Oct 29, 2001 CSE 373, Autumn 2001 1

External Storage• For large data sets, the computer

will have to access the disk.

• Disk access can take 200,000 times longer than a machine instruction.

• The RAM model does not account for disk I/O.

memory

disk

128 MBfast, expensive

60 GBslow, cheap

Oct 29, 2001 CSE 373, Autumn 2001 2

Disks, continued

• The difference between memory speed and disk speed is increasing.

• Example: State of Florida driving records (256 bytes). 10,000,000 items. 6 disk accesses per second on a time-sharing system.

• unbalanced binary search tree: possibly 10,000,000 accesses.

• BST: on avg. 32 accesses (5 sec.)

• AVL: worst: 1.44 log n

typical case: log n, 25 accesses (4 sec.)

Oct 29, 2001 CSE 373, Autumn 2001 3

Disk accesses

• Goal: reduce the number of disk accesses.

• We are willing to do more complicated computations in memory in order to save disk time.

• Idea: increase the branching of the tree so that the height is decreased.

• Defn: An M-ary search tree allows up to M children per node.

Oct 29, 2001 CSE 373, Autumn 2001 4

B-Trees1. All the data items are stored at

the leaves.

2. The non-leaf nodes store up to M-1 keys. The ith key represents the smallest key in subtree i+1.

3. The root is either a leaf of has between 2 and M children.

4. All non-leaf nodes (except the root) have between M/2 and M children.

5. All leaves are at the same depth and have between L/2 and L data items.

Oct 29, 2001 CSE 373, Autumn 2001 5

B-Trees: Choices

• Choose M and L based on the size of the keys K and on the size of the record R.

• Suppose a disk block is of size B (bytes). Choose M so that a non-leaf node fits into one block:

B (M-1) · K + M · 4

• Choose L so that a leaf node fits into one block:

B L · R

• accesses: log2 N vs. logM/2 N

Oct 29, 2001 CSE 373, Autumn 2001 6

Hash Tables

• Constant time accesses!

• A hash table is an array of some fixed size, usually a prime number.

• General idea:

key space (e.g., strings)

0

…

TableSize –1

hash func.h(K)

hash table

Oct 29, 2001 CSE 373, Autumn 2001 7

Desirable Properties

We want a hash function to:

1. be simple/fast to compute,

2. map different keys to different cells, (impossible – why?)

3. have keys distributed evenly among cells.

Idea: If #1 and #3 are true and the hash table is not very full, then it should be fast to do a find.

Oct 29, 2001 CSE 373, Autumn 2001 8

Example

• key space = integers

• h(K) = K mod 10

0

1 41

2

3

4 34

5

6

7 7

8 18

9

We lose all ordering information:findMin, findMax, inorder traversal, printing items in sorted order.

Oct 29, 2001 CSE 373, Autumn 2001 9

Example 2

• key space = strings

• s = s0 s1 s2 … s k-1

h(s) = s0 mod TableSize

BAD HASH FUNCTION

h(s) = mod TableSize

BETTER HASH FUNCTION

1

0

37k

i

iis

Oct 29, 2001 CSE 373, Autumn 2001 10

Collision Resolution• Separate chaining: All keys that

map to the same hash value are kept in a list.

0

1

2

3

4

5

6

7

8

9

10

107

22 12 42