Orderly broadcasting in multidimensional tori
Download
1 / 55

Orderly Broadcasting in Multidimensional Tori - PowerPoint PPT Presentation


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

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

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

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

  • Conclusion and Future Work


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

  • Computer networks

  • Parallel processing

  • Cache coherence


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

Broadcast models can be divided into two major categories:

  • Classical Broadcasting (Slater in 1977)

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


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

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

u

v1

v4

v2

v3


Example : Classical Broadcasting

1

u

v1

v4

v2

v3


Example : Classical Broadcasting

1

u

v1

2

2

v4

v2

v3


Example : Classical Broadcasting

1

u

v1

2

2

v4

3

v2

v3

b(u) = 3


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

u

v1

v4

v2

v3


Example : Messy Broadcasting

1

u

v1

v4

v2

v3


Example : Messy Broadcasting

1

u

v1

2

2

v4

v2

v3


Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

v2

v3


Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

4

4

v2

v3


Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

4

4

v2

v3

5


Example : Messy Broadcasting

1

u

v1

2

3

2

v4

3

4

4

6

v2

v3

5

bm(u) = 6


How hard is it?

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


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

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 Broadcasting

Problem:

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


1

2

3

Example : Orderly Broadcasting

u

v1

v4

v2

v3


1

2

3

Example : Orderly Broadcasting

1

u

v1

v4

v2

v3


1

2

3

Example : Orderly Broadcasting

1

u

v1

2

2

v4

v2

v3


1

2

3

Example : Orderly Broadcasting

1

u

v1

3

2

3

2

v4

v2

v3


1

2

3

Example : Orderly Broadcasting

1

u

v1

3

2

3

2

v4

4

v2

v3

b(u) = 4


Previous Results


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


A 2-dim Torus

Torus T69


123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


123 4

Ordering  on 2D Tori

0

1

2

……………

n

……………

0

1

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

m


The Variable ℓ


Upper Bound on 2-dim Tori


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 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 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 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 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 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 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 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 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 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 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


123 4

Ordering d for d-dim Tori

3

2

0

1

4


Upper Bound on d-dim Tori


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

Thank You !


ad
  • Login