Efficient Distributed Workload (Re-)Embedding Monika Stefan - - PowerPoint PPT Presentation

Efficient Distributed Workload (Re-)Embedding

Stefan Neumann Monika Henzinger Stefan Schmid

Many Years Ago

• Single server
• Systems were fixed and

workload-agnostic

• Simple communication

patterns (if at all),
 endpoints fixed

https://www.flickr.com/photos/jurvetson/157722937

2

Nowadays

• Large distributed systems


(even geographically distributed):

communication over network

• Virtualization technologies

enable workload-aware

• perations that improve system

efficiency

• Communicating processes can

be far away and
 re-locating them is costly

https://wikileaks.org/amazon-atlas/map/
 https://commons.wikimedia.org/wiki/File:Bacloud.com_data_center.JPG

3

Nowadays

• Large distributed systems


(even geographically distributed):

communication over network

• Virtualization technologies

enable workload-aware

• perations that improve system

efficiency

• Communicating processes can

be far away and
 re-locating them is costly

• Communication requests

contain patterns

https://wikileaks.org/amazon-atlas/map/
 https://commons.wikimedia.org/wiki/File:Bacloud.com_data_center.JPG

3

Nowadays

• Large distributed systems


(even geographically distributed):

communication over network

• Virtualization technologies

enable workload-aware

• perations that improve system

efficiency

• Communicating processes can

be far away and
 re-locating them is costly

• Communication requests

contain patterns

https://wikileaks.org/amazon-atlas/map/
 https://commons.wikimedia.org/wiki/File:Bacloud.com_data_center.JPG

How to exploit the patterns? When to
 re-locate workloads?

New challenge

3

The Model

4

The Model

ℓ servers

4

The Model

ℓ servers

4

data centers RACK scale computing

The Model

server

ℓ

ℓ servers

4

data centers RACK scale computing

The Model

server

• ccupied

VM slot

ℓ

n

n virtual machines (VMs)

The VMs are the workloads.

5

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

εn additional slots for VMs

6

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

7

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

7

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

7

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

7

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

7

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Old communication links stay forever

8

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

9

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

1

9

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

1 1

9

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Communication requests arrive online

1 1 1

9

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α > 1

re-location cost

10

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α

α > 1

re-location cost

11

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α

α > 1

re-location cost

12

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α > 1

re-location cost

13

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α

α > 1

re-location cost

14

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α

α > 1

re-location cost

15

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α > 1

re-location cost

16

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α

α > 1

re-location cost

17

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α

α > 1

re-location cost

18

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

Re-locate VMs to avoid cross-server communication

α > 1

re-location cost

19

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

• Internal server communication cost:
• Server-server communication cost:

1

• VM re-location cost:

➡ Given an online sequence of communication requests,


minimize total cost paid for communication

α

1

α

20

The Model

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

• Internal server communication cost:
• Server-server communication cost:

1

• VM re-location cost:

➡ Given an online sequence of communication requests,


minimize total cost paid for communication

α

20

After all communications finished: 1 server = 1 component

Analysis

• Competitive analysis comparing to OPT:
• OPT knows all communications in advance
• OPT computes solution with optimal cost
• (Strict) competitive ratio =

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

ALG OPT

21

Results

• For servers:
• Algorithm which is -competitive
• Lower bound: Any algorithm must be

• competitive

➡ Our results are almost tight for two servers

server

• ccupied

VM slot free
 VM slot

2

n εn

O ( log n ε )

Ω(1/ε + log n)

22

ℓ = 2

Results

• For servers:
• Algorithm which is -competitive

➡ Efficient when is small,


e.g., for communication across data centers

➡ Implementable for distributed computation


communication cost ≤ communication for re-locating VMs

server

• ccupied

VM slot free
 VM slot

ℓ

n εn

O ((ℓ log n log ℓ)/ε)

ℓ ℓ

ℓ = O ( εn)

(if )

23

Applications

• Distributed Union Find Data Structure


(with small cost for re-locating the sets across servers)

• Online Balanced k-way Partition


(with small cost for re-assigning numbers to balanced partitions)

24

Algorithm for Two Servers

Color each VM based on its initial server

25

Algorithm for Two Servers

26

Algorithm for Two Servers

26

Algorithm for Two Servers

Move small component to larger one

26

Algorithm for Two Servers

Move small component to larger one

26

Algorithm for Two Servers

Move small component to larger one

27

Algorithm for Two Servers

Move small component to larger one

28

Algorithm for Two Servers

Move small component to larger one

28

Algorithm for Two Servers

Move small component to larger one

28

Algorithm for Two Servers

Move small component to larger one

29

Algorithm for Two Servers

Move small component to larger one

30

Algorithm for Two Servers

30

Algorithm for Two Servers

31

Algorithm for Two Servers

Contains more yellow than green VMs

31

Algorithm for Two Servers

Majority-voting step

Contains more yellow than green VMs

31

Algorithm for Two Servers

Majority-voting step

Contains more yellow than green VMs assign to yellow server

31

Algorithm for Two Servers

Majority-voting step

Contains more yellow than green VMs assign to yellow server

32

Algorithm for Two Servers

Majority-voting step

Contains more yellow than green VMs assign to yellow server

Ensures that we stay close to initial assignment

32

For each new
 communication request:

• Move smaller component to the


server of the larger one


• If size of new component exceeds

a power of 2:
 Perform majority-voting step


• If server capacity exceeded:


Find cheapest balanced
 assignment using
 brute-force enumeration

Can only happen
 times

O ( log n ε )

33

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

34

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

34

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

34

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

Traverse tree from root downwards and
 perform majority voting step at each internal node

34

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

Traverse tree from root downwards and
 perform majority voting step at each internal node

34

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

Traverse tree from root downwards and
 perform majority voting step at each internal node

34

Generalization to Servers

ℓ

S0 S1 S2 S3 S4 S5 S6 S7

Traverse tree from root downwards and
 perform majority voting step at each internal node

34

Summary

• We introduced a new model for
• nline workload (re-)embedding
• Distributed algorithm which is

• competitive
• Applications to version of

fundamental problems such as union find and k-way partition O ((ℓ log n log ℓ)/ε)

35

S0 S1 S2 S3 S4 S5 S6 S7

Open Problems

• Our algorithm for servers has competitive-ratio


. Can we shave the -factor?

• Study generalized setting where communication patterns

can change arbitrarily over time

• Tuning our algorithms further to perform even better in

specific use cases

O ((ℓ log n log ℓ)/ε)

ℓ ℓ

36

Efficient Distributed Workload (Re-)Embedding

Distributed
 Union Find Data Structure


(with small cost
 for re-locating the sets)

Online
 Balanced k-way Partition


(with small cost
 for assigning numbers to partitions)

Applications

37

new model for distributed workload (re-)embedding

Thank you!

O ((ℓ log n log ℓ)/ε)

distributed algorithm with competitive ratio

Related Work

• Avin et al. (DISC’16, SIDMA’19):
• No ground-truth assumption
• -competitive algorithm
• Competitive ratio for deterministic algorithms is


38

Ω(n/ℓ) ˜ O(n/ℓ)