Self-Adjusting Grid Networks to Minimize Expected Path …borokhom/presentation_sirocco2013.pdf · Self-Adjusting Grid Networks to Minimize Expected Path Length Chen Avin,Michael

Self-Adjusting Grid Networks to MinimizeExpected Path Length

Chen Avin, Michael Borokhovich, Bernhard Haeupler, Zvi Lotker

Communication Systems Engineering, BGU, Israel

Computer Science and Artificial Intelligence Laboratory, MIT, USA

SIROCCO 2013

Motivation - Data Centers

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 1 / 21

Energy cost ($50B in US alone 2008,doubles every 5 years!)

Routing consumes about 20-30%

Need to adjust the network, i.e.,reduce the expected route length

Fixed infrastructure...

Move processes (e.g., VM) betweenmachines

Virtualization and SDN (softwaredefined networks), e.g., OpenFlowenable VM migration

















Simple Example


P1P1

P2P2

P3P3

P4P4

P6P6

P5P5P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1

P2P2

P3P3P4P4

P6P6

P5P5P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1 P2P2

P3P3P4P4

P6P6P5P5

12153

10

E [route length] = 124 + 1

56 + 3104 = 4.4


51 + 3101 = 1

Simple Example


P1P1

P2P2

P3P3

P4P4

P6P6

P5P5P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1

P2P2

P3P3P4P4

P6P6

P5P5P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1 P2P2

P3P3P4P4

P6P6P5P5

12153

10


56 + 3104 = 4.4


51 + 3101 = 1

Simple Example


P1P1

P2P2

P3P3

P4P4

P6P6

P5P5P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1

P2P2

P3P3P4P4

P6P6

P5P5P1P1 P2P2

P3P3P4P4

P6P6P5P5

P1P1 P2P2

P3P3P4P4

P6P6P5P5

12153

10


56 + 3104 = 4.4


51 + 3101 = 1

Model and Problem Definition


Host Graph: H(V ,E): Physical Infrastructure

Routing Requests: σ = (σ1, σ2, . . . , σm) σt = (u, v)

We assume the requests are i.i.d. from a given distribution














Requests Distribution

Guest Weighted Graph: G(P,W )

|V | = |P| = n

1 2 3 4 5

1

2

3

4

5

8/1

9/1

2/1

0

0

G(P, W)

1 2 3 4 5

1

2 1/8 0

3

4 0 1/2

5

1/8

1/2

1/9

p(u, v)




Requests Distribution

Guest Weighted Graph: G(P,W )

|V | = |P| = n

1 2 3 4 5

1

2

3

4

5

8/1

9/1

2/1

0

0

G(P, W)

1 2 3 4 5

1

2 1/8 0

3

4 0 1/2

5

1/8

1/2

1/9

p(u, v)

Expected Path Length


A placement (arrangement):ϕ : P → V

For H,G, ϕ:EPL(ϕ) =

∑u,v∈P

Pr(u, v)dH(ϕ(u), ϕ(v))

Minimum Expected Path Length Problem

MEPL = minϕ

EPL(ϕ)





∑u,v∈P



MEPL = minϕ

EPL(ϕ)





∑u,v∈P



MEPL = minϕ

EPL(ϕ)

Example


Find the best way to put processes on the graphto minimize expected path length

G(P,W )

H(V ,E)

ϕ : P → V

Related Work


VLSI layout

Minimum Linear Arrangement (MLA)

Known to be hard (NP-Complete)

11 22 33 44 66 77 88 99 101041 2 3 5 6 7 8 9 10

G(P, W)

1 2 3 4 5

1

2 1/8 0

3

4 0 1/2

5

1/8

1/2

1/9

MLA = minϕ

∑u,v ,∈P

w(u, v)|ϕ(u)− ϕ(v)|

Hardness of MEPL

LemmaIf G is a symmetric product distribution, MEPL is still hard.

LemmaIf H is a 2-dimensional grid, MEPL is still hard.


Host Graph – Grid

Guest Graph – Symmetric Product Distribution

Activity level: p(u)

Probability of request: p(u, v) = p(u) · p(v)

Hardness of MEPL




Host Graph – Grid

Guest Graph – Symmetric Product Distribution

Activity level: p(u)

Probability of request: p(u, v) = p(u) · p(v)

Is there a CLIQUE of size k in H ?



