Graph Structure via Exact Gaussian ... - Sushant Sachdeva › laplacian2.0 › schild.pdf · Graph...

Graph Structure via Exact Gaussian Elimination

Aaron [email protected]

University of California, Berkeley

October 6, 2018

Aaron Schild (UC Berkeley) Combinatorial Schur complements October 6, 2018 1

mailto:[email protected]

Relevant graph theoretic property of Schur complementsH := Schur(G , S): a weighted graph with V (H) = S and the propertythat the following two distributions are identical:

the list of vertices visited by a random walk in H

the list of vertices in S visited by a random walk in G

1

1

11

1

1

1 1

1

1

11/3

1/3

1/3⇒

4/3

1

1

1

1/3

1/3

⇒1/3

1/3

1/3


Goals of this talk

Use our graph theoretic view of Schur complements to

I prove that electrical flows are good `2-oblivious routersI sample uniformly random spanning trees in almost-linear time


Goals of this talk

Use our graph theoretic view of Schur complements toI prove that electrical flows are good `2-oblivious routers

I sample uniformly random spanning trees in almost-linear time


Goals of this talk

Use our graph theoretic view of Schur complements toI prove that electrical flows are good `2-oblivious routersI sample uniformly random spanning trees in almost-linear time


Part I: Localization of Electrical Flows


Joint work with Satish Rao and Nikhil Srivastava



Types of flows

Undirected graph G

Two vertices s, t ∈ G (always endpoints of an edge)

f(p)(s, t) ∈ RE(G): `p-minimizing s-t unit flow

Examples:

I f(1)(s, t) (s-t shortest path, left)I f(2)(s, t) (s-t electrical flow, middle)I f(∞)(s, t) (proportional to s-t max flow, right)

1

1

3/5

3/5

2/5

2/5

2/5

1/2

1/2

1/2

1/2

1/2

s

t

s s

t t



Types of flows

Undirected graph G



Examples:


1

1

3/5

3/5

2/5

2/5

2/5

1/2

1/2

1/2

1/2

1/2

s

t

s s

t t



Types of flows

Undirected graph G



Examples:


1

1

3/5

3/5

2/5

2/5

2/5

1/2

1/2

1/2

1/2

1/2

s

t

s s

t t



Types of flows

Undirected graph G



Examples:I f(1)(s, t) (s-t shortest path, left)I f(2)(s, t) (s-t electrical flow, middle)I f(∞)(s, t) (proportional to s-t max flow, right)

1

1

3/5

3/5

2/5

2/5

2/5

1/2

1/2

1/2

1/2

1/2

s

t

s s

t t



Main question: how concentrated are electrical flows?



Concentration of flows

Concentration of an edge e’s `p flow:∑

f ∈E(G) |f(p)f (e)|

Also the average length of flow paths



Concentration of flows

Concentration of an edge e’s `p flow:∑

f ∈E(G) |f(p)f (e)|

Also the average length of flow paths



`1-flows (shortest paths) are concentrated

Total flow = 1 =⇒ concentrated

e 1-1

+1

0



`∞-flows (max flows) can be spread out

Total flow = n/2 =⇒ spread out

e 1/2-1

+1

1/2



`2-flows (electrical flows) can be spread out

k = Θ(√n)

Total flow = Θ(√n) =⇒ spread out

.

.

.1/2

1/(2k)

k edgepaths

k paths

One edge



... but they aren’t on average!

k = Θ(√n)

Total flow = Θ(1) =⇒ concentrated!

.

.

.

k edgepaths

k paths

One edge

1 - θ(1/k)

θ(1/k)

θ(1/k2)

θ(1/k)



Result: electrical flows concentrate on average

Theorem

In any unweighted graph G,∑e∈E(G)

∑f ∈E(G)

|f(2)e (f )| ≤ O(m log2 n)



Applications

Electrical flows as oblivious routers

I `p-oblivious routing: given a set of demands d1, . . . , dk , how well canrouting demands independently do when compared with the optimal`p-multicommodity flow?

I `∞-oblivious routing on edge-transitive graphsI `2-oblivious routing on general graphs [HHN+08]

Almost-linear time random spanning tree generation

I Consider a process of the following form.I Fix two vertices s, t and repeatedly:

F Compute s-t electrical flowF Change edge weight with minimum energy by constant factor

I Localization says that low energy edges remain low for a whileI Thus we can do fewer electrical flow computations



Applications

Electrical flows as oblivious routersI `p-oblivious routing: given a set of demands d1, . . . , dk , how well can

routing demands independently do when compared with the optimal`p-multicommodity flow?








