1

Wide-Sense Nonblocking Under New Compound Routing Strategies

Junyi David Guo(郭君逸 )師範大學 數學系

2011/06/29Joint work with F.H. Chang

2

Applications

• Come from the need to interconnect telephones

• Interconnect processors with memories• Data transmission• Conference calls• Satellite communication

3

One frequently discussed topic in switching networks is its nonblocking property.

5

Multi-stage interconnection network

inputs outputs

stage

6

Symmetry

• C(n, m, r)• C(2, 4, 3)

n 1

2

r

1

2

m

n1

2

r

7

Definitions

• Request• Strictly nonblocking(SNB)• Wide-sense

nonblocking(WSNB)

8

Matrix

1

2

4

2

O1 O2 O3

I1

I2

I3

1

3

9

3-stage Clos Network

• [Clos 1953] C(n1, r1, m, n2, r2) is SNB iff m ≥ n1+n2-1.

• C(n, m, r) is SNB iff m ≥ 2n-1.

10

[n+5,2n-2]

n+4

n+3

[12,n-1]11[6,10]2n-14,51,2,3

n+2

n,n+1

11

Algorithms

• Cyclic dynamic (CD)• Cyclic static (CS)• Save the unused (STU)• Packing (P)• Minimum index (MI)• Compound

12

• Beneš(1965) proved that C(n, m, 2) is WSNB under P if and only if .

13

Literature review

• Smith(1977) proved that C(n, m, r) is not WSNB under P or MI if .

• Du et al.(2001) improved to .• Hwang(2001) extend it to cover CS and CD.• Yang and Wang(1999) proved that C(n, m, r)

is not WSNB under P if .• Chang et al.(2004) improved to .• Chang et al.(2007) extend it to Multi-logd.

16

Packing

• For P, hence STU, C(n, m, r), r ≥ 3, is WSNB iff m ≥ 2n-1.

[1,n]

[1,n]

n

[1,n-1]

n+1

n

[1,n-1]

n+1

n+1n

[1,n-1]

n+1

n+1n

[1,n-1]

n+1

n+1

n+2

n

[1,n-1]

n+1

n+2

n+2

n

[1,n-1]

n+1,n+2

n+2n

[1,n-1]

[n+1,2n-2]

17

MI

• C(n, m, r) is WSNB under MI iff m ≥ 2n-1

X

Y

2n-3

18

Packing+MI, STU+MI

• C(n, m, r), r ≥ 3, is WSNB iff m ≥ 2n-1.

19

3-stage Clos Network

• C(n1, r1, m, n2, r2)• C(2, 4, 3, 3, 2)

n1

n2

1

2

r1

r2

1

1

2

m

20

Asymmetry

• C(n1, r1, m, n2, r2) is WSNB iff m ≥ n1+n2-1 under every known routing strategies.

21

Multi-logd N Networks

• First proposed by Lea(1990).

Baseline

Baseline

Baseline

23

Baseline Example

• BL2(4)

26

Multi-logdN Network

• (Shyy & Lea 1991, Hwang 1998, Chang, Guo, and Hwang 2007) Multi-logdN network is SNB if and only if , where

odd. for 1d2even, for 1)1()(

21-n

2 1

nnddnp

n

• Maximal blocking configuration(MBC), is a set of requests blocking .

28

Graph Model

• The graph model of BL2(4)

1

2

3

4

5

6

1

2

3

4

1

2

3

4

5

6

1

2

3

4

5

6

29

P or STU

• Multi-logdN network is WSNB under P or STU iff p ≥ p(n).

30

MI

• Multi-logdN network is WSNB under MI iff p ≥ p(n).

I1

I2

O1

O2

I1

I2

O1

O2

stage

21n

21n

22n

22n

0

21n

d

0

1

0

21n

d2

1n

d

0

21-n

d

21-n

d

21-n

d

21-n

d

31

For n even

I1

I2

O1

O2

12

n

1I

2I

1O

2O

2n 1

2

n

0 0 0 0 0

12 n

d

12 n

d

2n

d

12)1( n

dd

12 2 n

d

1

1d

d

1d

12 d

1

12 n

d

12 n

d

12 n

d

12 n

d

2n

d

12 n

d

2n

d 2n

d

12)1( n

dd

12 2 n

d12 2

n

d

32

P+MI and STU+MI

• Multi-logdN network is WSNB under P+MI and STU+MI iff p ≥ p(n).

33

Generalizations

• General vertical-copy network.

37

Open Problems

• Find a single-selected case for P, even STU, under 3-stage Clos network or multi-logd network.

• Multi-log network with extra stage.• MI under vertical-copy network• Find a general argument independent of

any routing algorithm. • Conjecture: There is no good WSNB

algorithm under one-to-one traffic.

38

Thank you!

Fewest move

Alexander H. Frey, Jr. and David Singmaster. Handbook of Cubik Math. Enslow Publishers, 1982.◦最少 17步，最多 52步◦猜測 : 「上帝的數字 (God’s number)」為 20。

43

任意的魔術方塊，在幾步內可以完成呢？

Fewest move

Michael Reid.◦ Lower bounds: Superflip requires 20 face turns. (1995)

U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 ◦Upper bounds: New upper bounds. (1995)

Face turn metric(FTM): 29 moves. Quarter turn metric(QTM): 42 moves.

44

Fewest move

Silviu Radu.◦Rubik can be solved in 27f. (2006/04)◦Paper. (June 30, 2007)

Face turn metric(FTM): 27 moves. Quarter turn metric(QTM): 34 moves.

◦Using GAP(Groups, Algorithms, Programming):a System for Computational Discrete Algebra

45

Fewest moveDaniel Kunkle and Gene Cooperman, 26 Moves

Suffice for Rubik's Cube, Proc. of International Symposium on Symbolic and Algebraic Computation (ISSAC '07), ACM Press, 2007.◦FTM: 26 moves.◦Gene Cooperman, Larry Finkelstein, and Namita

Sarawagi. Applications of Cayleygraphs. In AAECC: Applied Algebra, Algebraic Algorithmsand Error-Correcting Codes, InternationalConference, pages 367-378 LNCS, Springer-Verlag, 1990. FTM: 11 moves. QTM: 14 moves.

46

Fewest move

Tomas Rokicki.◦25 Moves Suffice for Rubik’s Cube. (2008.3)◦23 Moves Suffice for Rubik’s Cube. (2008.4)◦22 Moves Suffice for Rubik’s Cube. (2008.8)◦God’s Number is 20. (2010.7)

47

3x3 方塊的變化數There are

Cube subgroups.

48

( 8 ! ∙ 38

3 )×( 12 ! ∙ 212

2 )× 12=43252003274489856000

¿ 4.3 ×1019

Partition into 2,217,093,120 sets of 19,508,428,800 positions.

Reduce to 55,882,296.About 35 CPU years.

49

God’s Number is 20

50

Distance Count of Positions0 11 182 2433 3,2404 43,2395 574,9086 7,618,4387 100,803,0368 1,332,343,2889 17,596,479,795

10 232,248,063,31611 3,063,288,809,01212 40,374,425,656,24813 531,653,418,284,62814 6,989,320,578,825,35815 91,365,146,187,124,31316 about 1,100,000,000,000,000,00017 about 12,000,000,000,000,000,00018 about 29,000,000,000,000,000,00019 about 1,500,000,000,000,000,00020 about 300,000,000

Number of Positions