nontrivial eigs 0. Challenge: construct infinite families. Alon-Boppana 86: Can t beat [ 2 d 1, 2 d 1]

17 The meaning of 2 d 1 The infinite d-ary tree λ A T = [ 2 d 1, 2 d 1]

18 The meaning of 2 d 1 The infinite d-ary tree λ A T = [ 2 d 1, 2 d 1] Alon-Boppana 86: This is the best possible spectral expander.

19 Ramanujan Graphs: Definition: G is Ramanujan if all non-trivial eigs have absolute value at most 2 d 1 [ [ ] ] -d 0 d -2 d 1 2 d 1

20 Ramanujan Graphs: Definition: G is Ramanujan if all non-trivial eigs have absolute value at most 2 d 1 [ [ ] ] -d 0 d -2 d 1 2 d 1 Friedman 08: A random d-regular graph is almost Ramanujan : 2 d 1 + o(1)

21 Ramanujan Graphs: Definition: G is Ramanujan if all non-trivial eigs have absolute value at most 2 d 1 [ [ ] ] -d 0 d -2 d 1 2 d 1 Friedman 08: A random d-regular graph is almost Ramanujan : 2 d 1 + o(1) Margulis, Lubotzky-Phillips-Sarnak 88: Infinite sequences of Ramanujan graphs exist for d = p + 1

22 Ramanujan Graphs: Definition: G is Ramanujan if all non-trivial eigs have absolute value at most 2 d 1 [ [ ] ] -d 0 d -2 d 1 2 d 1 Friedman 08: A random d-regular What graph about is almost Ramanujan : 2 d 1 + o(1) d p + 1? Margulis, Lubotzky-Phillips-Sarnak 88: Infinite sequences of Ramanujan graphs exist for d = p + 1

23 [Marcus-Spielman-S 13] Theorem. Infinite families of bipartite Ramanujan graphs exist for every d 3.

24 [Marcus-Spielman-S 13] Theorem. Infinite families of bipartite Ramanujan graphs exist for every d 3. Proof is elementary, doesn t use number theory. Not explicit. Based on a new existence argument: method of interlacing families of polynomials.

25 [Marcus-Spielman-S 13] Theorem. Infinite families of bipartite Ramanujan graphs exist for every d 3. Ep k = 1 1 k x n m x Proof is elementary, doesn t use number theory. Not explicit. Based on a new existence argument: method of interlacing families of polynomials.

26 Bilu-Linial 06 Approach Find an operation which doubles the size of a graph without blowing up its eigenvalues. [ [ ] ] -d 0 d -2 d 1 2 d 1

27 Bilu-Linial 06 Approach Find an operation which doubles the size of a graph without blowing up its eigenvalues. [ [ ] ] -d 0 d -2 d 1 2 d 1

28 Bilu-Linial 06 Approach Find an operation which doubles the size of a graph without blowing up its eigenvalues. [ [ ] ] -d 0 d -2 d 1 2 d 1

29 2-lifts of graphs b c a d e

30 2-lifts of graphs a a b b c e c e d d duplicate every vertex

31 2-lifts of graphs a 0 a 1 b 0 b 1 c 0 e 0 c 1 e 1 d 0 d 1 duplicate every vertex

32 2-lifts of graphs a 0 a 1 b 0 b 1 c 0 e 0 c 1 e 1 d 0 d 1 for every pair of edges: leave on either side (parallel), or make both cross

33 2-lifts of graphs a 0 a 1 b 0 b 1 c 0 e 0 c 1 e 1 d 0 d 1 for every pair of edges: leave on either side (parallel), or make both cross

34 2-lifts of graphs 2 m possibilities a 0 a 1 b 0 b 1 c 0 e 0 c 1 e 1 d 0 d 1 for every pair of edges: leave on either side (parallel), or make both cross

35 Eigenvalues of 2-lifts (Bilu-Linial) Given a 2-lift of G, create a signed adjacency matrix A s with a -1 for crossing edges and a 1 for parallel edges b 0 c 0 a 0 e 0 b 1 c 1 a 1 e d 0 d 1

36 Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are: λ 1,, λ n = eigs A λ 1 λ n = eigs(a s ) A s =

37 Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new) Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues have absolute value at most

38 Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new) Conjecture: Every d-regular adjacency matrix A has a signing A s with A S 2 d 1

39 Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new) Conjecture: Every d-regular adjacency matrix A has a signing A s with A S 2 d 1 Bilu-Linial 06: This is true with O( d log 3 d)

40 Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d-regular adjacency matrix A has a signing A s with A S 2 d 1 We prove this in the bipartite case.

