1. Treaps

2. Definition
A treap is a binary search tree in which each node has both a key and a priority. Nodes are ordered in an in-order fashion by their keys and are heap-ordered by their priorities.

3. Keys
a b c d e f g
Priorities d 1 a3 f 2 c 4 e 5 g 6 b 7

4. Existence Why must a treap with any priorities exist?
We can always create one by sorting by priority and then inserting as a binary search tree!

5. Insertion How do we insert into an existing treap?
Insert as a BST, then perform a series of tree rotations to enforce the heap ordering invariant.

6. Rotation Review – R to Rootx r B C C r x B A

7. Rotation Review – R to Rootx r B C C r x B A

8. Deletion Given an existing treap, how do we delete a node?
Rotate it down to a leaf, and then delete it.

9. Theorem
Expected # of comparisons for insert, search, delete is Θ(log n)
Equivalent to showing the expected height is Θ(log n)
We will prove a weaker statement:
Expected height of a single node is O(log n)

10. Oh god, a proof
Suppose we search
for a node m.

11. Oh god, a proof
Xi = 1 if i is an ancestor of m
0 otherwise

12. Oh god, a proof
depth(m) = Σ1 ≤ i ≤ nXi

13. E[depth(m)] = E[Σ1 ≤ i ≤ nXi]
Oh god, a proof
E[depth(m)] = E[Σ1 ≤ i ≤ nXi]

14. E[depth(m)] = Σ1 ≤ i ≤ nE[Xi]
Oh god, a proof
E[depth(m)] = Σ1 ≤ i ≤ nE[Xi]

15. Oh god, a proof
E[depth(m)] =
Σ1 ≤ i ≤ nPr[ i is an ancestor of m ]

16. Finding Pr[ i is an ancestor of m ]
Suppose i ≤ m.
Consider [i, i+1, …, m-1, m]
i must have the highest priority out of these

17. Finding Pr[ i is an ancestor of m ]
Suppose i > m.
By symmetry:

18. Oh god, a proof
E[depth(m)] =
Σ1 ≤ i ≤ nPr[ i is an ancestor of m ]

19. Oh god, a proof
E[depth(m)] =
Σ1 ≤ i ≤ m Σm+1 ≤ i ≤ n

20. Oh god, a proof
E[depth(m)] =
Σ1 ≤ i ≤ m Σm+1 ≤ i ≤ n
≤ 2 log n

21. COUNTING ROTATIONS
Theorem. The expected number of rotations for insertion or deletion is < 2
Right-most nodes in the left subtree – L(x)
Left-most nodes in the right subtree – R(x) x
R(x)
L(x)

22. COUNTING ROTATIONS
Lemma. Consider the treap obtained right after insertion of node x. Then the number of rotations performed during insertion of x is L(x)+R(x) x x

23. COUNTING ROTATIONS
Let Di be the event that i is in the left subtree (red) m
CLAIM:

24. Informal proof or Game of Darts1 i m n
Don’t care
Don’t care
Remaining
darts
Second dart
First dart

25. COUNTING ROTATIONS
Let D*i be the event that i is in the right subtree (blue) m
CLAIM:

26. COUNTING ROTATIONS
Lemma. Consider the treap obtained right after insertion of node x. Then the number of rotations performed during insertion of x is L(x)+R(x)