Orderly Broadcasting in Multidimensional Tori - PowerPoint PPT Presentation

Orderly broadcasting in multidimensional tori l.jpg
Download
1 / 55

Orderly Broadcasting in Multidimensional Tori. Presentation by Perouz Taslakian. May 6, 2004. Outline. Introduction and Motivation Types of Broadcasting Orderly Broadcasting Lower Bound on 2-dimensional Tori Upper Bound for 2-dimensional Tori Upper Bound for d -dimensional Tori

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Presentation

Orderly Broadcasting in Multidimensional Tori

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Orderly broadcasting in multidimensional tori l.jpg

Orderly Broadcasting in Multidimensional Tori

Presentation by

Perouz Taslakian

May 6, 2004


Outline l.jpg

Outline

  • Introduction and Motivation

  • Types of Broadcasting

  • Orderly Broadcasting

  • Lower Bound on 2-dimensional Tori

  • Upper Bound for 2-dimensional Tori

  • Upper Bound for d-dimensional Tori

  • Conclusion and Future Work


Introduction l.jpg

Introduction

Broadcasting : process of sending a message from one node of a communication network to the rest of the nodes.

Originator: the node of the network which initiates broadcasting by sending its message to the rest of the nodes.

Broadcast problem : the problem of determining the amount of time needed to transmit a message to every node in an interconnection network


Motivation l.jpg

Motivation

  • Computer networks

  • Parallel processing

  • Cache coherence


Constraints l.jpg

1

u

v

2

w

Constraints

Broadcasting is subject to the following constraints:

  • Each message transmission takes 1 time unit

  • A node can transmit only to an adjacent node

  • A node can transmit a message to 1 node in 1 time unit


Types of broadcasting l.jpg

Types of Broadcasting

Broadcast models can be divided into two major categories:

  • Classical Broadcasting (Slater in 1977)

  • Messy Broadcasting (Alshwede, Khatchadrian & Harutyunyan in 1994)


Notations l.jpg

Notations

A network is modeled as a connected graph G=(V, E)

b(u) : min number of time units required to complete broadcasting when u is the originator (classical)

bm(u): max number of time units required to complete broadcasting when u is the originator (messy)

b(G) = max {b(u) | u  V}

bm(G) = max {bm(u) | u  V}


Classical broadcasting l.jpg

Classical Broadcasting

Find a scheme so as information dissemination takes the least amount of time.

Assumption: each vertex knows the

  • graph topology

  • originator of the message

  • time the message was sent.

    When a vertex is informed, it transmits the message to its neighbor in the most cleverway.


Example classical broadcasting l.jpg

Example : Classical Broadcasting

u

v1

v4

v2

v3


Example classical broadcasting10 l.jpg

Example : Classical Broadcasting

1

u

v1

v4

v2

v3


Example classical broadcasting11 l.jpg

Example : Classical Broadcasting

1

u

v1

2

2

v4

v2

v3


Example classical broadcasting12 l.jpg

Example : Classical Broadcasting

1

u

v1

2

2

v4

3

v2

v3

b(u) = 3


Messy broadcasting l.jpg

Messy Broadcasting

Analyzes broadcast schemes that take the mostamount of time

Assumption: each vertex knows nothing about the graph topology and the originator.

When a vertex is informed, it transmits the message to a randomly chosen neighbor in each time unit.


Example messy broadcasting l.jpg

Example : Messy Broadcasting

u

v1

v4

v2

v3


Example messy broadcasting15 l.jpg

Example : Messy Broadcasting

1

u

v1

v4

v2

v3


Example messy broadcasting16 l.jpg

Example : Messy Broadcasting

1

u

v1

2

2

v4

v2

v3


Example messy broadcasting17 l.jpg

Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

v2

v3


Example messy broadcasting18 l.jpg

Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

4

4

v2

v3


Example messy broadcasting19 l.jpg

Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

4

4

v2

v3

5


Example messy broadcasting20 l.jpg

Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

4

4

6

v2

v3

5

bm(u) = 6


How hard is it l.jpg

How hard is it?

Finding b(u) for an arbitrary graph isNP-Complete


Practicality l.jpg

Practicality

Storing information about the graph topology and the originator is not efficient in practice

Sometimes, network nodes have primitive structures with small memories that cannot store such information.

Building networks where the vertices have no decision making responsibility is easier.


Orderly broadcasting l.jpg

Orderly Broadcasting

The neighbors of each vertex are assigned a unique number.

Assumption: each vertex knows nothing about the graph topology and the originator.

