all-to-all broadcast problems on cartesian product graphs jen-chun lin 林仁俊...
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![Page 1: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/1.jpg)
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
![Page 2: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/2.jpg)
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
![Page 3: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/3.jpg)
Introduction
![Page 4: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/4.jpg)
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
![Page 5: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/5.jpg)
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
![Page 6: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/6.jpg)
All-to-all broadcasting numbers of Cartesian product of
cycles and complete graphs
![Page 7: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/7.jpg)
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
![Page 8: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/8.jpg)
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
![Page 9: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/9.jpg)
3AH 3
BH
3CH3
DH
1AB
1BC
1CD
1DA
2BC
2DA
2CD
2AB
3BC
3DA
3AB
3CD
![Page 10: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/10.jpg)
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
![Page 11: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/11.jpg)
Theorem
32
1)C □ (
nmmn
Ct nm , all for
![Page 12: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/12.jpg)
![Page 13: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/13.jpg)
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
![Page 14: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/14.jpg)
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
![Page 15: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/15.jpg)
Lemma
Given a graph G, if and only if there
exists a t-broadcasting systm of G.
tGt
2
1)C □ (
mntCt nm
![Page 16: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/16.jpg)
Lemma
For any graph G with ,
m
nnGt
)1(
mGEnGV and
2
1
2
1)C □ (
mn
mn
mnmnCt nm
![Page 17: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/17.jpg)
Theorem
32
1)C □ (
nmmn
Ct nm , all for
![Page 18: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/18.jpg)
Theorem
1 ,3 ,, allfor 1
1)12()K □( 12
nmnmn
nmCt nm
![Page 19: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/19.jpg)
![Page 20: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/20.jpg)
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
![Page 21: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/21.jpg)
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
,
,
,
![Page 22: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/22.jpg)
![Page 23: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/23.jpg)
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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
![Page 25: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/25.jpg)
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
![Page 26: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/26.jpg)
All-to-all broadcasting numbers of hypercubes
![Page 27: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/27.jpg)
Theorem
114
)( 2
n
nQt
n
n all for
![Page 28: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/28.jpg)
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
![Page 29: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/29.jpg)
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
![Page 30: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/30.jpg)
• 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
![Page 31: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/31.jpg)
Definition
2 type is Let
and1 type is Α Let:vQVv
:vQVv
n
n
2
2 ,
![Page 32: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/32.jpg)
Theorem
Ν
n
nQt
n
n all for 14
)( 2
![Page 33: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/33.jpg)
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
![Page 34: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/34.jpg)
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
![Page 35: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/35.jpg)
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
![Page 36: All-to-all broadcast problems on Cartesian 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](https://reader033.vdocuments.net/reader033/viewer/2022061510/5697bff81a28abf838cbf77b/html5/thumbnails/36.jpg)
Thanks for your listening!