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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

- 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(log
_{2}n) 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.

- 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
- ...

- 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?

- 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?

- 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).

Wednesday, 23 August 2000

- Created as a blank outline.

Thursday, 24 August 2000

- Slight reorganization to page design.

Tuesday, 28 November 2000

- Filled in the details. Many come from outline 43 of CSC152 2000S.

Back to Binary Search Trees. On to No Class.

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