Applications



I `∞-oblivious routing on edge-transitive graphs

I `2-oblivious routing on general graphs [HHN+08]







Applications










Applications










Applications




Almost-linear time random spanning tree generationI Consider a process of the following form.I Fix two vertices s, t and repeatedly:





Applications






I Localization says that low energy edges remain low for a while

I Thus we can do fewer electrical flow computations



Applications









Linear algebraic version

Focus on unweighted graphs

A: n × n adjacency matrix, D = diag(A1): degree matrix

L = D − A: Laplacian matrix of G with pseudoinverse L+

be ∈ Rn: signed indicator of edge e

In unweighted graphs, f(2)e (f ) = |bTe L+bf |.

Restatement:∑

e∈E(G)

∑f ∈E(G) |bTe L+bf | ≤ O(m log2 n)



Relevant graph theoretic property of Schur complementsH := Schur(G , S): a weighted graph with V (H) = S and the propertythat the following two distributions are identical:



1

1

11

1

1

1 1

1

1

11/3

1/3

1/3⇒

4/3

1

1

1

1/3

1/3

⇒1/3

1/3

1/3



Projecting from G into Schur(G , S)

L+ = PT(Schur(G ,S)+ 0

0 M−1

)P

Applying Schur complement projection to a vector⇔ running a random walk until it hits S

P1v =

v

1/3

1/3

1/3

0

0

1



Proof outline: high level

Goal:∑

e∈E(G)


Pick a vertex, Schur complement it out, repeat (like Kyng-Sachdeva)

Let xi be the ith chosen vertex

Let Gi := Schur(G ,V (G ) \ x1, x2, . . . , xi) with Laplacian Li

Let Pi be the Schur complement projection from G to Gi

Let Vi =∑

e∈E(G)

∑f ∈E(G) |(Pibe)TL+i (Pibf )|

Suffices to show that Vi−1 ≤ Vi + O(m(log n)/(n − i))




Goal:∑

e∈E(G)






Let Vi =∑

e∈E(G)






Goal:∑

e∈E(G)






Let Vi =∑

e∈E(G)






Goal:∑

e∈E(G)






Let Vi =∑

e∈E(G)






Goal:∑

e∈E(G)






Let Vi =∑

e∈E(G)






Goal:∑

e∈E(G)






Let Vi =∑

e∈E(G)






Goal:∑

e∈E(G)






Let Vi =∑

e∈E(G)





Subproblem for bounding increments

For a vertex set S , v ∈ S , and a vertex x in G , let

pSv (x) be the probability that random walk from x hits S at v

For an edge e with endpoints x and y , let

qSv (e) = |pSv (x)− pSv (y)|

Lemma ∑v∈S

(∑

e∈E(G) qSv (e))2∑

e∈E(G) qSv (e)2

≤ O(m log n)








Lemma ∑v∈S

(∑


e∈E(G) qSv (e)2

≤ O(m log n)








Lemma ∑v∈S

(∑


e∈E(G) qSv (e)2

≤ O(m log n)








Lemma ∑v∈S

(∑


e∈E(G) qSv (e)2

≤ O(m log n)



Warmup: S = V (G )

qSv (e) = 1 if v is an endpoint of e, 0 otherwise. Therefore,

∑v∈S

(∑


e∈E(G) qSv (e)2

=∑v∈S

deg(v)2

deg(v)= 2m

e



General S

Edges can only contribute substantially to a small number of terms

e v



General S

Edges can only contribute substantially to a small number of terms

f



Lessons from Part I

electrical flows are concentrated on average

proof reduces to simpler objectives via vertex elimination

probabilistic interpretation reasons about change in original graph



Lessons from Part I






Lessons from Part I






Lessons from Part I





Part II: Random Spanning Trees




The weighted uniformly random spanning tree problem

Given an undirected graph G with weights (conductances) cee∈E(G) onits edges, sample a spanning tree T of G with probability proportional to∏

e∈E(T ) ce .

Direct applications

I Symmetric [GSS11] and asymmetric [AGM+10] traveling salesmanI Dependent roundingI Sparser cut sparsifiers [FH10, GRV09] and spectral sparsifiers [KS18]I Spectral sparsifiers via edge elimination [LS18]

Probability theory

Sampling from determinantal point processes and correlateddistributions

Algorithmic applications of electrical flows



Motivation


e∈E(T ) ce .

Direct applications

I Symmetric [GSS11] and asymmetric [AGM+10] traveling salesmanI Dependent roundingI Sparser cut sparsifiers [FH10, GRV09] and spectral sparsifiers [KS18]I Spectral sparsifiers via edge elimination [LS18]

Probability theory





Motivation


e∈E(T ) ce .

