Hashing, just a simple note about hashing.

1. Hashing
Dictionary
Direct-access Table
Chaining
Hash Function

2. Dictionary
● Abstract Data type, maintain set of items with keys.
– Insert (item) – also replacing existed item
– Delete (item)
– Search (key): return the item with given key or report if
doesn't exist (null).
Item → (key, value)

3. Simple Approach: DAT
Storing items directly into
giant array which is the
index of item as key.
0 NULL
1 ITEM 1
2 NULL
3 NULL
4 ITEM 2
...
...
m-1 ...

4. Direct-access
Binary search
items[11]
0 null
1 null
... ...
7 7
8 null
9 9
10 null
11 11
Direct access
i→ 1 2 4 5 7 9 11
Search for 11, low: 0, upper: 5, mid = 5/2 = 2
| check for 11
i→ 2 2 4 5 7 9 11
Low = 3, upper = 5, mid = 4
|check for 11
i→ 3 2 4 5 7 9 11
Low = 5, upper = 5, mid = 5
Found 11 |
i→ 4 2 4 5 7 9 11

5. Two Big Problem
(1)Keys may be negative integers
● Maps keys into non negative integers (such as String or
string of bits)
(2)Gigantic memory
● Hashing (cut into piece)

6. Big idea
Reduce universe () of all keys (integers)
down to reasonable size of m for table. 0 NULL
1 ITEM 1
2 NULL
3 NULL
4 ITEM 2
...
...
m-1 ...
k1
k2
k3
k4
keyspace
h(k2) = 1
h(k3) = 4
h(k)
Ideal: m = (n)

7. Problem
● There probably collision
– i.e:
h(kx) = h(kj) but kx ≠ kj
0 NULL
1 ITEM 1
2 NULL
3 NULL
4 ITEM 2
...
...
m-1 ...
k1
k2
k3
k4
keyspace
h(k1) = 4
h(k4) = 4
h(k)

8. Chaining
● Use linked list to store value with collide keys.
0 NULL /
1 ITEM 1 /
2 NULL /
3 NULL /
4 ITEM 2 ITEM 3 /
...
... /
m-1 ... /
k1
k2
k3
k4
keyspace
h(k1) = 4
h(k4) = 4
h(k) Worst cases:
- a bunch of keys is
mapped into the same
index

9. Length of chain
Expected length of chain for n keys with m slots:
n
m
=⍺
⍺ = load factor of table
⍺ = O(1) if m = O(n)

10. Hash functions
(1) Division Method:
h(k )=k modm
(2) Multiplication method
h(k )=[(a . k )mod 2w ]≫(w−r )
a = integer
w = bit of machine
r = bits to shift
m=2r

11. Cont...
(3) Universal hashing
h(k )=[(a . k+b)mod p ]modm
Random integer
∈ {0,1,..., p-1}
Prime > ||
Worst cases: k1 ≠ k2
Pr
[h (k 1)=h (k 2)]
=
1
m

12. Sample of code