41 Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d-regular adjacency matrix A has a signing A s with λ 1 (A S ) 2 d 1

42 Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d-regular bipartite adjacency matrix A has a signing A s with A S 2 d 1 Trick: eigenvalues of bipartite graphs are symmetric about 0, so only need to bound largest

43 Random Signings Idea 1: Choose s 1,1 m randomly.

44 Random Signings Idea 1: Choose s 1,1 m randomly. Unfortunately, (Bilu-Linial showed A is nearly Ramanujan ) when

45 Random Signings Idea 2: Observe that where Consider Usually useless, but not here! is an interlacing family. such that

46 Random Signings Idea 2: Observe that where Consider Usually useless, but not here! is an interlacing family. such that

47 Random Signings Idea 2: Observe that where Consider Usually useless, but not here! coefficient-wise average such that is an interlacing family.

48 Random Signings Idea 2: Observe that where Consider Usually useless, but not here! is an interlacing family. such that

49 3-Step Proof Strategy 1. Show that some poly does as well as the. such that

50 NOT λ max χ As Eλ max (χ As ) 3-Step Proof Strategy 1. Show that some poly does as well as the. such that

51 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial.

52 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. 3. Bound the largest root of the expected poly.

53 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. 3. Bound the largest root of the expected poly.

54 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. 3. Bound the largest root of the expected poly.

55 Step 2: The expected polynomial Theorem [Godsil-Gutman 81] For any graph G, the matching polynomial of G

56 The matching polynomial (Heilmann-Lieb 72) m i = the number of matchings with i edges

57

58 one matching with 0 edges

59 7 matchings with 1 edge

60

61

62 Proof that Expand using permutations x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x

63 Proof that Expand using permutations same edge: same value x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x

64 Proof that Expand using permutations same edge: same value x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x

65 Proof that Expand using permutations Get 0 if hit any 0s x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x

66 Proof that Expand using permutations x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x Get 0 if take just one entry for any edge

67 Proof that Expand using permutations x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x Only permutations that count are involutions

68 Proof that Expand using permutations x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x Only permutations that count are involutions

69 Proof that Expand using permutations x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x Only permutations that count are involutions Correspond to matchings

70 Proof that Expand using permutations x ±1 0 0 ±1 ±1 ±1 x ± ±1 x ± ±1 x ±1 0 ±1 0 0 ±1 x ±1 ± ±1 x Only permutations that count are involutions Correspond to matchings

71 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. [Godsil-Gutman 81] 3. Bound the largest root of the expected poly.

72 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. [Godsil-Gutman 81] 3. Bound the largest root of the expected poly.

73 The matching polynomial (Heilmann-Lieb 72) Theorem (Heilmann-Lieb) all the roots are real

74 The matching polynomial (Heilmann-Lieb 72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most

75 The matching polynomial (Heilmann-Lieb 72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most The number 2 d 1 comes by comparing to an infinite d ary tree [Godsil].

76 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. [Godsil-Gutman 81] 3. Bound the largest root of the expected poly. [Heilmann-Lieb 72]

77 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. [Godsil-Gutman 81] 3. Bound the largest root of the expected poly. [Heilmann-Lieb 72]

78 3-Step Proof Strategy 1. Show that some poly does as well as the. such that Implied by: is an interlacing family.

79 Averaging Polynomials Basic Question: Given when are the roots of the related to roots of?

80 Averaging Polynomials Basic Question: Given when are the roots of the related to roots of? Answer: Certainly not always

81 Averaging Polynomials Basic Question: Given when are the roots of the related to roots of? Answer: Certainly not always p x = x 1 x 2 = x 2 3x + 2 q x = x 3 x 4 = x 2 7x + 12 x i x 2.5 3i = x 2 5x + 7

82 Averaging Polynomials Basic Question: Given when are the roots of the related to roots of? But sometimes it works:

83 A Sufficient Condition Basic Question: Given when are the roots of the related to roots of? Answer: When they have a common interlacing. Definition. interlaces if

84 Theorem. If interlacing, have a common

85 Theorem. If interlacing, have a common Proof.

86 Theorem. If interlacing, have a common Proof.

87 Theorem. If interlacing, have a common Proof.

88 Theorem. If interlacing, have a common Proof.

89 Theorem. If interlacing, have a common Proof.

90 Theorem. If interlacing, have a common Proof.

91 Interlacing Family of Polynomials Definition: is an interlacing family if can be placed on the leaves of a tree so that when every node is the sum of leaves below & sets of siblings have common interlacings

92 Interlacing Family of Polynomials Definition: is an interlacing family if can be placed on the leaves of a tree so that when every node is the sum of leaves below & sets of siblings have common interlacings

93 Interlacing Family of Polynomials Definition: is an interlacing family if can be placed on the leaves of a tree so that when every node is the sum of leaves below & sets of siblings have common interlacings

94 Interlacing Family of Polynomials Definition: is an interlacing family if can be placed on the leaves of a tree so that when every node is the sum of leaves below & sets of siblings have common interlacings

95 Interlacing Family of Polynomials Definition: is an interlacing family if can be placed on the leaves of a tree so that when every node is the sum of leaves below & sets of siblings have common interlacings

96 Interlacing Family of Polynomials Theorem: There is an s so that

97 Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of, has

98 Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of, has

99 Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of, has

100 Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of, has.

