1 Wide-Sense Nonblocking Under New Compound Routing Strategies Junyi David Guo( 郭君逸 ) 師範大學 數學系 2011/06/29 Joint 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.

4 Preliminaries Crossbars, X nm inlets, outlets 1 2 n 1 2 m 1 2 n 1 2 m

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 n 1 2 r

7 Definitions Request Strictly nonblocking(SNB) Wide-sense nonblocking(WSNB)

8 Matrix O1O1 O2O2 O3O3 I1I1 I2I2 I3I3 1 3

9 3-stage Clos Network [Clos 1953] C(n 1, r 1, m, n 2, r 2 ) is SNB iff m ≥ n 1 +n 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

13 Literature review

14 Cyclic Dynamic CD can reach any state. Theorem C(n, m, r) is WSNB under CD iff m ≥ 2n-1. [1,n-1] [n,2n-2]2n-1

15 Cyclic Static C(n, m, r) is WSNB under CS iff m ≥ 2n-1 [1,n] [n,2n-2] [1,n-1] [n,2n-2]

16 Packing For P, hence STU, C(n, m, r), r ≥ 3, is WSNB iff m ≥ 2n-1. [1,n] n [1,n-1] n+1 n [1,n-1] n+1 n [1,n-1] n+1 n [1,n-1] n+1 n+2 n [1,n-1] n+1 n+2 n [1,n-1] n+1,n+2 n+2 n [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(n 1, r 1, m, n 2, r 2 ) C(2, 4, 3, 3, 2) n1n1 n2n2 1 2 r1r1 r2r m

20 Asymmetry C(n 1, r 1, m, n 2, r 2 ) is WSNB iff m ≥ n 1 +n 2 -1 under every known routing strategies.

21 Multi-log d N Networks First proposed by Lea(1990). Baseline

22 Baseline d-nary baseline network BL d (n) d n inputs, d n outputs n stages d n-1 d  d crossbars BL d (n-1) 1 2 d n X dd

23 Baseline Example BL 2 (4)

24 Multi-log d N Network Theorem(Shyy & Lea 1991, Hwang 1998) Multi- log d N network is SNB if p ≥ p(n), where

25 Theorem Multi-log d N network is SNB only if p ≥ p(n). x yxy

26 Multi-log d N Network

27 Theorem Multi-log d N network is WSNB under CD and CS iff p ≥ p(n).

28 Graph Model The graph model of BL 2 (4)

29 P or STU Multi-log d N network is WSNB under P or STU iff p ≥ p(n).

30 MI Multi-log d N network is WSNB under MI iff p ≥ p(n). I1I1 I2I2 O1O1 O2O2 I1I1 I2I2 O1O1 O2O2 stage

31 For n even

32 P+MI and STU+MI Multi-log d N network is WSNB under P+MI and STU+MI iff p ≥ p(n).

33 Generalizations General vertical-copy network.

34 Generalizations A vertical-copy network V is WSNB under the CS and CD routing iff V is SNB. Theorem A vertical-copy network V is WSNB under the P routing iff V is SNB.

35 Larger Perspective Theorem(Tsai, Wang, and Hwang 2001) For, C(n, m, r) is WSNB if and only if m ≥ 2n-1.

36 Conclusion In 3-stage Clos Networks, the conclusions for all r and all known algorithms are known. The proofs are not messy. The necessity condition of Multi-log d N network is SNB. Extended to multi-log n N netowrks and vertical-copy netowrks. Find a relation between multi-log n N netowrks and 3-stage Clos network. The concept of MBC

37 Open Problems Find a single-selected case for P, even STU, under 3-stage Clos network or multi-log d 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!

39 Beneš(1965) Theorem C(n, m, 2) is WSNB under STU if m ≥ 3n/2. a b c d |a  b  c  d| < 3n/2 |a  d| < n |b  c| < n

40 Minimum Index 1,3,4,5,6,7 2,8 4,5,6,7 1,2,3,8

41 Packing

42 Reduce the result of Tsai et al. to polynomial bound of r and extend to multi-log d N networks.

Fewest move Alexander H. Frey, Jr. and David Singmaster. Handbook of Cubik Math. Enslow Publishers, ◦ 最少 17 步，最多 52 步 ◦ 猜測 : 「上帝的數字 ( God’s number) 」為 20 。 43

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 F2Superflip requires 20 face turns ◦ Upper bounds: New upper bounds. (1995)New upper bounds  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) Rubik can be solved in 27f ◦ Paper. (June 30, 2007) Paper  Face turn metric(FTM): 27 moves.  Quarter turn metric(QTM): 34 moves. ◦ Using GAP(Groups, Algorithms, Programming): a System for Computational Discrete AlgebraGAP 45

Fewest move Daniel Kunkle and Gene Cooperman, 26 Moves Suffice for Rubik's Cube, Proc. of International Symposium on Symbolic and Algebraic Computation (ISSAC '07), ACM Press, ◦ FTM: 26 moves. ◦ Gene Cooperman, Larry Finkelstein, and Namita Sarawagi. Applications of Cayleygraphs. In AAECC: Applied Algebra, Algebraic Algorithmsand Error-Correcting Codes, InternationalConference, pages LNCS, Springer- Verlag,  FTM: 11 moves.  QTM: 14 moves. 46

Fewest move Tomas Rokicki. ◦ 25 Moves Suffice for Rubik’s Cube. (2008.3) 25 Moves Suffice for Rubik’s Cube ◦ 23 Moves Suffice for Rubik’s Cube. (2008.4) 23 Moves Suffice for Rubik’s Cube ◦ 22 Moves Suffice for Rubik’s Cube. (2008.8) 22 Moves Suffice for Rubik’s Cube ◦ God’s Number is 20. (2010.7) God’s Number is 20 47

3x3 方塊的變化數 There are Cube subgroups. 48

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 Number of Positions