Computer Science/61b/Homework/hw6

Files

• /dict/HashTableChained.java
• /SimpleBoard.java
• uses a DList implementation from hw4 in /list
• other files included as necessary but not releasable

Grade

10/10

Description

                             CS 61B  Homework 6
                     Due 4pm Friday, October 27, 2006

This homework will teach you about hash tables, hash codes, and compression functions. This is an individual assignment; you may not share code with other students.

Copy the Homework 6 directory by doing the following, starting from your home directory. Don't forget the "-r" switch in the cp command.

   cp -r $master/hw/hw6 .

Part I (6 points)

Implement a class called HashTableChained, a hash table with chaining. HashTableChained implements an interface called Dictionary, which defines the set of methods (like insert(), find() and remove()) that a dictionary needs. Both files appear in the "dict" package.

The methods you will implement are a subset of those listed on page 389 of Goodrich and Tamassia (no iterators are used in this assignment), plus a makeEmpty() method which removes every entry from a hash table. There are also two HashTableChained constructors. One lets applications specify an estimate of the number of entries that will be stored in the hash table; the other uses a default size. Both constructors should create a hash table that uses a prime number of buckets. (Several methods for identifying prime numbers were discussed early in the semester.) In the first constructor, shoot for a load factor between 0.5 and 1. In the second constructor, shoot for around 100 buckets. Descriptions of all the methods may be found in Dictionary.java and HashTableChained.java.

Do not change Dictionary.java. Do not change any prototypes in HashTableChained.java, or throw any checked exceptions. Most of your solution should appear in HashTableChained.java, but other classes are permitted. You will probably want to use a linked list code, of your choice. (Note that even though the hash table is in the "dict" package, it can still use linked list code in a separate "list" package. There's no need to move the list code into the "dict" package.)

Look up the hashCode method in the java.lang.Object API. Assume that the objects used as keys to your hash table have a hashCode() method that returns a "good" hash code between Integer.MIN_VALUE and Integer.MAX_VALUE (that is, between -2147483648 and 2147483647). Your hash table should use a compression function, as described in lecture, to map each key's hash code to a bucket of the table. Your compression function should be computed by the compFunction() helper method in HashTableChained.java (which has "package" protection so we can test it). insert(), find(), and remove() should all use this compFunction() method.

The methods find() and remove() should return (and in the latter case, remove) an entry whose key is equals() to the parameter "key". Reference equality (==) is NOT required for a match.

Compression functions

Besides the lecture notes, compression functions are also covered in Section 9.2.4 of Goodrich and Tamassia. Unfortunately, they make the erroneous claim that for a hash code i and an N-bucket hash table,

 h(i) = |ai + b| mod N

is "a more sophisticated compression function" than

 h(i) = |i| mod N.

Actually, the "more sophisticated" function causes exactly the same collisions as the less sophisticated compression function; it just shuffles the buckets to different indices. The better compression function, as I said in lecture, is

 h(i) = ((ai + b) mod p) mod N,

where p is a large prime that's substantially bigger than N. (You can replace the parentheses with absolute values if you like; it doesn't matter much.)

For this homework, the simplest compression function might suffice. The bottom line is whether you have too many collisions or not in Part II. If so, you'll need to improve your hash code or compression function or both.

Part II (4 points)

It is often useful to hash data structures other than strings or integers. For example, game tree search can sometimes be sped by saving game boards and their evaluation functions, so that if the same game board can be reached by several different sequences of moves, it will only have to be evaluated once. For this application each game board is a key, and the value returned by the minimax algorithm is the value stored alongside the key in the hash table. If our search encounters the same game board again, we can look up its value in the dictionary, so we won't have to calculate the value twice.

The class SimpleBoard represents an 8x8 checkerboard. Each position has one of three values: 0, 1, or 2. Your job is to fill in two missing methods: equals() and hashCode(). The equals() operation should be true whenever the boards have the same pieces in the same locations. The hashCode() function should satisfy the specifications described in the java.lang.Object API. In particular, if two SimpleBoards are equals(), they have the same hash code.

You will be graded on how "good" your hash code and compression function are. By "good" we mean that, regardless of the table size, the hash code and compression function evenly distribute SimpleBoards throughout the hash table. Your solution will be graded in part on how well it distributes a set of randomly constructed boards. Hence, the sum of all the cells is not a good hash code, because it does not change if cells are swapped. The product of all cells is even worse, because it's usually zero. What's better? One idea is to think of each cell as a digit of a base-3 number (with 64 digits), and convert that base-3 number to a single int. (Be careful not to use floating-point numbers for this purpose, because they round off the least significant digits, which is the opposite of what you want. Better to round off the most significant digits, which is what happens when an int gets too big.)

Do not change any prototypes in SimpleBoard.java, or throw any checked exceptions. The file Homework6Test.java is provided to help you test your HashTableChained and your SimpleBoard together. Note that Homework6Test.java does NOT test all the methods of HashTableChained; you should write additional tests of your own. Moreover, you will need to write a test to see if your hash code is doing a good job of distributing SimpleBoards evenly through the table. Our autograder will do extensive tests on that.

A tutorial on collision probability

Students are always surprised when they find out how many collisions occur in a working hash table. You might have the misimpression that there won't be many collisions at all until the table is nearly full. Let's analyze how many collisions you should expect to see if your hash code and compression function are good.

If you have N buckets and a good (pseudorandom) hash function, the probability of any two keys colliding is 1/N. So when you have i keys in the table and insert key i + 1, the probability that the new key does not collide with any old key is (1 - 1/N)i. If you insert n distinct items, the expected number that won't collide is

n-1
 ∑ (1 - 1/N)i = N - N (1 - 1/N)n
i=0

so the expected number of collisions is n - N + N (1 - 1/N)n.

Now, for any n and N you test, you can just plug them into this formula and see if the number of collisions you're getting is in the ballpark of what you should expect to get. For example, if you have N = 100 buckets and n = 100 keys, expect about 36.6 collisions.