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Lecture 10 Hashing

Explore the motivation, properties, collisions, solutions, universal hash function, modular arithmetic, and designing hash functions. Learn the concepts behind efficient lookup and space optimization in hashing.

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Lecture 10 Hashing

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  1. Lecture 10 Hashing

  2. Motivation: Set and Map • Goal: An array whose index can be any object. • Example: DictionaryDictionary[“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, 643523} • 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. 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

  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 • 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 • Claim: For every x, y (x≠y), any two numbers u, v in {0, 1, 2, …, p-1}, we have

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