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Wide-Sense Nonblocking Under New Compound Routing Strategies

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. Applications. Come from the need to interconnect telephones Interconnect processors with memories Data transmission Conference calls

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Wide-Sense Nonblocking Under New Compound Routing Strategies

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  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. Multi-stage interconnection network outputs inputs stage

  5. Symmetry • C(n, m, r) • C(2, 4, 3) 1 1 1 n n 2 2 2 r r m

  6. Definitions • Request • Strictly nonblocking(SNB) • Wide-sense nonblocking(WSNB)

  7. Matrix O1 O2 O3 1 4 I1 2 I2 1 3 2 I3

  8. 3-stage Clos Network • [Clos 1953] C(n1, r1, m, n2, r2) is SNB iffm ≥ n1+n2-1. • C(n, m, r) is SNB iffm ≥ 2n-1.

  9. n,n+1 n+2 1,2,3 4,5 2n-1 [6,10] 11 [12,n-1] n+3 n+4 [n+5,2n-2]

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

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

  12. Literature review • Smith(1977) proved that C(n, m, r) is not WSNB under P or MI if . • Du etal.(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 etal.(2004) improved to . • Chang etal.(2007) extend it to Multi-logd.

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

  14. MI • C(n, m, r) is WSNB under MI iffm ≥ 2n-1 2n-3 X Y

  15. Packing+MI, STU+MI • C(n, m, r), r ≥ 3, is WSNB iffm ≥ 2n-1.

  16. 3-stage Clos Network • C(n1, r1, m, n2, r2) • C(2, 4, 3, 3, 2) 1 n1 1 1 n2 2 2 r2 m r1

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

  18. Baseline Baseline Baseline Multi-logd N Networks • First proposed by Lea(1990).

  19. Baseline Example • BL2(4)

  20. Multi-logdN Network • (Shyy & Lea 1991, Hwang 1998, Chang, Guo, and Hwang 2007) Multi-logdN network is SNB if and only if , where • Maximal blocking configuration(MBC), is a set of requests blocking .

  21. 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

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

  23. stage O1 I1 I1 O1 I2 O2 I2 O2 MI • Multi-logdN network is WSNB under MI iffp ≥ p(n).

  24. I1 O1 I2 O2 For n even

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

  26. Generalizations • General vertical-copy network.

  27. 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.

  28. Thank you!

  29. Fewest move 任意的魔術方塊，在幾步內可以完成呢？ • 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。

  30. 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.

  31. Fewest move • SilviuRadu. • 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

  32. 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, 2007. • FTM: 26 moves. • Gene Cooperman, Larry Finkelstein, and NamitaSarawagi. 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.

  33. 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)

  34. 3x3方塊的變化數 • There are Cube subgroups.

  35. God’s Number is 20 • Partition into 2,217,093,120 sets of 19,508,428,800 positions. • Reduce to 55,882,296. • About 35 CPU years.

  36. Number of Positions

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