101 Interlacing Family of Polynomials Theorem: There is an s so that Proof: By common interlacing, one of, has.

102 An interlacing family Theorem: Let is an interlacing family

103 To prove interlacing family Let Leaves of tree = signings s 1,, s m Internal nodes = partial signings s 1,, s k

104 To prove interlacing family Let Leaves of tree = signings s 1,, s m Internal nodes = partial signings s 1,, s k Need to find common interlacing for every internal node

105 How to Prove Common Interlacing Lemma (Fisk 08, folklore): Suppose p(x) and q(x) are monic and real-rooted. Then: a common interlacing r of p and q convex combinations, αp + 1 α q has real roots.

106 To prove interlacing family Let Need to prove that for all, is real rooted

107 To prove interlacing family Let Need to prove that for all, is real rooted are fixed is 1 with probability, -1 with are uniformly

108 Generalization of Heilmann-Lieb Suffices to prove that is real rooted for every product distribution on the entries of s

109 Generalization of Heilmann-Lieb Suffices to prove that is real rooted for every product distribution on the entries of s:

110 Transformation to PSD Matrices Suffices to show real rootedness of

111 Transformation to PSD Matrices Suffices to show real rootedness of Why is this useful?

112 Transformation to PSD Matrices Suffices to show real rootedness of Why is this useful?

113 Transformation to PSD Matrices

114 Transformation to PSD Matrices E s det xi di A s = Edet xi v ij v ij T ij E where v ij = δ i δ j with probability λ ij δ i + δ j with probability (1 λ ij )

115 Master Real-Rootedness Theorem Given any independent random vectors v 1,, v m R d, their expected characteristic polymomial Edet has real roots. xi i v i v i T

116 Master Real-Rootedness Theorem Given any independent random vectors v 1,, v m R d, their expected characteristic polymomial Edet xi i v i v i T has real roots. How to prove this?

117 The Multivariate Method A. Sokal, 90 s-2005: it is often useful to consider the multivariate polynomial even if one is ultimately interested in a particular one-variable specialization Borcea-Branden 2007+: prove that univariate polynomials are real-rooted by showing that they are nice transformations of real-rooted multivariate polynomials.

118 Real Stable Polynomials Definition. p R x 1,, x n is real stable if every univariate restriction in the strictly positive orthant: p t f x + t y y > 0 is real-rooted. If it has real coefficients, it is called real stable.

119 Real Stable Polynomials Definition. p C x 1,, x n is real stable if every univariate restriction in the strictly positive orthant: p t f x + t y y > 0 is real-rooted.

120 Real Stable Polynomials Definition. p R x 1,, x n is real stable if every univariate restriction in the strictly positive orthant: p t f x + t y y > 0 is real-rooted.

121 Real Stable Polynomials Definition. p R x 1,, x n is real stable if every univariate restriction in the strictly positive orthant: p t f x + t y y > 0 is real-rooted.

122 Real Stable Polynomials Definition. p R x 1,, x n is real stable if every univariate restriction in the strictly positive orthant: p t f x + t y y > 0 is real-rooted.

123 Real Stable Polynomials Definition. p R x 1,, x n is real stable if every univariate restriction in the strictly positive orthant: p t f x + t y y > 0 is real-rooted. Not positive

124 A Useful Real Stable Poly Borcea-Brändén 08: For PSD matrices is real stable

125 A Useful Real Stable Poly Borcea-Brändén 08: For PSD matrices is real stable Proof: Every positive univariate restriction is the characteristic polynomial of a symmetric matrix. det x i A i + t y i A i = det(ti + S) i i

126 Excellent Closure Properties Definition: is real stable if Implies. for all i If is real stable, then so is 1. p(α, z 2,, z n ) for any α R 2. 1 zi p(z 1, z n ) [Lieb-Sokal 81]

127 A Useful Real Stable Poly Borcea-Brändén 08: For PSD matrices is real stable Plan: apply closure properties to this to show that Edet xi i v i v i T is real stable.

128 Central Identity Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0

129 Central Identity Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0 Key Principle: random rank one updates (1 z ) operators.

