EE382C Lecture 5 - Stanford University

EE 382C - S11 - Lecture 5

1

EE382C

Lecture 5

Introduction to Routing

4/12/11


2

Logistics

• Homework 1 due NOW

• Homework 2: 8.1, 8.9, 9.5, 10.4


3

Question of the day

What oblivious routing algorithm should you use on a 8-ary 2-

cube to maximize worst-case throughput?

What does this algorithm do to the latency of nearest-neighbor

traffic?

Can you optimize both at the same time?


4

Consider an 8-node Ring

1 2 3 4 5 6 70

How would you route on this network?


5


• Some possibilities:

– Greedy – always go in the shortest direction

– Random – pick a random direction

– Aimless – pick a random direction at each hop

– Weighted – by how much?

– Adaptive – based on what?

– Other?

• How do you choose?

1 2 3 4 5 6 70


6


• Some possibilities:

– Greedy – always go in the shortest direction

– Random – pick a random direction

– Weighted – by (k-d/k)

• How do each of these algorithms do on

– Worst case traffic (tornado)

– Benign traffic

1 2 3 4 5 6 70


7

Report Card

Algorithm Locality Uniform

traffic

NN traffic Tornado

traffic

Greedy A 1 (A) 2 (A) 0.33 (F)

Random F 0.5 (F) 0.5 (F) 0.4 (C)

Weighted C 0.76 (C) 1.14 (C) 0.53 (A)

Non-adaptive algorithms have a fundamental

tradeoff between load balance and locality


8

Routing basics (1)

• Topology sets a “roadmap” between nodes

– Specific topology chosen to exploit packaging technology, meet

application requirements (bandwidth, latency, scalability, etc.) at the

minimum cost

• Topology fixes performance bounds

– Efficient routing (& flow control) strives to achieve these bounds

• Routing determines how data navigates through this

roadmap

– Determines a path (ordered set of channels) from source to

destination


9

Topology & Routing


10

Routing basics (2)

• Routing Relation

– All at once

• R: N x N (P)

– Incremental (per hop)

• R: N x N (C)

– Incremental with channel history

• R: C x N (C)


11

Goals of efficient routing

Capacity

Late

ncy (

sec)

Offered Load (bits/s)

Hop count

UR

Optimally routed

Uniform Random

(UR) traffic

• Load balance over all channels

• Minimize latency (reduce hop count)


12

Capacity bound

• B is bisection bandwidth

• N is number of nodes

• With UR traffic, each node sends half its traffic across the

bisection

• Capacity, Θ(UR) = 2B/N bits/s

N/2 N/2B


13

Worst-case upper bound

• B is bisection bandwidth

• N is number of nodes

• Each node sends all its traffic across the bisection

• Worst-case throughput, ΘWC B/N bits/s

N/2 N/2B


14

Performance measures

Late

ncy (

sec)

Offered Load (bits/s)URUR

Worst-case upper

bound


15

Routing: figures of merit

Late

ncy (

sec)

Offered Load (bits/s)URUR

Benign

High ben

Adversarial

High adv

Average

High avg

Low Tlow

Low Tint


16

Taxonomy of routing algos

• Deterministic

– Fixed path from source to destination

• Oblivious

– Can use randomization to select between different paths

• Adaptive

– Can use network state to make routing decisions


17

Deterministic routing (1)

0

1

2

3

4

5

6

7

00

01

02

03

10

11

12

13

20

21

22

23

0

1

2

3

4

5

6

7

(a) Routing from 3 to 5 in a 2-ary 3-fly

00

01

02

03

10

11

12

13

0

3

4

7

8

11

12

15

0

3

4

7

8

11

12

15

(b) Routing from 7 to 11 in a 4-ary 2-fly

Destination tag routing in flys


18

Deterministic routing (2)

• Dimension order routing

• Route to destination one

dimension at a time

• In 2-D,

– X first

– Y second

Route from 03 to 22

Route from 45 to 30


19

Oblivious routing

• Path from source A to destination B chosen using

randomization

• Advantages

– Simple analysis and implementation

– Can provide good load balancing

• Disadvantages

– Cannot have both low latency and good load balancing

– Packet reordering


20

Valiant’s oblivious routing

• Valiant’s randomized routing (VAL) [Valiant ’81]

– Route from src random node dest

– Minimal routes in each phase

• Optimal adv; Poor ben


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Connection to Clos / fat tree

1st butterfly2nd butterfly

• If we fold the network in implementation …


22

VAL’s analysis

• Converts every traffic into two phases of UR traffic

• Thus, channel load, , is twice the load on UR traffic for

minimal routing

• Hence throughput, is half the throughput of minimal

routing on UR traffic

– i.e. val = 0.5 capacity for all traffic


23

Adding VAL to routing example

Algorithm Locality Uniform traffic Tornado traffic

Greedy A 1 (A) 0.33 (F)

Random F 0.5 (F) 0.4 (C)

RLB C 0.76 (C) 0.53 (A)

VAL F 0.5 (F) 0.5 (A-)

• What about the worst-case traffic ?