Direct applicationsI Symmetric [GSS11] and asymmetric [AGM+10] traveling salesmanI Dependent rounding

I Sparser cut sparsifiers [FH10, GRV09] and spectral sparsifiers [KS18]I Spectral sparsifiers via edge elimination [LS18]

Probability theory





Motivation


e∈E(T ) ce .

Direct applicationsI Symmetric [GSS11] and asymmetric [AGM+10] traveling salesmanI Dependent roundingI Sparser cut sparsifiers [FH10, GRV09] and spectral sparsifiers [KS18]

I Spectral sparsifiers via edge elimination [LS18]

Probability theory





Motivation


e∈E(T ) ce .

Direct applicationsI Symmetric [GSS11] and asymmetric [AGM+10] traveling salesmanI Dependent roundingI Sparser cut sparsifiers [FH10, GRV09] and spectral sparsifiers [KS18]I Spectral sparsifiers via edge elimination [LS18]

Probability theory





Motivation


e∈E(T ) ce .


Probability theory





Motivation


e∈E(T ) ce .


Probability theory





Motivation


e∈E(T ) ce .


Probability theory





Matrix-based algorithms

m: number of edgesn: number of verticesIdea: go through edges one by one and flip coins conditioned on priorchoices

Matrix-based algorithms (runtimes for weighted graphs)

I [Gue83] O(mnω)I [CMN96] O(nω)I [DKP+17] O(n4/3m1/2 + n2)I [DPPR17] O(n2δ−2)I No runtime dependence on weights (,)I Quadratic for sparse graphs (/)





Matrix-based algorithms (runtimes for weighted graphs)I [Gue83] O(mnω)I [CMN96] O(nω)

I [DKP+17] O(n4/3m1/2 + n2)I [DPPR17] O(n2δ−2)I No runtime dependence on weights (,)I Quadratic for sparse graphs (/)





Matrix-based algorithms (runtimes for weighted graphs)I [Gue83] O(mnω)I [CMN96] O(nω)I [DKP+17] O(n4/3m1/2 + n2)

I [DPPR17] O(n2δ−2)I No runtime dependence on weights (,)I Quadratic for sparse graphs (/)





Matrix-based algorithms (runtimes for weighted graphs)I [Gue83] O(mnω)I [CMN96] O(nω)I [DKP+17] O(n4/3m1/2 + n2)I [DPPR17] O(n2δ−2)

I No runtime dependence on weights (,)I Quadratic for sparse graphs (/)





Matrix-based algorithms (runtimes for weighted graphs)I [Gue83] O(mnω)I [CMN96] O(nω)I [DKP+17] O(n4/3m1/2 + n2)I [DPPR17] O(n2δ−2)I No runtime dependence on weights (,)

I Quadratic for sparse graphs (/)





Matrix-based algorithms (runtimes for weighted graphs)I [Gue83] O(mnω)I [CMN96] O(nω)I [DKP+17] O(n4/3m1/2 + n2)I [DPPR17] O(n2δ−2)I No runtime dependence on weights (,)I Quadratic for sparse graphs (/)



Aldous-Broder: a random walk-based algorithm

Algorithm:

Pick an arbitrary vertex u ∈ V (G ).

Do a random walk until all vertices have been visited.

Return the edges used to visit each vertex for the first time.

Generates a weighted uniformly random spanning tree of G !

Runtime: cover time, which can be Ω(mn) /

Only need first visits (at most n such visits) ,




Algorithm:










Algorithm:










Algorithm:









History

m: number of edgesn: number of vertices

Matrix-based algorithms (runtimes for weighted graphs)I [Gue83]I [CMN96] O(nω)I [DKP+17] O(n4/3m1/2 + n2)I [DPPR17] O(n2δ−2)I No runtime dependence on weights (,)I Quadratic for sparse graphs (/)

Random-walk-based algorithms (runtimes for unweighted graphs)

I [Bro89, Ald90] O(mn)I [KM09] O(m

√n)

I [MST15] O(m4/3)I Polynomial dependence on weights (/)I Subquadratic running time (,)I This work O(m1+o(1))



History



Random-walk-based algorithms (runtimes for unweighted graphs)

I [Bro89, Ald90] O(mn)I [KM09] O(m

√n)




History



Random-walk-based algorithms (runtimes for unweighted graphs)I [Bro89, Ald90] O(mn)I [KM09] O(m

√n)

I [MST15] O(m4/3)

I Polynomial dependence on weights (/)I Subquadratic running time (,)I This work O(m1+o(1))



History




√n)

I [MST15] O(m4/3)I Polynomial dependence on weights (/)

I Subquadratic running time (,)I This work O(m1+o(1))



History




√n)

I [MST15] O(m4/3)I Polynomial dependence on weights (/)I Subquadratic running time (,)

I This work O(m1+o(1))



History




√n)




Aldous-Broder remixFor each v ∈ V (G ), let S

(0)v = v

and pick shortcutters S (i)v σ0i=1 with v ∈ S

(i)v ⊆ V (G )

Pick an arbitrary vertex u ∈ V (G ).While there is an unvisited vertex

I

I Sample the edge that the random walk starting at u uses to exit S(0)u .

