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Correctness of Constructing Optimal Alphabetic Trees Revisited

Correctness of Constructing Optimal Alphabetic Trees Revisited. Theoretical computer science 180 (1997) 309-324. Marek Karpinski, Lawrence L. Larmore, Wojciech Rytter. Outline. Definitions General version of Garsia-Wachs (GW) algorithm Proof of GW Hu-Tacker (HT) algorithm

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Correctness of Constructing Optimal Alphabetic Trees Revisited

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

  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. Definitions Binary tree: Every internal node has exactly two sons

  4. Definitions

  5. The Move Operator

  6. The Move Operator

  7. The Move Operator

  8. The Move Operator

  9. Definitions

  10. Theorem 1 (correctness of GW)

  11. Garsia-Wachs Algorithm

  12. Definitions

  13. Theorem 2

  14. Shift Operations

  15. Shift Operations

  16. LeftShift Example

  17. LeftShift Example

  18. LeftShift Example

  19. LeftShift Example

  20. LeftShift Example

  21. LeftShift Example

  22. LeftShift Example

  23. LeftShift Example

  24. LeftShift Example

  25. Theorem 2

  26. Proof of Point 2 in Theorem 2

  27. Proof of Point 2 in Theorem 2

  28. Proof of Point 3 Theorem 2

  29. Definition of Well Shaped Segments

  30. Definition of Well Shaped Segments Active Window

  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. Movability Lemma

  33. Movability Lemma

  34. Movability Lemma

  35. Movability Lemma

  36. Movability Lemma

  37. Theorem 3

  38. Point 1 in Theorem 2

  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. Hu-Tucker Algorithm

  41. Hu-Tucker Algorithm

  42. Hu-Tucker Algorithm

  43. Hu-Tucker Algorithm

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

  45. Definitions Normal sequence – sequence of weights Special sequence – sequence of weights, each one is either transparent or opaque 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. MoveTransparent

  47. MoveTransparent

  48. MoveTransparent

  49. MoveTransparent

  50. MoveTransparent

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