130 Proof of Master Real- Rootedness Theorem Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0

131 Proof of Master Real- Rootedness Theorem Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0

132 Proof of Master Real- Rootedness Theorem Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0

133 Proof of Master Real- Rootedness Theorem Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0

134 Proof of Master Real- Rootedness Theorem Suppose v 1,, v m are independent random vectors with A i Ev i v i T. Then Edet xi v i v i T = m i=1 i 1 z i det xi + i z i A i z 1 = =z m =0

135 The Whole Proof Edet xi i v i v i T is real-rooted for all indep. v i.

136 The Whole Proof rank one structure naturally reveals interlacing. Eχ As (d x) is real-rooted for all product distributions on signings. Edet xi i v i v i T is real-rooted for all indep. v i.

137 The Whole Proof Eχ As (x) is real-rooted for all product distributions on signings. Edet xi i v i v i T is real-rooted for all indep. v i.

138 The Whole Proof χ As x x ±1 m is an interlacing family Eχ As (x) is real-rooted for all product distributions on signings. Edet xi i v i v i T is real-rooted for all indep. v i.

139 The Whole Proof such that χ As x x ±1 m is an interlacing family Eχ As (x) is real-rooted for all product distributions on signings. Edet xi i v i v i T is real-rooted for all indep. v i.

140 3-Step Proof Strategy 1. Show that some poly does as well as the. such that 2. Calculate the expected polynomial. 3. Bound the largest root of the expected poly.

141 Infinite Sequences of Bipartite Ramanujan Graphs Find an operation which doubles the size of a graph without blowing up its eigenvalues. [ [ ] ] -d 0 d -2 d 1 2 d 1

142 Main Theme Reduced the existence of a good matrix to: 1. Proving real-rootedness of an expected polynomial. 2. Bounding roots of the expected polynomial.

143 Main Theme Reduced the existence of a good matrix to: 1. Proving real-rootedness of an expected polynomial. (rank-1 structure + real stability) 2. Bounding roots of the expected polynomial. (matching poly + combinatorics)

144 Beyond complete graphs Unweighted sparsifiers of general graphs?

145 Beyond complete graphs G H 1 O(n)

146 Weights are Required in General G This edge has high resistance. H 1 O(n)

147 What if all edges are equally important? G?

148 Unweighted Decomposition Thm. G H 1 H 2 Theorem [MSS 13]: If all edges have resistance O(n/m), there is a partition of G into unweighted 1 + εsparsifiers, each with O n ε 2 edges.

149 Unweighted Decomposition Thm. G H 1 H 2 Theorem [MSS 13]: If all edges have resistance α, there is a partition of G into unweighted O(1)- sparsifiers, each with O mα edges.

150 Unweighted Decomposition Thm. G H 1 H 2 Theorem [MSS 13]: If all edges have resistance α, there is a partition of G into two unweighted 1 + αapproximations, each with half as many edges.

151 Unweighted Decomposition Thm. v i v i T I 2 i i v i v i T = I i v i v i T I 2 Theorem [MSS 13]: Given any vectors i v i v T i = I and ε, there is a partition into approximately v i 1/2-spherical quadratic forms, each I 2 ± O ε.

152 Unweighted Decomposition Thm. Proof: Analyze expected charpoly of a random partition: Edet xi i T i I i v i v T T i det xi i v i v i 2 i i v i v i T = I i v i v i T I 2 Theorem [MSS 13]: Given any vectors i v i v T i = I and ε, there is a partition into approximately v i 1/2-spherical quadratic forms, each I 2 ± O ε.

153 Unweighted Decomposition Thm. Other applications: Kadison-Singer Problem Uncertainty principles. i v i v i T I 2 i v i v i T = I i v i v i T I 2 Theorem [MSS 13]: Given any vectors i v i v T i = I and ε, there is a partition into approximately v i 1/2-spherical quadratic forms, each I 2 ± O ε.

154 Summary of Algorithms Result Edges Weights Time Spielman-S 08 O(nlogn) Yes O~(m) Random Sampling with Effective Resistances

155 Summary of Algorithms Result Edges Weights Time Spielman-S 08 O(nlogn) Yes O~(m) Batson-Spielman-S 09 O(n) Yes O(n 4 )

156 Summary of Algorithms Result Edges Weights Time Spielman-S 08 O(nlogn) Yes O~(m) Batson-Spielman-S 09 O(n) Yes O(n 4 ) Marcus-Spielman-S 13 O(n) No O(2 n ) Edet xi v i v i T det xi v i v i T i i

157 Open Questions Non-bipartite graphs Algorithmic construction (computing generalized μ G is hard) More general uses of interlacing families

158 Open Questions Nearly linear time algorithm for 4n/ε 2 size sparsifiers Improve to 2n/ε 2 for graphs? Fast combinatorial algorithm for approximating resistances