Host Harbitrary graph

Guest Gsymmetric product distribution

p(u, v) = p(u) · p(v)

k nodes Cliquep(u) = 1/k

(n− k) disjoint nodesp(u) = 0

1k2

?←−

H has a clique of size k if and only if MEPL = k(k−1)k2 = 1− 1

k

Is there a CLIQUE of size k in H ?



Host Harbitrary graph

Guest Gsymmetric product distribution

p(u, v) = p(u) · p(v)

k nodes Cliquep(u) = 1/k

(n− k) disjoint nodesp(u) = 0

1k2

?←−

H has a clique of size k if and only if MEPL = k(k−1)k2 = 1− 1

k

Can we embed Tree into Grid?



Guest G

?←−

Host H

1k−1

Host Hgrid (k2 nodes)

Guest Gtree (k nodes)

Can we embed G to H?

This is hard [Bhatt et al., 1987]

G can be embedded into H if and only if MEPL = k−1k−1 = 1

Can we embed Tree into Grid?



Guest G

?←−

Host H

1k−1

Host Hgrid (k2 nodes)

Guest Gtree (k nodes)

Can we embed G to H?

This is hard [Bhatt et al., 1987]

G can be embedded into H if and only if MEPL = k−1k−1 = 1

Main Result


TheoremFor d-dimensional grid (H) and a symmetric productdistribution (G) there is a simple distributed algorithmwith a local switching policy between processes and theirneighbors that achieves a constant approximation to MEPL

Expected Distance to Center


Expected center: c∗(ϕ) = arg minx

∑u

p(u)d(ϕ(u), ϕ(x))

Expected distance to center: C(ϕ) =∑

u

p(u)d(ϕ(u), c∗(ϕ))

Minimum expected distance: Cmin = minϕ

C(ϕ)

Center (for SPD)

•ϕ(u)

ϕ(v)

c∗




∑u

p(u)d(ϕ(u), ϕ(x))


u

p(u)d(ϕ(u), c∗(ϕ))


C(ϕ)

Center (for SPD)

•ϕ(u)

ϕ(v)

c∗




∑u

p(u)d(ϕ(u), ϕ(x))


u

p(u)d(ϕ(u), c∗(ϕ))


C(ϕ)

Center (for SPD)

•ϕ(u)

ϕ(v)

c∗

Switching Rule – Optimize Expected Distance to the Center


ϕ1(u)

ϕ1(v)

c∗switch u and v?−→

ϕ2(v)

ϕ2(u)

c∗

Switch only if: C(ϕ2) ≤ C(ϕ1)

Assumptions:

Recall that: C(ϕ) =∑

u

p(u)d(ϕ(u), c∗(ϕ))

Every node knows current ϕ (locations of all nodes in H)centralized directory

Every node knows activity level p(u) of all nodesobserving requests over time



ϕ1(u)

ϕ1(v)


ϕ2(v)

ϕ2(u)

c∗


Assumptions:


u

p(u)d(ϕ(u), c∗(ϕ))





ϕ1(u)

ϕ1(v)


ϕ2(v)

ϕ2(u)

c∗


Assumptions:


u

p(u)d(ϕ(u), c∗(ϕ))





ϕ1(u)

ϕ1(v)


ϕ2(v)

ϕ2(u)

c∗


Assumptions:


u

p(u)d(ϕ(u), c∗(ϕ))





Greedy approach

Every switch decreases C(ϕ)

Will stop at some local minimum placement ϕ̂

How far local minimum C(ϕ̂) from global minimum Cmin?

What can we say about EPL(ϕ̂)?

