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

Graph Structure via Exact Gaussian Elimination

Aaron Schildaschild@berkeley.edu

University of California, Berkeley

October 6, 2018

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

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

1/3⇒

⇒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)

Types of flows

Undirected graph G

Examples:

Types of flows

Undirected graph G

Examples:

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)

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

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

Total flow = n/2 =⇒ spread out

e 1/2-1

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

k = Θ(√n)

Total flow = Θ(√n) =⇒ spread out

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

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:

1/3⇒

⇒1/3

Projecting from G into Schur(G , S)

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

0 M−1

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

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

General S

Edges can only contribute substantially to a small number of terms

General S

Edges can only contribute substantially to a small number of terms

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

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:

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

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

History

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

I [MST15] O(m4/3)

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

History

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

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

History

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

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

History

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 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 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

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]

[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

[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

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

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

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

best tradeoff for R =√m.

Algorithm:

Runtime:

Algorithm:

Runtime:

Algorithm:

Runtime:

Algorithm:

Runtime:

Online shortcutting

0Shortcutting method Preprocessing Query

Online None O(|E (SC )|)

Online shortcutting

tShortcutting method Preprocessing Query

Online shortcutting

s t1 0

Using online shortcutting

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

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

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

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:

1/3⇒

⇒1/3

Charging shortcutter uses to Schur complement crossings

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)

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

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

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

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

Baruch Awerbuch, Bonnie Berger, Lenore Cowen, and David Peleg.Near-linear time construction of sparse neighborhood covers.SIAM J. Comput., 28(1):263–277, February 1999.

Arash Asadpour, Michel X. Goemans, Aleksander Madry,Shayan Oveis Gharan, and Amin Saberi.An o(log n/ log log n)-approximation algorithm for the asymmetrictraveling salesman problem.In Proceedings of the Twenty-first Annual ACM-SIAM Symposium onDiscrete Algorithms, SODA ’10, pages 379–389, Philadelphia, PA,USA, 2010. Society for Industrial and Applied Mathematics.

David J. Aldous.The random walk construction of uniform spanning trees and uniformlabelled trees.SIAM J. Discret. Math., 3(4):450–465, November 1990.

A. Broder.Generating random spanning trees.In Proceedings of the 30th Annual Symposium on Foundations ofComputer Science, SFCS ’89, pages 442–447, Washington, DC, USA,1989. IEEE Computer Society.

Charles J. Colbourn, Wendy J. Myrvold, and Eugene Neufeld.Two algorithms for unranking arborescences.Journal of Algorithms, 20(2):268–281, 3 1996.

David Durfee, Rasmus Kyng, John Peebles, Anup B. Rao, andSushant Sachdeva.Sampling random spanning trees faster than matrix multiplication.In Proceedings of the 49th Annual ACM SIGACT Symposium onTheory of Computing, STOC 2017, pages 730–742, New York, NY,USA, 2017. ACM.

David Durfee, John Peebles, Richard Peng, and Anup B. Rao.Determinant-preserving sparsification of SDDM matrices withapplications to counting and sampling spanning trees.CoRR, abs/1705.00985, 2017.

Wai Shing Fung and Nicholas J. A. Harvey.Graph sparsification by edge-connectivity and random spanning trees.CoRR, abs/1005.0265, 2010.

Navin Goyal, Luis Rademacher, and Santosh Vempala.Expanders via random spanning trees.In Proceedings of the Twentieth Annual ACM-SIAM Symposium onDiscrete Algorithms, SODA ’09, pages 576–585, Philadelphia, PA,USA, 2009. Society for Industrial and Applied Mathematics.

Shayan Oveis Gharan, Amin Saberi, and Mohit Singh.A randomized rounding approach to the traveling salesman problem.In Proceedings of the 2011 IEEE 52Nd Annual Symposium onFoundations of Computer Science, FOCS ’11, pages 550–559,Washington, DC, USA, 2011. IEEE Computer Society.

A. Guenoche.Random spanning tree.Journal of Algorithms, 4:214–220, 1983.

Prahladh Harsha, Thomas P. Hayes, Hariharan Narayanan, HaraldRacke, and Jaikumar Radhakrishnan.Minimizing average latency in oblivious routing.In Proceedings of the Nineteenth Annual ACM-SIAM Symposium onDiscrete Algorithms, SODA ’08, pages 200–207, Philadelphia, PA,USA, 2008. Society for Industrial and Applied Mathematics.

