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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!