Self-Adjusting Grid Networks to Minimize Expected Path …borokhom/presentation_sirocco2013.pdf ·...

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

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

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 2 / 21

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

E [route length] = 121 + 1

51 + 3101 = 1

Simple Example

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 2 / 21

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

E [route length] = 121 + 1

51 + 3101 = 1

Simple Example

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 2 / 21

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

E [route length] = 121 + 1

51 + 3101 = 1

Model and Problem Definition

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 3 / 21

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

Model and Problem Definition

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 3 / 21

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

Model and Problem Definition

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 3 / 21

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

Model and Problem Definition

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 4 / 21

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

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)

Model and Problem Definition

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 4 / 21

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

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 5 / 21

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)

Expected Path Length

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 5 / 21

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)

Expected Path Length

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 5 / 21

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)

Example

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 6 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 7 / 21

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.

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 8 / 21

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

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

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 8 / 21

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 ?

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 9 / 21

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

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 ?

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 9 / 21

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

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?

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 10 / 21

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

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?

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 10 / 21

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

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 11 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 12 / 21

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∗

Expected Distance to Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 12 / 21

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∗

Expected Distance to Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 12 / 21

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∗

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21

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

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21

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

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21

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

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 13 / 21

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

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21

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)

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21

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)

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21

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)

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21

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)

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21

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)

Switching Rule – Optimize Expected Distance to the Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 14 / 21

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)

MEPL and Minimum Expected Distance to Center

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 15 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 15 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 16 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 17 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 17 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 18 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 19 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 19 / 21

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

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 20 / 21

900 nodes50% inactive8 clusters

900 nodes16 clusters

Animation

Simulations – Clustered Requests

Michael Borokhovich Self-Adjusting Grid Networks to Minimize Expected Path Length 20 / 21

900 nodes50% inactive8 clusters

900 nodes16 clusters

Animation

Summary

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

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

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

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

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

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!