I Replace u with the non-S(i∗)u endpoint of this edge.


0laoblogger.comAaron Schild (UC Berkeley) Combinatorial Schur complements October 6, 2018 27


Wishful thinkingFor each v ∈ V (G ), let S

(0)v = v


(i)v ⊆ V (G )


I

I Sample the edge that the random walk starting at u uses to exit theset of visited vertices.



0laoblogger.comAaron Schild (UC Berkeley) Combinatorial Schur complements October 6, 2018 27


Shortcutting meta-algorithmFor each v ∈ V (G ), let S

(0)v = v


(i)v ⊆ V (G )


ILet i∗ ∈ 0, 1, . . . , σ0 be the maximum value of i for which all

vertices in S(i)u have been visited.

I Sample the edge that the random walk starting at u uses to exit S(i∗)u .





Example

For each v ∈ V (G ), let S(0)v = v

and pick shortcutters S (i)v 3i=1 with v ∈ S

(i)v ⊆ V (G )


ILet i∗ ∈ 0, 1, 2, 3 be the maximum value of i for which all







ExampleFor each v ∈ V (G ), let S

(0)v = v

and pick shortcutters S(1)v to be the 2-neighborhood of v for all

v ∈ V (G )Pick an arbitrary vertex u ∈ V (G ).While there is an unvisited vertex









(0)v = v











(0)v = v










Example on walk step 1



(i)v ⊆ V (G )


While there is an unvisited vertexI i∗ ← 0I Sample the edge that the random walk starting at u uses to exit S

(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .





Example on walk step 65 → 120



(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )



(i∗)u .








(i)v ⊆ V (G )


While there is an unvisited vertexI DONEI Sample the edge that the random walk starting at u uses to exit S

(i∗)u .





Summary of existing shortcutting-based algorithms

σ0: number of shortcuttersm: number of edgesn: number of vertices

Algorithm σ0 Shortcutting method Runtime

[Bro89, Ald90]

0 N/A

O(mn)

[KM09]

1 Offline

O(m√n)

[MST15]

2 Offline

O(m4/3)

This work

Θ(log log n) Online

O(m1+1/σ0)






[Bro89, Ald90] 0

N/A

O(mn)

[KM09]

1 Offline

O(m√n)

[MST15]

2 Offline

O(m4/3)

This work


O(m1+1/σ0)






[Bro89, Ald90] 0 N/A O(mn)

[KM09]

1 Offline

O(m√n)

[MST15]

2 Offline

O(m4/3)

This work


O(m1+1/σ0)







[KM09] 1

Offline

O(m√n)

[MST15]

2 Offline

O(m4/3)

This work


O(m1+1/σ0)







[KM09] 1

Offline

O(m√n)

[MST15] 2

Offline

O(m4/3)

This work


O(m1+1/σ0)







[KM09] 1

Offline

O(m√n)

[MST15] 2

Offline

O(m4/3)

This work Θ(log log n)

Online

O(m1+1/σ0)







[KM09] 1 Offline O(m√n)

[MST15] 2 Offline O(m4/3)

This work Θ(log log n)

Online

O(m1+1/σ0)







[KM09] 1 Offline O(m√n)

[MST15] 2 Offline O(m4/3)

This work Θ(log log n) Online O(m1+1/σ0)



Using Laplacian solvers to calculate hitting probabilities

pu = Pru[ random walk starting at u hits s before t]

s t1 0

upu

Can compute all pus in O(m) time! [ST14]



Shortcutting methods

Given: a shortcutter SC with C ⊆ SC ⊆ V (G )

Goal: sample the SC -escape edge for random walk starting at u

SC

Cu

Shortcutting method Preprocessing Query

Offline O(|E (SC )||∂SC |) O(log n)



Offline shortcutting



SCs t1

0Computes pu for all u in SC





Offline shortcutting



SC

s

t

1

0Computes pu for all u in SC





Graph Partitioning

Theorem (e.g. LR99)

A partition with diameter R and O(m(logm)/R) intercluster edges exists.

