Held: Friday, 21 November 2014

Back to Outline 44 - Tree Traversal. On to Outline 46 - Hash Tables, Continued.

Summary

We consider an essential implemntation of the Dictionary ADT - Hash tables. Hash tables provide such a common implementation that many programmers often refer to dictionaries as "hashes".

Related Pages

• EBoard 45 (Source 45) (HTML 45)

Overview

• An introduction to hash tables.
• Hash functions.
• An exercise.
• Handling collisions.
• Hashing in Java.
• Handling removal.

Administrivia

• New lab partners!

Upcoming Work

• Homework: Phase 2 of project (Class design, Algorithm design, I/O subsystem) due Tuesday night. (Yes, a few of you need extra time. If that's the case, let me know.)
• Reading for Monday: Hash Tables

Fun Things

• "Jingle Bell Holiday" in town on Friday night. (Yes, crash commercialism has moved this before Thanksgiving.) http://www.grinnellchamber.org/whats_new.cfm?newsID=142

Extra Credit

Academic

• CS Table today: ShellShock
• Learning from Alums, next Tuesday: Alex Cohn '11 (in person)
• CS Extras next Tuesday: Summer Opportunities in CS

Peer Support

• Ajuna's Radio show Mondays at 8pm. Listen to African music.
• Ezra's Radio show Thursdays at midnight. Radio melodrama.
• Charlie's Friday Night "War in Animated Film" ExCo. (Not this week)
• Copehagen, 7:30 p.m. on the 21st, 22nd, afternoon 23rd

Miscellaneous

• VP Student Affairs Candidate Open Sessions
  • Today, 4:15, JRC 209

Hash Tables

• As you may have noticed, dictionaries are "a lot like arrays" (except that you index dictionaries by object and arrays by number).
• However, in terms of implementation, that claim hasn't held.
  • The unordered list implementation has O(n) set and O(n) get.
  • An ordered array implementation has O(n) set and O(log₂n) get.
  • The binary search tree implementation has O(logn) set and O(logn) get, if we're lucky (or if we learn how to keep the tree balanced).
  • The skip list implementation has expected O(logn) set and O(logn) set.
• The corresponding array operations are O(1).
• Is there a dictionary implementation with O(1) get and set?
• Let's remind ourselves about the implementation strategies we know.
  • We can use a simple linked structure.
  • We can use a linked structure in which we do something clever (e.g., skip lists, binary search trees).
  • We can use an array (with extra space at the ends).
  • We can do clever things with the array (e.g., randomized linear structures)
• Which strategy might give us more efficient implementations?
• 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 set.
• How? Well, we know that arrays provide O(1) get and set, so use arrays (or vectors).
• 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 set a value, determine the number corresponding to the key and add 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 computing 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 (when two keys map to the same index).

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, it isn't possible to guarantee that no two objects have the same position (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
  • constant1firstLetter + constant2secondLetter
  • ...

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 first three 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 about 20 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 40.
  
  • Once you've computed your result, mod it by 40 (compute the remainder after dividing by 40)
• For my first name (Samuel),
• The sum is 19 (S) + 1 (A) + 13 (M) = 33
• The index is therefore 33.
• Is this a good hash function?
  
  • Why?
  • Why not?

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 for java.util.Hashtable
• Why are there three constructors?
• What methods are there other than get and set?
• Where's the hash function?
  • Answer: Every class is expected to provide its own.
• Why is there also a java.util.HashMap](http://docs.oracle.com/javase/8/docs/api/java/util/HashMap.html)

Handling Conflicting Keys

• Since we typically have many more possible keys than places in the array, we will eventually hit a point in which two keys hash to the same value.
• What should we do when this conflict happens?
• It is not reasonable to disallow the second key, since it's a different key.
• We could put a list of key/value pairs, rather than a singleton, at every position. This technique is sometimes called chaining.
• We could step through the array until we find an empty space. This technique is sometimes called linear probing. (You don't need to offset by 1; you can offset by a constant amount.)
• I thought people tended to use linear probing, but I see that Java uses chaining.

Removing Elements from Hash Tables

• Our analysis of Hash Tables to date has been based on two simple operations: get and set.
• What happens if we want to remove elements? This operation can significantly complicate matters.
• If we've chosen linear probing to resolve 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 remove it later when it's convenient to do so (e.g., when we grow the table).
  • 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).