All-to-all broadcast problems onCartesian product graphs

Jen-Chun Lin林仁俊

指導教授：郭大衛教授國立東華大學

應用數學系碩士班

Outline• Introduction• Main result

• Reference

3, allfor 2

1)C □ ( .1

nmmn

Ct nm

1 ,3 ,, allfor 1

1)12()K □( .2 12

nmnmn

nmCt nm

1 ,, allfor 224

)K □( .3 1212

nmnmnm

nmmnKt nm

Ν

n

nQt

n

n allfor 14

)( .4 2

Introduction

Suppose each vertex has a private message needed to send to every other vertices(all-to-all broadcast). At each time unit, vertices exchange their messages under the following constraints:

(1) only one message can travel a link at a time unit

(2) a message requires one time unit to be transferred between two nodes

(3) each vertex can use all of its links at the same time

Chang et al.

They gave upper and lower bounds for all-to-all broadcast number (the shortest time needed to complete the all-to-all broadcast) of graphs and give formulas for the all-to-all broadcast number of trees, complete bipartite graphs and double loop networks under this model.

Gt

All-to-all broadcasting numbers of Cartesian product of

cycles and complete graphs

Given a graph G and a positive integer t, we use G◦t to denote the multigraph obtained from G by replacing each edge in E(G) with k edges

uv .,...,, 21 tuvuvuv

1 2

34

①

②

③

①

②

③

①

①②

②③

③

1AB 2AB 3AB

3G 1AD

1CD

1BC

2AD

2CD

2BC

3AD

3CD

3BC

3AH 3

BH

3CH3

DH

1AB

1BC

1CD

1DA

2BC

2DA

2CD

2AB

3BC

3DA

3AB

3CD

For a graph G with a set ofedge-disjoint subgraphs of G◦t isa t-broadcasting system of G.

,,...,, 110 nvvvGVtv

tv

tv n

HHH110

,...,,

system ngbroadcasti-3

Theorem

32

1)C □ (

nmmn

Ct nm , all for

0,0 2,01,0

0,1 1,1

1,2

0,3

3,0

0,4

1,3

1,4 2,4

2,2

2,3

3,1

3,4

3,3

3,2

2,1

0,2

①

①

②

②

③

③

④

④

⑤

⑤

⑥

⑥ ⑦

⑦

⑧

⑧

⑨⑨ ⑩

The broadcasting tree .10C □ 5410

0,0 0,0a for of CH

0,0 2,01,0

0,1 1,1

1,2

0,3

3,0

0,4

1,3

1,4 2,4

2,2

2,3

3,1

3,4

3,3

3,2

2,1

0,2 ①

①

②

②

⑨

⑧

④

⑤ ⑥

③

⑨

③

④

⑧

⑩

⑦

⑤ ⑥ ⑦

The broadcasting tree .10C □ 5410

2,10,0 5,4 1,2a for of CH

Lemma

Given a graph G, if and only if there

exists a t-broadcasting systm of G.

tGt

2

1)C □ (

mntCt nm

Lemma

For any graph G with ,

m

nnGt

)1(

mGEnGV and

2

1

2

1)C □ (

mn

mn

mnmnCt nm

Theorem

32

1)C □ (

nmmn

Ct nm , all for

Theorem

1 ,3 ,, allfor 1

1)12()K □( 12

nmnmn

nmCt nm

0,0 2,01,0

0,1 1,1

1,2

0,3

3,0

0,4

1,3

1,4 2,4

2,2

2,3

3,1

3,4

3,3

3,2

2,1

0,2

4,0

4,1

4,4

4,3

4,2

The perfect broadcasting tree .8K □ 557

0,0 0,0a for of CH

① ② ③ ④

⑤ ⑥

⑦ ⑧

①

①

②

②

③

③

④

④

⑤

⑤ ⑥

⑥

⑦

⑦

⑧

⑧

Theorem

1224

)K □( 1212

nmnmnm

nmmnKt nm ,, all for

For positive integers m and n withWe set

.1nm

when when

when

nmn

nmnm

nmn

nm

nm

nm

3,3

32,

2,2

,

,

,

3

1

The subgraph .6K □ 596

0,0 KH of

0,0

0,2

0,1

0,4

0,3

6,03,02,01,0 7,04,0 8,05,0

1

1

2

2

1 1 12 2 2 2

3 3 3 3

3

4

4 4 4

4

4

5

5 5 5

5

5

6

6 6 6

6

6

0,0

0,1

6,03,02,01,0 7,04,0 8,05,0

0,2

0,4

0,3

7

7

7

7 7

7

88

All-to-all broadcasting numbers of hypercubes

Theorem

114

)( 2

n

nQt

n

n all for

ijabandabbbbaaaQ

niiaaaaQ

PPPPPQ

PPPQ

jjiinnn

inn

n

n

4

41101102

1102

44

222222

222

1:,...,,,...,,)E(

1,3,2,1,0:,...,,)V(let

C ...□ □C □C

□... □ □ □... □ □

□... □ □

For a vertex the weight of ,denoted by , is defined by

Definition

.1

0

n

iiavw

,,...,, 2110 nn QVaaav in v

vw

0

3 2

1

n個

2Pn個

2P2n個

For the set the right-shift of elements in is a function from to definedBy =

Definition

,AAAAn nA

nA nA

.,...,, 110n

n Aaaa

110 ,...,, nk aaa 11011 ,...,,,,...,, knkk aaaaaa

nk k .10,

1,0,3,2,13 =ex:

3,2,1,1,0

• the orbit of v, denoted by

• A vertex v in is said to be

Definition

.vO

1,...,2,1,0: nkvO kv

nQV 2

nO

nO

v

v

if : type2if :type1

2

nOv

Definition

2 type is Let

and1 type is Α Let:vQVv

:vQVv

n

n

2

2 ,

Theorem

Ν

n

nQt

n

n all for 14

)( 2

Lemma

(0,0,………,0)

(1,0,………,0) (0,1,………,0) (0,0,………,1)

………方向1 方向

2

方向n

1vw

2vw (2,0,………,0) (0,2,………,0) (0,0,………,2)

(1,0,………,1) (1,1,………,0)

vwuw

andQEvn A with uvertex u i

ists a , there exvwwith in ΑFor each v

n

.1

2

2

Lemma

is in Aaaaau

ni i then is in BaaaIf v

nii

n

.,...,1,...,,

,10,,,...,,

1410

110

(1,1,………,1)

(0,1,………,1)

B

A

6

2

0,0

0,1 1,0

0,2 1,1 2,0

0,3 1,2 3,0 2,1

3,1 1,3 2,2

3,2 2,3

3,3

1 1

2

443 3

5 5

6

7

7

8

.848

0,0 of Qtree H

ting t broadcasThe perfec

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