Graph Sparsification by Effective Resistances

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-Karger96

• 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 BLH ?

• 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
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 BK96 Improves O(nlogc n) sparsifiers
ST04
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 qO(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
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 qO(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 qO(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 qO(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 qO(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 qO(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
51

• Nearly Linear Time

52
The Algorithm

• Sample edges of G with probability
• If chosen, include in H with weight
• Take qO(nlogn/?2) samples with replacement
• Divide all weights by q.

53
The Algorithm

• Sample edges of G with probability
• If chosen, include in H with weight
• Take qO(nlogn/?2) samples with replacement
• Divide all weights by q.

54
Nearly Linear Time
55
Nearly Linear Time

• So care about distances between cols. of BL-1

56
Nearly Linear Time

• So care about distances between cols. of BL-1
• Johnson-Lindenstrauss! Take random Qlogn m
• Set ZQBL-1

57
Nearly Linear Time
58
Nearly Linear Time

• Find rows of Zlog n n by
• ZQBL-1
• ZLQB
• ziL(QB)i

59
Nearly Linear Time

• Find rows of Zlog n n by
• ZQBL-1
• ZLQB
• ziL(QB)i
• Solve O(logn) linear systems in L using
  Spielman-Teng 04 solver
• which uses combinatorial O(nlogcn) sparsifier.
• Can show approximate Reff suffice.

60
Main Conjecture

• Sparsifiers with O(n) edges.

61
Example Another edge to include
m-1
1
k-by-k complete bipartite
0
m
k-by-k complete bipartite
1
m-1
61
62
The Projection Matrix

• Lemma.
• ? is a projection matrix
• im(?)im(B)
• Tr(?)n-1
• ?(e,e)?(e,-)2

63
Last Steps
64
Last Steps
65
Last Steps
66
Last Steps
67
Last Steps

• We also have
• and
• since ?e2?(e,e).

68
Reduction to ?

• Goal

69
Reduction to ?

• Goal
• Write
• Then

70
Reduction to ?

• Goal
• Write
• Then
• Goal

71
Reduction to ?
72
Reduction to ?
73
Reduction to ?
74
Reduction to ?
75
Reduction to ?
76
Reduction to ?

• Lemma.
• Proof. ? is the projection onto im(B).