Class 42: Hash Tables

Held Friday, November 12, 1999

Overview

Today, we'll consider another (and somewhat different) implementation of dictionaries: the hash table. Hash tables usually provide O(1) get and put operations.

Notes

• Don't forget that I expect mostly-working code for your portions of the project on Monday.
• I need a volunteer to collect Pizza orders (I'll pick it up and pay for it once I have details).

Contents

• Detour: The Options Group
• Hash Tables
  • Hash Functions
  • An Exercise in Hashing
  • Handling Collisions
• Hashing in Java
• Removing Elements from Hash Tables

Summary

• Possible visit from prospectives
• Hash tables: efficient dictionaries
  • Designing hash functions

Handouts

• Documentation for java.util.Hashtable

Detour: The Options Group

• Since we are employing dictionary-like objects in the mail project (in particular, the Options class), it's appropriate to spend a few more minutes considering that part.
• Presumably, options needs to be stored between invocations of the email package.
• Where should those options be stored?
  • And whose responsibility is it to remember the location?
• For our purposes, it will be easiest to store the options in a file. The group and I have decided on one of the following formats:

  Option-Name:Option-Type:Option-Value

  or

  Option-Name
  Option-Type
  Option-Value

• The group is willing to deal with the basic types: booleans, integers, doubles, and Strings.
• However, they need the cooperation of other groups creating classes that might be stored as options, such as the Ordering class
  • You need to provide a reasonable toString method that generates a string with no embedded carriage returns.
  • You need to provide a constructor that takes a string (as generated by your toString method) as its only parameter.
• Initially, they will only support a prespecified set of classes.
  • I've written some code that they can use for using any class that supports these two operations.

Hash Tables

• Surprisingly, if you're willing to sacrifice some space and increase your constant, it is possible to build a dictionary with O(1) get and put.
• How? By using an array, and numbering your keys in such a way that
  • all numbers are between 0 and array.length-1
  • no two keys have the same number (or at least few have the same number).
• If there are no collisions (keys with the same number), the system is simple
  • To put a value, determine the number corresponding to the key and put it in that place of the array. This is O(1+cost of computing that number).
  • To get a value, determine the number corresponding to the key and look in the appropriate cell. This is O(1+cost of finding that number).
• Implementations of dictionaries using this strategy are called hash tables.
• The function used to convert an object to a number is the hash function.
• To better understand hash tables, we need to consider
  • The hash functions we might develop.
  • What to do about collisions.

Hash Functions

• The goal in developing a hash function is to come up with a function that is unlikely to map two objects to the same position.
  • Now, this isn't possible (particularly if we have more objects than positions).
  • We'll discuss what to do about two objects mapping to the same position later.
• Hence, we sometimes accept a situation in which the hash function distributes the objects more or less uniformly.
• It is worth some experimentation to come up with such a function.
• In addition, we should consider the cost of computing the hash function. We'd like something that is relatively low cost (not just constant time, but not too many steps within that constant).
• We'd also like a function that does (or can) give us a relatively large range of numbers, so that we can get fewer collisions by increasing the size of the hash table.
  • We might want to make the size of the table a parameter to the hash function.
  • We might strive for a hash function that uses the range of positive integers, and mod it by the size of the table.
• What are some hash functions you might use for strings?
  • Sum the ASCII values in the string
  • N*first letter + M*second letter
  • ...

An Exercise in Hashing

• Let's try an exercise. We'll come up with a hash value for everybody's first name. We'll then put things in the hash table.
• We'll use ``sum the values of the letters in the name''.
• We'll use the following table:

  A: 1   F: 6   K: 11  P: 16  U: 21  Z: 26
  B: 2   G: 7   L: 12  Q: 17  V: 22
  C: 3   H: 8   M: 13  R: 18  W: 23
  D: 4   I: 9   N: 14  S: 19  X: 24
  E: 5   J: 10  O: 15  T: 20  Y: 25
  

• There are 24 of you in the class. Typically, our hash tables are somewhat bigger than the size of the collection we're working with, so we'll use a hash table of size 50.
  • Once you've computed your result, mod it by 50.
• For my name (Samuel), the hash value is 19 (S) + 1 (A) + 13 (M) + 21 (U) + 5 (E) + 12 (L) = 71. Mod that by 50 to get 21.
• For one son's name (William), the hash value is 23 (W) + 9 (I) + 12 (L) + 12 (L) + 9 (I) + 1 (A) + 13 (M) = 79. Mod that by 50 to get 29.
• For the other son's name (Jonathan), the hash value is 10 (J) + 15 (O) + 14 (N) + 1 (A) + 20 (T) + 8 (H) + 1 (A) + 14 (N) = 83. Mod that by 50 to get 33.
• What if the hash table has only twelve cells (which would be large if I were storing only three objects)? Then both Jonathan and I hash to cell 11.

Handling Collisions

• What do you do when you try to insert an object into a hash table, and there's already an object at the position (but with a different key)?
  • You can put multiple objects in the same position (by using a list, a tree, or even another hash table with a different hash function).
  • You can find another place in the table for the object by changing the hash value.
  • You can expand the hash table.
• The first method requires us to provide a dynamic data structure for each cell of the table. It also means that the cost for getting an element also involves searching through the structure.
• The second method requires us to do more than one ``step'' when looking up a value (as we might need to repeatedly skip already filled spaces).
• The third method requires us to grow the table regularly, and at a significant cost (proportional to the number of elements in the table).
• The second method also assumes a limited number of objects will be placed in the table. when you delete an object, you need to consider whether one of the objects in other cells really belonged there.
• There are a number of ways you can compute the ``next'' space in the second method.
  • You can offset by a fixed amount.
  • You can use a second function to determine the offset (making the offset object-dependent).
  • You can can use a sequence of hash functions.
  • ...
• Despite all this joyous freedom, I'll admit that I prefer the first method.

Hashing in Java

• Hash tables are so useful that Java includes them as a standard library class, java.util.Hashtable.
• Let's look over the documentation
• Why are there three constructors?
• What methods are there other than get and put?
• Where's the hash function?

Removing Elements from Hash Tables

• Our analysis of Hash Tables to date has been based on two simple operations: get and put.
• What happens if we want to remove elements? This can significantly complicate matters.
• If we've chosen the ``shift into a blank space'' technique for resolving collisions, what do we do when it comes time to remove elements?
  • Do we shift everything back? If so, think about how far we may have to look.
  • Do we leave the thing there as a blank? We might then then remove it later when it's convenient to do so.
  • Do we do something totally different?
• Note also that there are different ways of specifying ``remove''. We might remove the element with a particular key. We might instead remove elements based on their value. The second is obviously a much slower operation than the first (unless we've developed a special way to handle that problem - see if you can think of one).