1. d-heaps Same as binary heaps, except d children instead of 2.
array implementation
insert: O(logd N)
delete: O(d logd N)
Why d-heaps? 10 16 12 80 45 27 78 23 95 15 90 95 87 54 59
Nov 7, 2001
CSE 373, Autumn 2001

2. Quadtrees Application: image representation (graphics)
Idea: recursively represent the image into finer and finer detail.
Nov 7, 2001
CSE 373, Autumn 2001

3. Octrees Similar idea for 3-D images.
Hidden surface removal: Draw cells that are “farther back” first.
Nov 7, 2001
CSE 373, Autumn 2001

4. Other applications Hash tables: B-trees:
compilers: keeping track of symbols
finding out what is known about a genetic sequence
passwords: hash functions
B-trees:
drivers license database
telephone directory
Nov 7, 2001
CSE 373, Autumn 2001

5. Optimal BSTs Suppose each key has a different frequency of access.
unsorted list: keep the most frequently accessed items at the front of the list.
By analogy, in a BST, more frequently accessed keys ought to be kept closer to the root.
A: 0.35, B: 0.3, C: 0.35 A A B C C B C A C A B C B B A
Nov 7, 2001
CSE 373, Autumn 2001

6. Finding the Optimal Tree
Brute force: (4n / n3/2) trees.
Let K1 < K2 < … < Kn be the keys in their dictionary order, and let pi be the probability of accessing Ki.
Let T be any tree constructed from the keys Kj, …, Kk, where 1  j  k  n.
T is optimal if its cost is as small as the cost of any other tree with the same keys.
Nov 7, 2001
CSE 373, Autumn 2001

7. Observation: Every subtree of an optimal tree is itself optimal.
Suppose d  0, if we know the optimal trees for Ki, …, Ki+c for c < d and 1  i  n-d, then we can compute the optimal tree for the keys Kj, …, Kj+d for all j  n-d. T TR TL
Nov 7, 2001
CSE 373, Autumn 2001

8. How? For all possible roots, find the left and right subtrees that are optimal. Choose the root that results in the tree with the lowest cost. Recursion? No! n C 1 1
(n3)
running time n
C[j,k] – cost of the optimal tree on keys j through k
r[j,k] – root of the optimal tree
Nov 7, 2001
CSE 373, Autumn 2001

9. Probability-balanced trees
Choose the root Kl so that
p(1, l-1)  p(l+1, n)
Can build the tree in linear time.
Can prove that the expected search time is within a factor of 2 of optimal.
In practice: within a few percent of optimal.
Nov 7, 2001
CSE 373, Autumn 2001