Optimal Network Locality in Distributed Services

Optimal Network
Locality in Distributed
Services
Gwendal Simon
Department of Computer Science
Institut Telecom - Telecom Bretagne
2010

Telecom Bretagne

Institut Telecom: graduate engineering schools
Telecom Bretagne: 1200 students (200 PhD)
Computer Science: 20 full-time research lecturers
2 / 27 Gwendal Simon Network Locality in Distributed Services

Credits
Funding:
Thomson R&D (now Technicolor)
French grant with Orange, NDS Tech. and INRIA


Credits
Funding:
Thomson R&D (now Technicolor)
French grant with Orange, NDS Tech. and INRIA

Co-authors:
Jimmy Leblet (post-doc)
Yiping Chen (PhD student)
Zhe Li (PhD student)
Gilles Straub (senior researcher Thomson)
Di Yuan (Ass. Professor: Linkopping Univ. Sweden)

Service Delivery Network

CLOUD

end user



data-center

CLOUD

end user



data-center CDN

CLOUD

end user



data-center CDN

in-network servers
CLOUD

end user



data-center CDN

in-network servers
set-top-box
CLOUD
set-top-box
set-top-box

set-top-box

end user


Toward a Decentralized Architecture
servers’ capacities scale down1

services scale up2

=⇒ multi-servers multi-component architectures3
1
J. He, A. Chaintreau and C. Diot. “A performance evaluation of scalable
live video streaming with nano data centers” Computer Networks, 2009
2
J. Pujol, V. Erramilli, and P. Rodriguez, “Divide and Conquer:
Partitioning Online Social Networks” Arxiv preprint arXiv:0905.4918, 2009.
3
R. Baeza-Yates, A. Gionis, F. Junqueira, V. Plachouras, and L. Telloli,
“On the feasibility of multi-site web search engines”, in ACM CIKM 2009

Problem Modelling
and Analysis


Problem Formulation (Assumptions)
About the n servers and the k components:
only one component per server
no capacity bounds
components are uniformly accessed


Problem Formulation (Assumptions)
About the n servers and the k components:
only one component per server
no capacity bounds
components are uniformly accessed

About the global service architecture:
client requests are routed toward the closest server
characteristics of links between servers are known
a generic distance (cost) function dĳ


Problem Formulation (Deﬁnition)
Rainbow distance for a server i:
total cost to fetch all missing components

j4
5
=
dĳ4 j3
=4
dĳ3
3 j2
dĳ2 =
i dĳ1 = 2 j1


Problem Formulation (Deﬁnition)
Rainbow distance for a server i:
total cost to fetch all missing components

j4
d(i) = dĳ1 + dĳ4 = 7 =
5
j3
dĳ4
=4
dĳ3
3 j2
dĳ2 =
i dĳ1 = 2 j1


Problem Formulation (Objective)
Global goal: assign components to servers

Optimization: minimize sum of rainbow distances

d(i)
0<i≤n


Problem Formulation (Objective)
Global goal: assign components to servers

Optimization: minimize sum of rainbow distances

d(i)
0<i≤n

Motivations:
network operator: reduce cross-domain traﬃc
service provider: reduce overall latency
academic: funny unknown problem

Problem Complexity
The problem is NP-complete:
closely related with domatic partition

6 4
7
3
8
9
5
1 0
2


Integer Programming
1 if component c is allocated at server i
xic =
0 otherwise

c 1 if i obtains component c from j
yĳ =
0 otherwise

n k n
c
Minimize d(i, j)yĳ
i=1 c=1 j=1
k
Subject to xic = 1, only one component per server
c=1
c
yĳ = 1 − xic , a server has c or has exactly one pointer to c
j=i
c
yĳ ≤ xjc , a server has c from another server if this latter has c


Related Works


Related Works
Facility Location Problem
⇒ open a subset of facilities with minimal overall cost

c1 = 9 c2 = 3 c3 = 5

u1 u2 u3

Related Works

c1 = 9 c2 = 3 c3 = 5

u1 u2 u3

i1
i2

Related Works

c1 = 9 c2 = 3 c3 = 5

u1 u2 u3

6 8
3

i1
i2

Related Works

c1 = 9 c2 = 3 c3 = 5

u1 u2 u3

6 8 12 11
3

7
i1
i2


Related Works
most variants are NP-complete
3
close variant is k-PUFLP: a 2 k − 1 -approx. algo4
possible transformation from our prob. to k-PUFLP

4
H. C. Huang and R. Li, “A k-product uncapacitated facility location
problem”, European Journal of Op. Res., vol. 185, no. 2, 2008.