When a vertex is informed, it transmits the message first to vertex numbered 1, then to vertex numbered 2, …etc.


Orderly broadcasting24 l.jpg

Orderly Broadcasting

Problem:

Find an orderingof the neighbors of all the vertices of a given graph that will minimize the broadcast time.


Example orderly broadcasting l.jpg

1

2

3

Example : Orderly Broadcasting

u

v1

v4

v2

v3


Example orderly broadcasting26 l.jpg

1

2

3

Example : Orderly Broadcasting

1

u

v1

v4

v2

v3


Example orderly broadcasting27 l.jpg

1

2

3

Example : Orderly Broadcasting

1

u

v1

2

2

v4

v2

v3


Example orderly broadcasting28 l.jpg

1

2

3

Example : Orderly Broadcasting

1

u

v1

3

2

3

2

v4

v2

v3


Example orderly broadcasting29 l.jpg

1

2

3

Example : Orderly Broadcasting

1

u

v1

3

2

3

2

v4

4

v2

v3

b(u) = 4


Previous results l.jpg

Previous Results


2 dimensional tori l.jpg

2-dimensional Tori

Torus Tmn: wrap-around grid with m rows & n columns

Each vertex in a 2-dimensional torus has degree 4

Diameter of Tmn is :

Every vertex in Tmn has : 1 vertex at distance D(Tmn) if m and n are even 2 vertices at distance D(Tmn) if one of m or n is odd 4 vertices at distance D(Tmn) if m and n are odd


Lower bound l.jpg

Lower Bound


A 2 dim torus l.jpg

A 2-dim Torus

Torus T69


Ordering on 2d tori l.jpg

123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


Ordering on 2d tori35 l.jpg

123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


Ordering on 2d tori36 l.jpg

123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


Ordering on 2d tori37 l.jpg

123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


Ordering on 2d tori38 l.jpg

123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


The variable l.jpg

The Variable ℓ


Upper bound on 2 dim tori l.jpg

Upper Bound on 2-dim Tori


How it works an example t 8 13 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

2

1

2

3

4

3

4

5

6

7


How it works an example t 8 1342 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

3

2

1

2

3

4

3

4

5

6

7


How it works an example t 8 1343 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

6

5

4

3

2

1

2

3

4

3

4

5

6

7


How it works an example t 8 1344 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

6

5

4

3

2

7

1

2

3

4

3

8

4

5

6

7


How it works an example t 8 1345 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

6

5

4

3

2

7

9

10

11

1

2

3

4

3

8

4

5

6

7


How it works an example t 8 1346 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

P1

1

6

5

4

3

2

7

9

10

11

1

2

3

4

3

8

12

12

14

4

5

6

7


How it works an example t 8 1347 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

2

3

4

P1

1

6

5

4

3

2

7

9

10

11

1

2

3

4

3

8

12

12

14

4

5

6

7


How it works an example t 8 1348 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

2

3

4

P1

1

6

5

4

3

2

7

9

10

11

1

2

3

4

3

8

12

12

14

9

4

8

5

7

6

6

7

5


How it works an example t 8 1349 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

2

3

4

P1

1

6

5

4

3

2

7

9

10

11

P2

1

2

3

4

3

8

12

12

14

13

12

11

9

4

8

5

7

6

6

7

5


How it works an example t 8 1350 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

2

3

4

P1

1

6

5

4

3

2

7

9

10

11

P2

P3

1

2

3

4

3

8

12

12

14

13

12

11

9

12

11

10

4

8

5

7

6

6

7

5


How it works an example t 8 1351 l.jpg

How it works: an example T813

0

1

3

4

6

7

10

11

2

5

8

9

12

0

0

1

2

3

4

P1

1

6

5

4

3

2

7

9

10

11

P2

P3

1

2

3

4

3

8

12

12

14

13

12

11

9

12

12

12

11

10

4

8

9

10

P4

5

7

6

6

7

5


Ordering d for d dim tori l.jpg

123 4

Ordering d for d-dim Tori

3

2

0

1

4


Upper bound on d dim tori l.jpg

Upper Bound on d-dim Tori


Conclusion and future work l.jpg

Conclusion and Future Work

  • Improve the upper and lower bound

  • Find orderings for other graph types – for example Butterfly BFm, Shuffle-Exchange SEm, DeBruijn DBm, Cube-Connected Cycles CCCm, …etc)

  • Study the relationship between the Classical, Messy and Orderly broadcast models.


The end l.jpg

The End

Thank You !


  • Login