1. Correctness of Constructing Optimal Alphabetic Trees Revisited Marek Karpinski, Lawrence L. Larmore, Wojciech Rytter Theoretical computer science 180 (1997) 309-324

2. 18/12/2000 2 Outline Definitions General version of Garsia-Wachs (GW) algorithm Proof of GW Hu-Tacker (HT) algorithm Proof of HT by similarity to GW

3. 18/12/2000 3 Definitions Binary tree: Every internal node has exactly two sons

4. 18/12/2000 4 Definitions

5. 18/12/2000 5 The Move Operator

6. 18/12/2000 6 The Move Operator

7. 18/12/2000 7 The Move Operator

8. 18/12/2000 8 The Move Operator

9. 18/12/2000 9 Definitions

10. 18/12/2000 10 Theorem 1 (correctness of GW)

11. 18/12/2000 11 Garsia-Wachs Algorithm

12. 18/12/2000 12 Definitions

13. 18/12/2000 13 Theorem 2

14. 18/12/2000 14 Shift Operations

15. 18/12/2000 15 Shift Operations

16. 18/12/2000 16 LeftShift Example

17. 18/12/2000 17 LeftShift Example

18. 18/12/2000 18 LeftShift Example

19. 18/12/2000 19 LeftShift Example

20. 18/12/2000 20 LeftShift Example

21. 18/12/2000 21 LeftShift Example

22. 18/12/2000 22 LeftShift Example

23. 18/12/2000 23 LeftShift Example

24. 18/12/2000 24 LeftShift Example

25. 18/12/2000 25 Theorem 2

26. 18/12/2000 26 Proof of Point 2 in Theorem 2

27. 18/12/2000 27 Proof of Point 2 in Theorem 2

28. 18/12/2000 28 Proof of Point 3 Theorem 2

29. 18/12/2000 29 Definition of Well Shaped Segments

30. 18/12/2000 30 Definition of Well Shaped Segments Active Window

31. 18/12/2000 31 Movability Lemma If the segment [i,…,j] is left well shaped, then the active pair (i,i+1) can be moved to the other side of the segment by locally rearranging sub-trees in the active window without changing the relative order of the other items and without changing the level function of the tree.

32. 18/12/2000 32 Movability Lemma

33. 18/12/2000 33 Movability Lemma

34. 18/12/2000 34 Movability Lemma

35. 18/12/2000 35 Movability Lemma

36. 18/12/2000 36 Movability Lemma

37. 18/12/2000 37 Theorem 3

38. 18/12/2000 38 Point 1 in Theorem 2

39. 18/12/2000 39 Hu-Tucker Algorithm Transparent items and opaque items Compatible pair – No opaque items in the middle Minimal compatible pair (mcp) – compatible pair (i,i+1) where Weight(i) + weight(i+1) is minimal Tie Breaking Rule

40. 18/12/2000 40 Hu-Tucker Algorithm

41. 18/12/2000 41 Hu-Tucker Algorithm

42. 18/12/2000 42 Hu-Tucker Algorithm

43. 18/12/2000 43 Hu-Tucker Algorithm

44. 18/12/2000 44 GW` Algorithm gmp – Globaly Minimal Pair GW` - the same as GW but always choose gmp instead of some other lmp.

45. 18/12/2000 45 Definitions Special sequence – sequence of weights, each one is either transparent or opaque Normal sequence – sequence of weights MoveTransparent operator – converts a special sequence into a normal sequence and moves all transparent items to their RightPos. (first it moves the rightmost item, then the one to its left, etc…)

46. 18/12/2000 46 MoveTransparent

47. 18/12/2000 47 MoveTransparent

48. 18/12/2000 48 MoveTransparent

49. 18/12/2000 49 MoveTransparent

50. 18/12/2000 50 MoveTransparent

51. 18/12/2000 51 MoveTransparent

52. 18/12/2000 52 MoveTransparent

53. 18/12/2000 53 MoveTransparent

54. 18/12/2000 54 MoveTransparent

55. 18/12/2000 55 MoveTransparent

56. 18/12/2000 56 MoveTransparent

57. 18/12/2000 57 MoveTransparent

58. 18/12/2000 58 MoveTransparent

59. 18/12/2000 59 MoveTransparent

60. 18/12/2000 60 MoveTransparent

61. 18/12/2000 61 MoveTransparent

62. 18/12/2000 62 The Simulation Lemma Assuming there are no ties

63. 18/12/2000 63 Proof of the Simulation Lemma Claim A Claim B

64. 18/12/2000 64 Proof of Claim A By contradiction assume that w,u are not visible to each other. This means there is an opaque item q between them. Two cases:

65. 18/12/2000 65 Proof of Claim A Contradiction – no place for q

66. 18/12/2000 66 Proof of Claim A

67. 18/12/2000 67 Proof of Claim A Contradiction – no place for q

68. 18/12/2000 68 Proof of Claim B Claim B Proof

69. 18/12/2000 69 Proof of Claim B The order of movements is from right to left, thus w is Processed first then the items between u and w, and then u. At this point u and w must be adjacent, thus q must be processed later, and thus it is to the left of u. q is not visible from u because (u,w) is mcp

70. 18/12/2000 70 Proof of Claim B Let q` be the opaque item between q and u visible from u Contradiction

71. 18/12/2000 71 Conclusion

72. 18/12/2000 72 Claim C

73. 18/12/2000 73 Proof of claim C

74. 18/12/2000 74 Proof of Simulation Lemma The proof of the lemma is by induction

75. 18/12/2000 75 Tie Braking Rule Theorem 4 The Tie Breaking Rule (TBR) is correct Proof

76. 18/12/2000 76 Proof of TBR Case 1: All weights are strictly positive

77. 18/12/2000 77 Proof of TBR Case 2: Some of the original weights are zero