1. 17CS1102
DATA STRUCTURES
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2. Hashing Goal Topics Collision handling Rehashing
Perform inserts, deletes, and finds in constant average time
Topics
Hash table, hash function, collisions
Collision handling
Separate chaining
Open addressing: linear probing,
quadratic probing, double hashing
Rehashing
Load factor

3. Tree Structures
Binary Search Trees
AVL Trees
Splay Trees
B-Trees

4. Tree Structures insert / delete / find worst average
Binary Search Trees N log N
AVL Trees log N
Splay Trees amortized log N*
B-Trees amortized log N*
*Individual operations may require more than O(log N) time, but a sequence of M operations will average O(M log N) time.

5. Goal
Develop a structure that will allow user to insert/delete/find records in
constant average time
structure will be a table (relatively small)
table completely contained in memory
implemented by an array
capitalizes on ability to access any element of the array in constant time

6. Hash Tables
When K is much smaller than U, a hash table requires much less space than a direct-address table
Can reduce storage requirements to |K|
Can still get O(1) search time, but on the average case, not the worst case

7. Hash Tables
Idea:
Use a function h to compute the slot for each key
Store the element in slot h(k)
A hash function h transforms a key into an index in a hash table T[0…m-1]:
h : U → {0, 1, , m - 1}
We say that k hashes to slot h(k)
Advantages:
Reduce the range of array indices handled: m instead of |U|
Storage is also reduced

8. Example: HASH TABLES U (universe of keys) h(k1) h(k4) k1 K (actualU
(universe of keys)
h(k1)
h(k4) k1 K
(actual
keys) k4 k2
h(k2) = h(k5) k5 k3
h(k3)
m - 1

9. Revisit Example 2

10. Do you see any problems with this approach?U
(universe of keys)
h(k1)
h(k4) k1 K
(actual
keys) k4 k2
h(k2) = h(k5)
Collisions! k5 k3
h(k3)
m - 1

11. Hash Function Thus, since f(x) = x % 15, if x = 25 129 35 2501 47 36
Storing the keys in the array is not a problem.
Array:
_ _ _ _ _ _ _ _ _

12. Hash Function What happens when you try to insert: x = 65 ? x = 65
f(x) = 5
Array:
_ _ _ _ _ _ _ _ _
65(?)

13. Hash Function What happens when you try to insert: x = 65 ? x 65
f(x) 5
Array:
65(?)
This is called a collision.