We show:

C(ϕ̂)

Cmin= O(1) and

EPL(ϕ̂)MEPL

= O(1)



Greedy approach





We show:

C(ϕ̂)

Cmin= O(1) and

EPL(ϕ̂)MEPL

= O(1)



Greedy approach





We show:

C(ϕ̂)

Cmin= O(1) and

EPL(ϕ̂)MEPL

= O(1)



Greedy approach





We show:

C(ϕ̂)

Cmin= O(1) and

EPL(ϕ̂)MEPL

= O(1)



Greedy approach





We show:

C(ϕ̂)

Cmin= O(1) and

EPL(ϕ̂)MEPL

= O(1)



Greedy approach





We show:

C(ϕ̂)

Cmin= O(1) and

EPL(ϕ̂)MEPL

= O(1)

MEPL and Minimum Expected Distance to Center


Lemma

∀ϕ : C(ϕ) ≤ EPL(ϕ) ≤ 2C(ϕ)

d(ϕ(u), ϕ(v)) ≤ d(ϕ(u), c∗) + d(c∗, ϕ(v))

ϕ(u)

ϕ(v)

c∗

MEPL and Minimum Expected Distance to Center


Lemma

∀ϕ : C(ϕ) ≤ EPL(ϕ) ≤ 2C(ϕ)

d(ϕ(u), ϕ(v)) ≤ d(ϕ(u), c∗) + d(c∗, ϕ(v))

ϕ(u)

ϕ(v)

c∗

Expected Rank


Rank of a node r(u) is the position of the node in theordered list of nodes’ activity levels.

Node with the highest activity level has rank 0.

E[R] =∑

u

p(u)r(u)

Line


For any local optimum ϕ̂: C(ϕ̂) ≤ E[R]

For the global optimum ϕ̃: Cmin ≥ 12E[R]

d(ϕ̂(u), c∗) ≤ r(u)

d(ϕ̃(u), c∗) ≥ r(u)/2

c∗ ϕ̃(u)

p(u)

c∗ ϕ̂(u)

p(u)

Line


For any local optimum ϕ̂: C(ϕ̂) ≤ E[R]

For the global optimum ϕ̃: Cmin ≥ 12E[R]

d(ϕ̂(u), c∗) ≤ r(u)

d(ϕ̃(u), c∗) ≥ r(u)/2

c∗ ϕ̃(u)

p(u)

c∗ ϕ̂(u)

p(u)

2-Dimensional Grid


C(ϕ̂) ≈√

n

Cmin ≈ 4√

n

v

Allow chess knightmoves

66 188 13 openclipart.org/peoplee portablee immportablee im_Chess_tile_-_Knight_2.svv

v

2-Dimensional Grid


C(ϕ̂) ≈√

n Cmin ≈ 4√

n

v



v

2-Dimensional Grid


C(ϕ̂) ≈√

n Cmin ≈ 4√

n

v



v

2-Dimensional Grid


C(ϕ̂) ≈√

n Cmin ≈ 4√

n

v



v

2-Dimensional Grid


For any local optimum ϕ̂

C(ϕ̂) ≤ 4√6E[√

R]

d(ϕ̂(v), c∗) ≤ 4√6

√r(v)

x1

x2x3

x4

x5

x6

x7

z6 z5

z4

z3

z2

z1

v

c∗

For the global optimum ϕ̃

Cmin ≥ 1√2E[√

R]

d(ϕ̃(v), c∗) ≥√

r(v)2

v

c∗

2-Dimensional Grid


For any local optimum ϕ̂

C(ϕ̂) ≤ 4√6E[√

R]

d(ϕ̂(v), c∗) ≤ 4√6

√r(v)

x1

x2x3

x4

x5

x6

x7

z6 z5

z4

z3

z2

z1

v

c∗

For the global optimum ϕ̃

Cmin ≥ 1√2E[√

R]

d(ϕ̃(v), c∗) ≥√

r(v)2

v

c∗

Simulations – Clustered Requests


900 nodes50% inactive8 clusters

900 nodes16 clusters

Animation

Simulations – Clustered Requests


900 nodes50% inactive8 clusters

900 nodes16 clusters

Animation

Summary


MEPL is hard for general graphs and requests patterns

For grids and symm. product distr. we showed greedyapproach that is a constant approximation

Future work:

Real datacenters infrastructureMore requests patterns

THANK YOU!

Summary




Future work:Real datacenters infrastructureMore requests patterns

THANK YOU!

Summary




Future work:Real datacenters infrastructureMore requests patterns

THANK YOU!

Documents

Self-Adjusting Grid Networks to Minimize Expected Path …borokhom/presentation_sirocco2013.pdf · Self-Adjusting Grid Networks to Minimize Expected Path Length Chen Avin,Michael