Related Works
3
a 2k − 1 -approx. algo


Related Works
3

Content Delivery Networks
k-median problem: no multiple servers


Related Works
3

Content Delivery Networks
k-median problem: no multiple servers

Nano data centers powered by set-top-boxes
uniform random allocation of components to servers


Our Algorithms


Approximate Algorithm
For a server i:
¯
1. compute distance d(i) to k − 1 closest servers
¯
2. wait until every server j with smaller d(j) are OK
3. try to optimize locally −→ optimized state
4. if impossible −→ saved state
5. uncolored saved nodes get furthest components



sorted list nearest neighbors
2 1,4,5
3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9



2 1,4,5
3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 optimized 4
16
2,5,7
1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9



2 1,4,5
3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11

8 3
optimized 17
13
5,14,16
11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9



2 1,4,5
conﬂict 3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9

saved but colored


2 1,4,5
conﬂict 3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9

saved and uncolored


2 1,4,5
3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9

colored by node 10


2 1,4,5
3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9

only node uncolored


2 1,4,5
3 1,8,16
1 2,3,16
6 8 3,11,12
7 5 1,2,4
11 8,12,13
10 4 2,5,7
16 1,3,5
13 4 12 8,9,11
15 2
15 1,10,11
5 10 2,6,15
1 18 14 3,16,17
11
17 5,14,16
8 3 13 11,12,15
16 7 2,4,6
12
6 2,7,10
9 8,12,14
17 18 4,5,17
14
9

choose farthest component

Proof
¯
d(i), cost to i’s k − 1 nearest neighbors.
d(i), rainbow cost of i.

6
7

10
13 4
15 2
5
1 18
11
8 3
16
12

17
14
9


Proof
¯

¯
optimized node: d(i) = d(i)
6
7

10
13 4
15 2
5
1 18
11
8 3
16
12

17
14
9


Proof
¯

¯
6
7

10 i node conﬂicting with its nearest neighbor:
13 4
15 2
optimized node at one hop
5 i
1 18
11 ¯
d(i) ≤ (k − 2)d(i)
8 3
16
12

17
14
9


Proof
¯

¯
6
7

10 node conﬂicting with its nearest neighbor:
13 4
15 2
optimized node at one hop
5
1 18
11 ¯
d(i) ≤ (k − 2)d(i)
3 i
8
16
12

17
node with two conﬂicting nearest neighbors:
14
9 j1 optimized node at two hops
i
¯
d(i) ≤ ( 3 k − 5 )d(i)
2 2


A Heuristic Algorithm
Idea: use the similarity with domatic partition
domatic coloring of a proximity graph


A Heuristic Algorithm
Idea: use the similarity with domatic partition
domatic coloring of a proximity graph

1. build a k-nearest neighbor graph O(n · log n)
2. augment it into an interval graph O(n)
3. build the domatic partition O(n)


Another Heuristic Algorithm
Based on a k-nearest neighbor graph, two rounds
1. explore surroundings: do not pick a component
hosted by a direct neighbor
hosted by a peer that considers you as a direct neighbor
hosted by one of its direct neighbors


Another Heuristic Algorithm
Based on a k-nearest neighbor graph, two rounds
1. explore surroundings: do not pick a component
hosted by a direct neighbor
hosted by a peer that considers you as a direct neighbor
hosted by one of its direct neighbors

2. try to maximize the beneﬁts
pick component satisfying in average the direct neighbors


Simulations


Conﬁgurations
Several contexts have been considered:
network of latencies
select randomly n peers among 20, 000 entries
⇒ minimize the global latency

network of Autonomous Systems
put µ peers into every AS
inter-AS routing
⇒ minimize the cross-domain traﬃc


Comparing to Exact Solutions


Going Further


Cross-Domain Gain

6

5
1,346
Average Hops

4
3,778 1,072 1,074 1,085
3
2,695 2,764 2,746
2

1

0
Random k-nearest Topo k-nearest Rela k-PUFLP

Peering Transit


Conclusion


Only Preliminary Works
Many theoretical results can be obtained:
relax assumptions (esp. capacity, number of
components)
study families of instances
better approximation


Only Preliminary Works
Many theoretical results can be obtained:
relax assumptions (esp. capacity, number of
components)
study families of instances
better approximation

Many realistic variants can be formulated:
take into account network architecture
objective of fairness


Any question?
<no image>


Optimal Network Locality in Distributed Services

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