1. Lecture 10 Hashing

2. Motivation: Set and Map
Goal: An array whose index can be any object.
Example: Dictionary Dictionary[“hash”] = “a dish of diced or chopped meat and often vegetables…”
Properties: 1. Efficient lookup: Hope lookup is O(1) 2. Space: space is within constant factor to a list.

3. Naïve implementation of a set
Method 1: Maintain a linked list.
Problem: Lookup takes O(n) time.
Method 2: Use a large array
a[i] = 1 if i is in the set
Problem: Needs huge amount of memory.

4. Hashing Idea: for each number, assign a random location
Example: {3, 10, 3424, }
Store number i in a[f(i)]
f(i): hash function.

5. Collisions Problem: want to add 123, f(123) = 4 = f(3424).
(This will always happen because of pigeon hole principle)
Solution: 123 and 3424 will share this location.
null 10 3
3424
643523
123

6. Fixed Hash Function
If the hash function is fixed, then it can be very slow for some bad examples.
Example: We can try to find n numbers x1, x2, …, xn such that f(xi) = y for some fixed y (always possible by pigeon hole principle)
Then hash table degenerates into a linked list.
Solution: Use a family of random hash functions.

7. Universal Hash Function
Hash function should be as “random” as possible.
Ideally: Choose a random function out of all functions!
However: cannot store a totally random function.
Can use modular arithmetic to construct good hash functions!
Goal: Construct a family of hash functions F, such that for any x ≠ y, we have Pr 𝑓∼𝐹 𝑓 𝑥 =𝑓 𝑦 = 1 𝑛 .

8. Recap: Modular Arithmetic
For a prime number p, only consider numbers {0, 1, 2, 3, …, p-1}
Can do addition, subtraction, multiplication the usual way (take mod p at the end).
Inverse: For any integer 0 < x < p, there is an integer 0 < y < p such that
𝑥𝑦≡1(𝑚𝑜𝑑 𝑝)
Example: p = 7, x = 2, then y = 4. We call y = x-1
Inverse can be computed efficiently.

9. Designing the Hash function
Pick a prime number p, construct a hash family with p2 functions
For every a, b in {0,1,2,…, p-1}, we have
𝑓 𝑎,𝑏 𝑥 =𝑎𝑥+𝑏 (𝑚𝑜𝑑 𝑝)
Claim: For every x, y (x≠y), any two numbers u, v in {0, 1, 2, …, p-1}, we have
Pr 𝑓 𝑥 =𝑢, 𝑓 𝑦 =𝑣 = 1 𝑝2