Class 51: Hash Tables

Back to Binary Search Trees. On to No Class.

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).

History

Wednesday, 23 August 2000

• Created as a blank outline.

Thursday, 24 August 2000

• Slight reorganization to page design.

Tuesday, 28 November 2000

Back to Binary Search Trees. On to No Class.

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