CSC 2300 Data Structures & Algorithms February 27, 2007 Chapter 5. Hashing

Today – Splay Trees and Hashing Splay Trees – Delete a Node Hashing – Overview Hashing – Separate Chaining

Splay Tree – Delete a Node How do we delete a node? First, access the node. This puts the node at the root. After we delete the root, we get T L and T R. Which node should we select as the new root? The largest element in T L. Find this element (easy), and rotate it to the root position of T L. Will this new root of T L have a right child? If not, what can we do?

Examples in Class We will illustrate the idea with a few examples in class:

Zig-zag and Zag-zig The web site uses a different (and perhaps more intuitive) way to describe the actions of Zig-zag and Zag-zig. We will use the definitions in the text. Zig-zag as in text: The web site calls it zag-zig – we do a zag first, and then a zig.

Hashing – Overview

Hash Functions

Integer Input Keys If the input keys are integers, then a good strategy is to compute key mod tableSize. When may this strategy become a bad choice? As an example, let tableSize=10. Can you suggest a sequence of input keys that will all be mapped to the same cell? What is then a good choice for tableSize?

String Input Keys Choose tableSize as a prime number – a good choice according to the previous slide. Since memory is cheap, construct a large table. Let tableSize=10,007. Suppose that all keys are eight or fewer characters long. Use this hash function: What can happen to the hash table? Hint. What is the maximum integer value of an ASCII character?

String Input Keys So, we want hashVal to be large – greater than 10,007. Multiplication may be more appropriate than addition. Try this hash function: This function uses the first three characters of the input string. We can check that 26 x 27 2 > 10,007. Why do we use 26 x 27 2 and not 26 3 ? What may go wrong with distribution in this hash table?

String Input Keys Include also the ten numbers 0 to 9. Get this hash function: This function uses Horner’s rule. What is it?

Horner’s Rule Problem 2.14 in text. Evaluate f(x) = a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 x + a 0 Code: poly = 0; for( i=n; i>=0; i-- ) poly = x * poly + a[i]; Why does this algorithm work? What is its running time?

Compromises A hash function needs not be the best with respect to table distribution. But it should be simple and reasonably fast. If the keys are very long, the hash function will take too long to compute. A common practice is not to use all the characters. As an example, consider a complete street address. The hash function may include only a couple of characters from the street address, and a couple of characters from the city name and the zip code. The idea is that the time saved in computing the hash function will make up for a slightly less evenly distributed function.

Collision Resolution If, when an element is inserted, it hashes to the same value as an already inserted element, then we have a collision and need to resolve it. There are several methods for dealing with collision, we will discuss the simplest approach: separate chaining.

How to Handle Collisions

Separate Chaining Keep a list of all elements that hash to the same value. Example. The keys are the first 10 perfect squares and the hash function is hash(x) = x mod 10.

Hash Table with Separate Hashing

Example

Discussion Another data structure could be used to resolve the collisions; for example, binary search trees. Why do we use linked lists instead? We define the load factor, λ, of a hash table to be the ratio of the number of elements in the table to the table size. The average length of a list is λ. The effort to perform a search is the constant time required to evaluate the hash function plus the time to traverse the list. In an unsuccessful search, what is the average number of nodes to examine? In a successful search, what is the average number of nodes to examine? Why 1+(λ/2) and not (λ/2)?

Hashing Applet In class, we will run some hashing examples using this web site: