## Overview

**Brute force approach.** All data is just a sequence of bits. Can treat key as a gigantic number and use it as an array index. Requires exponentially large amounts of memory.

**Hashing.** Instead of using the entire key, represent entire key by a smaller value. In Java, we hash objects with a hashCode() method that returns an integer (32 bit) representation of the object.

**hashCode() to index conversion.** To use hashCode() results as an index, we must convert the hashCode() to a valid index. Modulus does not work since hashCode may be negative. Taking the absolute value then the modulus also doesn't work since Math.abs(Integer.MIN_VALUE) is negative. Typical approach: use hashCode & 0x7FFFFFFF instead before taking the modulus.

**Hash function.** Converts a key to a value between 0 and M-1. In Java, this means calling hashCode(), setting the sign bit to 0, then taking the modulus.

**Designing good hash functions.** Requires a blending of sophisticated mathematics and clever engineering; beyond the scope of this course. Most important guideline is to use all the bits in the key. If hashCode() is known and easy to invert, adversary can design a sequence of inputs that result in everything being placed in one bin. Or if hashCode() is just plain bad, same thing can happen.

**Uniform hashing assumption.** For our analyses below, we assumed that our hash function distributes all input data evenly across bins. This is a strong assumption and never exactly satisfied in practice.

**Collision resolution.** Two philosophies for resolving collisions discussed in class: Separate (a.k.a. external) chaining and 'open addressing'.

**Separate-chaining hash table.** Key-value pairs are stored in a linked list of nodes of length M. Hash function tells us which of these linked lists to use. Get and insert both require potentially scanning through entire list.

**Resizing separate chaining hash tables.** Understand how resizing may lead to objects moving from one linked list to another. Primary goal is so that M is always proportional to N, i.e. maintaining a load factor bounded above by some constant.

**Performance of separate-chaining hash tables.** Cost of a given get, insert, or delete is given by number of entries in the linked list that must be examined.

- The expected amortized search and insert time (assuming items are distributed evenly) is N / M, which is no larger than some constant (due to resizing).

**Linear-probing hash tables.** We didn't go over this in detail in 61B, but it's where you use empty array entries to handle collisions, e.g. linear probing. Not required for exam.

## Recommended Problems

## Example Implementations

### C level

[Adapted from Textbook 3.4.5] Is the following implementation of hashCode() legal?

`public int hashCode() { return 17; }`

If so, describe the effect of using it. If not, explain why.

### B level

- HW7.
- Problem 2 of the Fall 2014 midterm.

### A level

One strategy discussed in class for hashing objects containing multiple pieces of data is as follows (in pseudocode), where x.get(i) returns the ith piece of data in x:

int hashCode = 0; for (int i = 0; i < x.length; i += 1) {

`hashCode *= R; hashCode += x.get(i).hashCode(); hashCode = hashCode % M;`

} return hashCode;

Effectively, we're summing each piece of data multiplied by a different power of R. In class, we used R = 31, and M is the number of buckets. Explain why the idea above does not work well if M = 31 (or some power of 31).

- If we start with a hash table of size 2 and double when the load factor exceeds some constant. Why is this procedure for setting sizes suboptimal from the perspective of utilizing all of the bits of the hashCode?
- [Adapted from textbook 3.4.23] Suppose that we hash strings using R = 256 and M = 255, treating each string as a sequence of characters where the ith piece of data is the ith character. Show that any permutation of letters with a string hashes to the same value. Why is this a bad thing?
- Find 2 strings in Java that hash to the same value (writing code is probably best).
- CS61B Fall 2009 midterm, #4 (really beautiful problem)

### A+ level

- Give a simple procedure that can be carried out by hand that takes a Java string X and finds another Java string Y with the same hashCode().