Class 51: Hash Tables

Held Wednesday, November 29, 2000

Summary

Today we visit yet another implementation of dictionaries: hash tables. Hash tables are interesting in that they provide operations that are likely to have O(1) time.

Notes

• Are there any final questions on exam 3?
  • I'd hope to receive many of them today so that I can grade them while I'm away (ho ho ho).
• Update on search.
• I'm still thinking about the last assignment. Do you really want another assignment?
• Form of the final (in-class).
• No class Friday. I think most of you need a break.

Overview

• Dictionaries vs. arrays
• An idea: Turn objects into numbers
• Implications of idea: Hash tables
• Writing hash functions
• Hashing in Java

Hash Tables

• As you may have noticed, although dictionaries are "a lot like arrays" [AP] (except that you index dictionaries by object and arrays by number), the best implementation we've seen so far for dictionaries has O(log₂n) add and get while the corresponding array operations are O(1).
• Is there a dictionary implementation with O(1) get and put?
• Surprisingly, if you're willing to sacrifice some space and increase your constant, it is possible to build a dictionary that is likely to have O(1) get and put.
• How? Well, we know that arrays provide O(1) get and put, so use arrays.
• How do we use an array? We "number" the 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 N 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 2*N.
  • Once you've computed your result, mod it by 2*N.
• For my name (Samuel), the hash value is 19 (S) + 1 (A) + 13 (M) + 21 (U) + 5 (E) + 12 (L) = 71.
• 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.
• 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.
• Is this a good hash function?

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).