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

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

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

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

5. Definitions

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

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

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

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

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

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

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

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

25. Theorem 1 (correctness of GW)

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

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

29. B
lexi trees over π
lexi trees over π’ such that i, i+1, are at the same depth
lexi trees over π such that i, i+1, are at the same depth
opt
lexi trees over π’ such that i, i+1, are siblings
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 shaped iff it is h-isolated and depthT(i) =depthT(i+1) = h+1
S=[i,…j] is right well shaped iff 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

54. Homework
To finish we have to consider one more case where i, and i+1 are siblings.
We also have to prove that depthT(LCA(j,j+1)) ≤ h
Case (b) of the THM is similar

55. 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

56. Hu-Tucker Algorithm

57. Hu-Tucker Algorithm

58. Hu-Tucker Algorithm

59. Hu-Tucker Algorithm