• We have shown that VAL yields optimal WC

• Greedy is sub-optimal in WC, optimal for UR

• VAL is WC optimal but sub-optimal for UR

• RLB better than VAL for UR; is RLB WC optimal?


24

c(0,0)

c (1,1)

c (N-1,N-1)

c (0,1)c(1,0)

0

1

N-1

0

1

N-1

Source

NodesDestination

Nodes

Finding the worst case traffic


25

Example 2,3-ary 2-cube - DORc d e

gf

ba

3

0

4

1

5

2

By symmetry only

need to test

channels a, c, and d

FromTo

a b c d e f g

0 1 1 0 0 0 0 0 0

0 2 1 1 0 0 0 0 0

0 3 0 0 1 0 0 0 0

0 4 1 0 0 1 0 0 0

0 5 1 1 0 0 1 0 0

1 0 0 0 0 0 0 0 0

1 2 0 1 0 0 0 0 0

1 3 0 0 1 0 0 0 0

1 4 0 0 0 1 0 0 0

1 5 0 1 0 0 1 0 0

2 0 0 0 0 0 0 0 0

2 1 0 0 0 0 0 0 0

2 3 0 0 1 0 0 0 0

2 4 0 0 0 1 0 0 0

2 5 0 0 0 0 1 0 0

3 0 0 0 0 0 0 0 0

3 1 0 0 0 0 0 1 0

3 2 0 0 0 0 0 1 1

3 4 0 0 0 0 0 1 0

3 5 0 0 0 0 0 1 1

4 0 0 0 0 0 0 0 0

4 1 0 0 0 0 0 0 0

4 2 0 0 0 0 0 0 1

4 3 0 0 0 0 0 0 0

4 5 0 0 0 0 0 0 1

5 0 0 0 0 0 0 0 0

5 1 0 0 0 0 0 0 0

5 2 0 0 0 0 0 0 0

5 3 0 0 0 0 0 0 0

5 4 0 0 0 0 0 0 0

Load on

0

1

2

3

4

5

0

1

2

3

4

5

Table shows only half the

channels (right and up)

Graph for channel b

Maximal matching is

(0,2), (1,5) or (0,5),

(1,2) gamma = 2.


26

RLB: worst-case optimal on ring

• Construct a channel-load (bipartite) graph for a channel

• The Maximum Weight Matching gives the worst traffic pattern

1 2 3 4

5670

0 0

2 2

3 3

5 5

6 6

1 1

4 4

7 7

7/81/8

7/8

5/83/8

5/8

7/8 1/8 0

• Max weight = (7+5+3+1)/8 = 2

• WC throughput, wc= 0.5

1/8

3/8

5/8

7/8


27

sd(3,4)

0 1 2 3 4 5 6 7

0 0 0 0 0 4 3 2 1

1 1 0 0 0 5 4 3 2

2 2 1 0 0 6 5 4 3

3 3 2 1 0 7 6 5 4

4 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0

In eighths (i.e 1 implies 1/8)


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sd(3,4)

0 1 2 3 4 5 6 7

0 0 0 0 0 4 3 2 1

1 1 0 0 0 5 4 3 2

2 2 1 0 0 6 5 4 3

3 3 2 1 0 7 6 5 4

4 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0



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sd(3,4)

0 1 2 3 4 5 6 7

0 0 0 0 0 4 3 2 1

1 1 0 0 0 5 4 3 2

2 2 1 0 0 6 5 4 3

3 3 2 1 0 7 6 5 4

4 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0



30

Question of the day

What oblivious routing algorithm should you use on a 8-ary 2-

cube to maximize worst-case throughput?

What does this algorithm do to the latency of nearest-neighbor

traffic?

Can you optimize both at the same time?


31

Summary

• Routing basics

– Taxonomy – deterministic, oblivious, adaptive

– Routing relation – NxN → P, NxN → C, CxN → C

– Performance measures

• Deterministic Routing

– Dimension order

– Destination tag

• Oblivious Routing

– Valiant’s algorithm

– RLB

• Analysis of worst-case traffic

– Linearity allows us to find worst-case traffic pattern

– Maximum weight matching of a bipartite graph

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EE382C Lecture 5 - Stanford University

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