The algorithm of Garsia and Wachs

1. The algorithm of Garsia and Wachs Presentation by a more recent proof of Karpinski et al.

2.  wi The problem We will identify the items with their weights w1, w2 . . . . wn ,wi ≥ 0 We are after a binary tree with w1, w2 . . . . wn at the leaves from left to right such that depth(i) is minimized

3. 1 2 3 Observation 1 If we new the di’s (depths) then we could build the tree easily in O(n) time 2 3 4 4 1

4. Observation 1 (Cont) So we will focus on how to find the depths 1 2 3 2 3 4 4 1

5. Definitions

6. 13 9 14 5 7 8 6 13 8 6 9 14 12 12 14 14 12 5 7 21 13 12 13 15 23 The algorithm 13 8 5 7 6 9 14

7. The algorithm (in words) Combine an LMP put the resulting node right before the first node to the right which is larger or equal Claim: The depths defined by the resulting tree are the depths of an optimal alphabetic tree. Ok, great, suppose we believe it, how do we implement this efficiently ?

8. Implementation (Bob Tarjan) We will always combine the rightmost LMP. • Maintain the current list of weights broken into sublists and singletons. The following invariant should hold • In a list of length at least 3: aj ≤ aj+2 • In a list of length at least 2 the next to last element is smaller than the element following the list.

9. Where can we have LMP • All pairs in a list except the leftmost are not LMPs. • If the list is of length at least 2 then the Rightmost element and the following on are not LMP.

10. Implementation (Cont) We pick the last two lists A and B and check whether b1 and b2 are LMP. If not we simply catenate A and B and repeat. Otherwise, we delete b1 and b2 from B and combine them to create b’. We search for the first item in B’ which is no smaller than b’ We split B’ just before that item that item to B1 and B2 and add b’ to the end of B1

11. Implementation (Cont) We represent sublists as search tree

12. Implementation (Cont) 13 8 5 7 6 9 14 7 6 5 9 13 8 14 5 7 7 9 14 We can find the first item from the left that is greater than or equal to some value in logarithmic time

13. 6 9 13 8 14 6 9 14 13 13 7 5 6 9 13 8 14 6 9 14 7 5 Implementation (Cont) 7 6 5 9 13 8 14 5 7 7 9 14

14. Implementation Each combination of LMP triggers a constant number of search tree operations. (2 deletion, search, split, insertion) Number of sublists is O(n), since there are n items and n-1 combinations of LMPs. Therefore the number of catenations is O(n) Total running time is therefore O(nlogn)

15. But why does this algorithm produce the optimal tree ??

16. Notation

17. Definitions (Cont)

18. Theorem 1 (correctness of GW) By induction on the length of the sequence, correctness reduces to:

19. lexi trees over π such that i, i+1, are at the same depth opt We will prove more lexi trees over π opt

20. lexi trees over π such that i, i+1, are at the same depth opt lexi trees over π A opt

21. lexi trees over π’ such that i, i+1, are siblings lexi trees over π such that i, i+1, are at the same depth opt lexi trees over π lexi trees over π’ such that i, i+1, are at the same depth opt opt

22. lexi trees over π‘ such that i, i+1, are siblings lexi trees over π such that i, i+1, are at the same depth opt B lexi trees over π lexi trees over π‘ such that i, i+1, are at the same depth opt opt

23. lexi trees over π‘ such that i, i+1, are siblings lexi trees over π such that i, i+1, are at the same depth opt C lexi trees over π lexi trees over π‘ such that i, i+1, are at the same depth opt opt

24. lexi trees over π’ such that i, i+1, are siblings lexi trees over π such that i, i+1, are at the same depth opt C lexi trees over π lexi trees over π’ such that i, i+1, are at the same depth opt opt

25. Theorem 1 (correctness of GW)

26. lexi trees over π such that i, i+1, are at the same depth opt lexi trees over π A opt

27. Proof of A There is an optimal lexi tree in which i, and i+1 are at the same level.

29. lexi trees over π’ such that i, i+1, are siblings lexi trees over π such that i, i+1, are at the same depth opt B lexi trees over π lexi trees over π’ such that i, i+1, are at the same depth opt opt

30. Proof of B Among optimal trees over π’ in which i, and i+1 are on the same level there is one in which they are siblings. So T’ is also optimal

31. Definition of Well Shaped Segments • A set S of leaves of T is h-isolated iff: • For any uS, depthT(u) ≥ h • For any uS, wS, depthT(LCA(u,w)) ≤ h

32. Definition of Well Shaped Segments S=[i,…j] is left well shapediff it is h-isolated and depthT(i) =depthT(i+1) = h+1 S=[i,…j] is right well shapediff it is h-isolated and depthT(j) =depthT(j+1) = h+1 Active Window

33. 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 depth function of the tree. (similar lemma holds for segments which are right well shaped)

34. Movability Lemma

35. Movability Lemma

36. Movability Lemma

37. Movability Lemma

38. Movability Lemma

39. Movability Lemma

40. Movability Lemma

41. Movability Lemma

42. Movability Lemma

43. Movability Lemma

44. Movability Lemma

45. Movability Lemma

46. The main theorem to establish C THM: (a) if T is optimal over π such that i, i+1 are at the same depth then the segment [i,….,j] is left well shaped in T (b) if T’ is optimal over π’ such that i, i+1 are at the same depth then the segment [i+2,….,j,i,i+1] is right well shaped in T Focus on (a)

47. Proof First we have to show that every leaf u[i,…,j], depth(u) ≥ h where h = depth(i)-1= depth(i+1)-1