10/20/10
18:26:13 1
23
CS 61B: Lecture 23
Wednesday, October 20, 2010
Today’s reading: Goodrich & Tamassia, Chapter 5.
DICTIONARIES (continued)
============
Hash Codes
----------
Since hash codes often need to be designed specially for each new object,
you’re left to your own wits. Here is an example of a good hash code for
Strings.
private static int hashCode(String key) {
int hashVal = 0;
for (int i = 0; i < key.length(); i++) {
hashVal = (127 * hashVal + key.charAt(i)) % 16908799;
}
return hashVal;
}
By multiplying the hash code by 127 before adding in each new character, we
make sure that each character has a different effect on the final result. The
"%" operator with a prime number tends to "mix up the bits" of the hash code.
The prime is chosen to be large, but not so large that 127 * hashVal +
key.charAt(i) will ever exceed the maximum possible value of an int.
The best way to understand good hash codes is to understand why bad hash codes
are bad. Here are some examples of bad hash codes on Words.
[1] Sum up the ASCII values of the characters. Unfortunately, the sum will
rarely exceed 500 or so, and most of the entries will be bunched up in
a few hundred buckets. Moreover, anagrams like "pat," "tap," and "apt"
will collide.
[2] Use the first three letters of a word, in a table with 26^3 buckets.
Unfortunately, words beginning with "pre" are much more common than
words beginning with "xzq", and the former will be bunched up in one
long list. This does not approach our uniformly distributed ideal.
[3] Consider the "good" hashCode() function written out above. Suppose the
prime modulus is 127 instead of 16908799. Then the return value is just
the last character of the word, because (127 * hashVal) % 127 = 0.
That’s why 127 and 16908799 were chosen to have no common factors.
Why is the hashCode() function presented above good? Because we can find no
obvious flaws, and it seems to work well in practice. (A black art indeed.)
Resizing Hash Tables
--------------------
Sometimes we can’t predict in advance how many entries we’ll need to store. If
the load factor n/N (entries per bucket) gets too large, we are in danger of
losing constant-time performance.
One option is to enlarge the hash table when the load factor becomes too large
(typically larger than 0.75). Allocate a new array (typically at least twice
as long as the old), then walk through all the entries in the old array and
_rehash_ them into the new.
Take note: you CANNOT just copy the linked lists to the same buckets in the
new array, because the compression functions of the two arrays will certainly
be incompatible. You have to rehash each entry individually.
You can also shrink hash tables (e.g., when n/N < 0.25) to free memory, if you
think the memory will benefit something else. (In practice, it’s only
sometimes worth the effort.)
Obviously, an operation that causes a hash table to resize itself takes
more than O(1) time; nevertheless, the _average_ over the long run is still
O(1) time per operation.
Transposition Tables: Using a Dictionary to Speed Game Trees
-------------------------------------------------------------
An inefficiency of unadorned game tree search is that some grids can be reached
through many different sequences of moves, and so the same grid might be
evaluated many times. To reduce this expense, maintain a hash table that
records previously encountered grids. This dictionary is called a
_transposition_table_. Each time you compute a grid’s score, insert into the
dictionary an entry whose key is the grid and whose value is the grid’s score.
Each time the minimax algorithm considers a grid, it should first check whether
the grid is in the transposition table; if so, its score is returned
immediately. Otherwise, its score is evaluated recursively and stored in the
transposition table.
Transposition tables will only help you with your project if you can search to
a depth of at least three ply (within the five second time limit). It takes
three ply to reach the same grid two different ways.
After each move is taken, the transposition table should be emptied, because
you will want to search grids to a greater depth than you did during the
previous move.
STACKS
======
A _stack_ is a crippled list. You may manipulate only the item at the top of
the stack. The main operations: you may "push" a new item onto the top of the
stack; you may "pop" the top item off the stack; you may examine the "top" item
of the stack. A stack can grow arbitrarily large.
| | | | | | -size()-> 2 |d| -top()-> d | |
|b| -pop()-> | | -push(c)-> |c| |c| | | -top()--
|a| | |a| |a| -push(d)--> |a| --pop() x 3--> | | |
--- v --- --- --- --- v
b EmptyStackException
public interface Stack {
public int size();
public boolean isEmpty();
public void push(Object item);
public Object pop() throws EmptyStackException;
public Object top() throws EmptyStackException;
}
In any reasonable implementation, all these methods run in O(1) time.
A stack is easily implemented as a singly-linked list, using just the front(),
insertFront(), and removeFront() methods.
Why talk about Stacks when we already have Lists? Mainly so you can carry on
discussions with other computer programmers. If somebody tells you that
an algorithm uses a stack, the limitations of a stack give you a hint how
the algorithm works.
Sample application: Verifying matched parentheses in a String like
"{[(){[]}]()}". Scan through the String, character by character.
o When you encounter a lefty--’{’, ’[’, or ’(’--push it onto the stack.
o When you encounter a righty, pop its counterpart from atop the stack, and
check that they match.
If there’s a mismatch or exception, or if the stack is not empty when you reach
the end of the string, the parentheses are not properly matched.