Applied throughout the metric embedding and LP rounding literature



Graph Partitioning via region growing

Theorem (e.g. LR99)





Graph Partitioning via region growing

Theorem (e.g. LR99)





Applying region growing to construct shortcutters [KM09]

Algorithm:

Apply region-growing.

Let S(1)v be the unique cluster containing v .

Runtime:

preprocessing time: (boundary)(cluster size)≤ O((m/R)m) ≤ O(m2/R)

normal random walk steps: O(mR)

shortcut random walk steps: O(m/R)n ≤ m2

R

best tradeoff for R =√m.




Algorithm:



Runtime:




R





Algorithm:



Runtime:




R





Algorithm:



Runtime:




R





Algorithm:



Runtime:




R




Online shortcutting



SCu

s

t

1

0Shortcutting method Preprocessing Query


Online None O(|E (SC )|)



Online shortcutting



SCu

s

tShortcutting method Preprocessing Query





Online shortcutting



SCu

s t1 0






Using online shortcutting

Shortcutter work: O(|E (Su)|), random walk work Ω(|E (Su)|2) ,

u

Su




Shortcutter work: Ω(|E (Su)|), random walk work can be O(1) /

u

Su




Core should be “well-separated” from the boundary of the shortcutter

SC

C



Bounding work of online shortcutting

Most random walk steps occur far away from an unvisited vertex

Lemma (Random walk bound)

Consider a random walk starting at an arbitrary vertex in a graph I and anedge u, v = f ∈ E (I ). The

expected number of times the random walk traverses f from u → v

before the distance R-neighborhood of u is covered

is at most O(cf R), where cf is the conductance of the edge f





















Recap of the Schur complement graph interpretationH := Schur(G , S): a weighted graph with V (H) = S and the propertythat the following two distributions are identical:



1

1

11

1

1

1 1

1

1

11/3

1/3

1/3⇒

4/3

1

1

1

1/3

1/3

⇒1/3

1/3

1/3



Charging shortcutter uses to Schur complement crossings

SC

C

u

Let Ri := mi/(σ0+1). To get almost-linear time,

total green conductance ≤ 1Ri

for S(i)C

and distance to unvisited vertex ≤ Ri+1

uses/shortcutter≤ (distance to unvisited vertex)(weight of edges)≤ O(Ri+1/Ri ) = mo(1) by edge crossing bound

work ≤ (total shortcutter size)(uses/shortcutter)≤ (m1+o(1))mo(1) = m1+o(1)




SC

C

u

LetRi := mi/(σ0+1). To get almost-linear time,


for S(i)C







SC

C

u



for S(i)C







SC

C

u



for S(i)C







SC

C

u



for S(i)C







SC

C

u



for S(i)C







SC

C

u



for S(i)C






Obtaining shortcutters that satisfy the first and thirdproperties




Build cores first for each Ri , i ∈ 1, 2, . . . , σ0 independently:

I Cover the graph with mo(1) well-separated families in the effectiveresistance metric

I Construction similar to sparse neighborhood covers [ABCP99]

Build shortcutters around the cores

I Voronoi diagram in probability space




Build cores first for each Ri , i ∈ 1, 2, . . . , σ0 independently:I Cover the graph with mo(1) well-separated families in the effective

resistance metric




Ri Ri Ri>>Ri >>Ri





resistance metric




Ri Ri Ri>>Ri >>Ri





resistance metric




Ri Ri>>Ri >>Ri





resistance metricI Construction similar to sparse neighborhood covers [ABCP99]










C1 C2 C3SC1 SC2 SC3






Build shortcutters around the coresI Voronoi diagram in probability space

C1 C2 C3SC1 SC2 SC3




First property: Schur complement conductance of S(i)C is at most

mo(1)

Ri

I Follows from well-separatedness

Third property: Each vertex in G is only in mo(1) shortcutters

I Follows from ≤ mo(1) families

O(1/Ri) conductance from Ri separation





mo(1)

Ri




O(1/Ri) conductance from Ri separation





mo(1)

Ri








mo(1)

Ri


Third property: Each vertex in G is only in mo(1) shortcuttersI Follows from ≤ mo(1) families



Lessons from Part II

An m1+o(1)αo(1)-time algorithm for generating weighted uniformlyrandom spanning trees

Overcame barriers in graph-partitioning based approaches from beforeby using Schur complements



Conclusion

Probabilistic interpretation of Laplacian Gaussian eliminationI Obtained a new `2 property of graphsI Random spanning trees in almost-linear time

Paradigm relevant for other problems?

Questions?



Bibliography

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