1. Hashing
Jeff Chastine

2. Hashing Many applications require INSERT, SEARCH and DELETE functions
Hashing on average time can do all of these in O (1)
Based on keys
Falls under two general categories:
Direct-Address Tables
Hash Tables
Jeff Chastine

3. Direct-Addressing Good for when universe U of keys is small
U = {0, 1, …, m – 1 | m is not large}
All elements have unique keys
Table T [0..m -1] | each slot corresponds to a key
All operations take only O (1)
Jeff Chastine

4. Direct Implementation
key
satellite data 1 U
(universe of keys) 2 2 3 9 6 3 7 4 4 1 2 5 K
(actual
keys) 5 3 6 5 8 7 8 8 9
Jeff Chastine

5. Direct-Addressing Operations
DIRECT-ADDRESS-SEARCH (T, k) return T[k] DIRECT-ADDRESS-INSERT (T, x) T[key[x]] ← x DIRECT-ADDRESS-DELETE (T, x) T[key[x]] ← NIL
Jeff Chastine

6. Hash Tables What are potential problems with direct addressing?
|U| may be impractical
Set of actual keys may be small
Example SSNs
Here, hash tables require much less storage
Only catch: O (1) is average time instead of worst-case !
Jeff Chastine

7. How it works
With direct-addressing, something with key k goes into slot k
With hashing it goes into h (k) | h is a hash function
Hash functions try to “randomize”
Hash function maps U to T [0..m – 1]
h :U → {0, 1, …, m – 1}
Instead of |U| values,need only m values
Jeff Chastine

8. Hash Implementation T U (universe of keys) K (actual keys) k1 k4 k5 k2U
(universe of keys)
h (k1)
h (k4) k1
h (k2) = h (k5) K
(actual
keys) k4 k5 k2 k3
h (k3)
m - 1
Jeff Chastine

9. Collisions Have two keys hash to the same slot
Because |U| > m, pigeon hole principle
Therefore, collisions must exist
We often talk of the load factor (α = n/m)
Pick a good hash function
Near random, yet deterministic
Can chain collisions together
This is where the worst-case comes from
Can use open addressing
Jeff Chastine

10. Chaining T U (universe of keys) K (actual keys) k1 k7 k4 k7 k1 k5 k2
Jeff Chastine

11. Hash Functions What makes a good hash function?
Equally likely to hash to any of the m slots
If keys are random numbers [0 … 1} then take floor of km
Convert strings to ASCII to hash?
Most usually involve mod
Jeff Chastine

12. Hash Functions Division method: Multiplication method:
h (k ) = k mod m
Multiplication method:
Let 0 < A < 1
h (k ) = floor(m (k A mod 1) ) // Fractional part
Jeff Chastine

13. Open Addressing
Systematically examine or probe slots until item is found
No lists and no elements stored outside the table; thus α <= 1
Instead of following pointers, we compute the sequence
Instead of fixed order – is based off of key
Jeff Chastine

14. Kinds of Open Addressing
Linear Probing
h (k, i ) = (h’ (k ) + i ) mod m
Quadratic Probing
h (k, i ) = (h’ (k ) +c1i + c2i 2) mod m
Double Hashing
h (k, i ) = (h1(k ) + i h2(k )) mod m
Jeff Chastine

15. Jeff Chastine