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Pancake Problem

Dive into the Pancake Problem, investigating sorting strategies using reversible prefixes. Learn about the magic number 4 and optimize moves with a dual linear program.

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Pancake Problem

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  1. Pancake Problem 許凱平 & 張傑生

  2. Resources • Google • “Bounds for Sorting by Prefix Reversal”(63) • “Pancake Problem” (218) • Slides • [S1] CMU’s “Great Theoretical Ideas In Computer Science” course • http://www-2.cs.cmu.edu/~15251/Materials/Lectures/Lecture01/lecture01.ppt • Steven Rudich

  3. Resources • Slides • [S2] On the Generalization of the Pancake Problem • http://hsu14.cis.nctu.edu.tw/upload/on%20the%20generalization%20of%20the%20pancake%20network.ppt • Marissa P. Justan, Felix P. Muga II,Ivan Hal Sudborough, • [S3] Pancake problems with restricted prefix reversals and some corresponding Cayley networks • http://hsu14.cis.nctu.edu.tw/upload/pancake.ppt • Douglas W. Bass and I. Hal Sudborough

  4. Resources • Papers • [P1]Gates W.H., and Papadimitriou, C.H. Bounds for sorting by prefix reversal. Discrete Math. 27 (1979), 47--57. • 數學系系圖有 • [P2] H. Heydari and I. Hal Sudborough. On the Diameter of the Pancake Network. Journal of Algorithms 25, 67-94 (1997). • http://www.atcminc.com/mPublications/EP/EPATCM98/ATCMP003/paper.pdf

  5. History • [P1] • n+(1/16)n<= f(n) <=(5n+5)/3 • Guess 19n/16 <=f(n) • [P2] • N+(1/14)n<=f(n) • Disprove “Guess 19n/16 <=f(n)”

  6. Pre-Knowledge • Symmetric Group • S2#8 • S3#5 • Dual Linear Program • Skip!

  7. Terms • Adjacency • Two neighbor (position) element x,y are said to be adjacent (value) • Iff |x-y| = 1 • Free, Singleton • No neighbor of x is adjacent to x • x is a singleton • Block • Block is a group of adjacent element

  8. Terms • Positions • Values

  9. Q’ Q P P’ P Q Q’ P P’ Q P Q Q P

  10. Ideas • Algebra • Use two-sided pancake • to simulate block • in one-sided pancake problem • Capital for block • Non-Capital for singleton

  11. Notations

  12. Lemma • P’-P • PQ--Q’P’

  13. Operations

  14. State Chart

  15. What it means? Q’P QP P’Q PQ P’Q’ PQ PQ PQ

  16. What will be PQR? • R’Q’P’

  17. Conjecture • 4 is the magic number

  18. Notations • adj: # of adjacency • blk: # of block • Move, flip

  19. t is free, t+o is also free • t is head of string, t is glue • t is free • t-1, t+1 is not the second element • o =1 or o=-1 • adj++; blk++

  20. t is free, t+o is 1st of a block • adj++;

  21. t is in block, t+o is free • Adj++

  22. t is in block, t+o is 1st of blk • Move++; • Adj++; • Blk--;

  23. t is free, t+o is also free • t#s--#ts,o=-1 • t#u--#tu,o=1

  24. t is free, t+o is 1st of a block • t#S' -- #tS, o=-1 • t#U -- #tU, o=1

  25. Case 3: t is free, S, U’ • For o=-1 • Steps • t#S#U’#--- • S’#t#U’ #-- • #St#U’ #----- • U#tS’##-- • #U’tS’## • Observation • Move+=4 • Adj+=2 • Blk--;

  26. Case3: t is free, S, U’ • For o=-1 • Steps • t1S2U’3 --- • S’1’t2U’3-- • 1St2U’3 ----- • U2’tS’1’3-- • 2U’tS’1’3

  27. New Notation • Case1, fig2(a) • t#s--#ts • t#s-(2)-#ts

  28. New Notation • Case3, fig2(c) t#S#U’#-(3)- S’#t#U’ #-(2)- #St#U’ #-(5)- U#tS’##-(2)- #U’tS’## • Briefly • t#S#U’#-(3,2,5,2)-#U’tS’##

  29. 1’3 Chances：無心插柳柳成蔭

  30. Case 6, fig2(f) or 2(g) • 2(f) • T#u#S#-(3,2,5,2)-#STu## • 2(g) • T#U’#s#-(3,2,5,2)-#sTU## • Move+=4;Adj+=2;Blk--; • Case 6 share the same operation property of Case 3, ground…

  31. Table2

  32. Bill’s Typos or I’m wrong • Case 5 rename to Case7, and Case7 rename to Case5 • Case 7, 2(e) • Case 5, 2(h) or 2(k)

  33. 3 basic equation • When string is sorted • move = Sum( move(i) * #case(i) ) • Let x(i) = case(i) • Adj = n-1 • Blk =1

  34. Total move: Use Table2 • z=X(1)+X(2)+4X(3)+X(4)+2X(5)+X(7)

  35. adj: Use Table2 • a: initial adjacency • a+X(1)+X(2)+2X(31)+3X(32)+3X(33)+3X(34)+X(4)+X(5)+X(7) • Eq(1) • n-1=a+X(1)+X(2)+2X(31)+3X(32)+3X(33)+3X(34)+X(4)+X(5)+X(7)

  36. blk : Use Table2 • b: initial blk • B+x(1)-x(31)-x(33)-2x(34)-x(5)-x(7) • Eq(2) • 1=B+x(1)-x(31)-x(33)-2x(34)-x(5)-x(7)

  37. (1)(3), b<=a • Eq(1) Eq(3) • n-1=a+X(1)+X(2)+2X(31)+3X(32)+3X(33)+3X(34)+X(4)+X(5)+X(7) • b<=a • N-1>=b+X(1)+X(2)+2X(31)+3X(32)+3X(33)+3X(34)+X(4)+X(5)+X(7) • Eq(3)

  38. Upper bound • 1=B+x(1)-x(31)-x(33)-2x(34)-x(5)-x(7) • N-1>= b+X(1)+X(2)+2X(31)+3X(32)+3X(33)+3X(34)+X(4)+X(5)+X(7) • z=X(1)+X(2)+4X(3)+X(4)+2X(5)+X(7) • Goal: Max(z) • X1=(n+1)/3, x2=x4=x5=x7=b=0 • X3=x31=(n-2)/3 • Z = (5n-7)/3 • Upper bound = z+4=(5n+5)/3

  39. Dual Linear Program • Duality Theorem of Von Neumna, Kuhn and Tucker, Gale, and Dantzig • I did not study this…

  40. Amazing Network

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