Jonathan A. Kelner and Aleksander Madry.Faster generation of random spanning trees.In 50th Annual IEEE Symposium on Foundations of ComputerScience, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA,pages 13–21, 2009.

Huan Li and Aaron Schild.Spectral subspace sparsification.2018.

Aleksander Madry, Damian Straszak, and Jakub Tarnawski.Fast generation of random spanning trees and the effective resistancemetric.In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposiumon Discrete Algorithms, SODA 2015, San Diego, CA, USA, January4-6, 2015, pages 2019–2036, 2015.

Daniel A. Spielman and Shang-Hua Teng.Nearly linear time algorithms for preconditioning and solvingsymmetric, diagonally dominant linear systems.SIAM J. Matrix Analysis Applications, 35(3):835–885, 2014.

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

Documents

- Bangladesh Air Force · 2019-04-04 · 12. 13. 15. . This formula is exact for polynomials upto degree 3. Three points Gaussian quadrature. f f(x)dx = — This formula is exact

Graphical Models with Symmetry - Oxford Statisticssteffen/seminars/waldsymmetry.pdf · 2012-07-09 · Introduction Examples Graphical Gaussian models Special graph types Adding mean

state space models - Jyväskylän yliopistousers.jyu.fi/~jovetale/posters/KFASposter.pdf · non-Gaussian models Exact diffuse initialization of unknown states Multivariate models

Gaussian Process Regression with Location Errors · problem of Gaussian process regression with location errors addressed in this paper is to predict xat unobserved (exact) locations

An exact algorithm for graph partitioning William W. Hager, Dzung T

Real-Time Exact Graph Matching with Application in Human ... · Real-Time Exact Graph Matching with Application in Human Action Recognition Oya C¸eliktutan 1; 2, Christian Wolf ,

Large-Scale Malware Indexing Using Function-Call Graphshuxin/papers/xin_SMIT_extend.pdf · query processing methods cannot scale to a large graph database; ... exact graph or subgraph

Deep Clustering by Gaussian Mixture Variational ...lijiaying.github.io/papers/iccv19.pdf · mixture variational autoencoder (VAE) with Graph embed-ding. To facilitate clustering,

Safe Exploration in Gaussian Policy Gradientguarantees for Gaussian policies to the adaptive-variance case, providing safe exact-gradient updates. In section 5, we generalize these

A Class of Non-Gaussian State Space Models with Exact ...faculty.chicagobooth.edu/drew.creal/research/papers/creal2015.pdfA Class of Non-Gaussian State Space Models with Exact Likelihood

MCMC for Variationally Sparse Gaussian Processes€¦ · (MCMC) approaches provide asymptotically exact approximations. Murray and Adams [11] and Filippone et al. [12] examine schemes

Faster Algorithms and Graph Structure via Gaussian Elimination€¦ · Faster Algorithms and Graph Structure via Gaussian Elimination by Aaron Victor Schild ... it yields sparsi ers

Web viewTides. Amount of daylight. Energy waves. Sine Graph: Fill in the important points using exact angles and heights: What is the period? Amplitude? Cosine Graph:

Graph and Hypergraph Decompositions for Exact …ISBN 978-952-10-9639-6 (PDF) Abstract This thesis studies exact exponential and ﬁxed-parameter algorithms for hard graph and hypergraph

Second Order Results for Nodal Sets of Gaussian Random ...people.maths.ox.ac.uk/belyaev/RandomWaveOxford/slides/...Theorem (Marinucci, P., Rossi & Wigman, 2016) 1. (Exact Cancellation)

Exact Solution of Graph Coloring Problems via Constraint ... · PDF fileExact Solution of Graph Coloring Problems via Constraint Programming and Column Generation Stefano Gualandi,

Zheng Zhang, 27.05.2010 Exact Matching slide 1 Outline Graph matching – Definition – Manifold application areas Exact matching – Definition – Different

An exact algorithm for graph partitioningAn exact algorithm for graph partitioning 533 sufﬁcient conditions for a local minimizer. Section 6 compares the performance of the new branch-and-bound

Exact Algorithms for Hard Graph Problems (Algoritmi Esatti - Idsia

On the Minimum of True Matches in Exact Graph Matching