“Sense shines with a double luster when it is set in humility. An able and yet humble man is a jewel worth a kingdom.” – William Penn Thought for the Day

External Hash Table: Iterators Slightly more complicated: –need to work through array –and work through linked lists table... key value key value key value

The HashTableIterator Class private class HashTableIterator... { private int index; private EntryNode nextEntry; public HashTableIterator () { for (index = 0; index < table.length; index++) if (table[index] != null) break; // First non-empty bucket if (index < table.length) // Have data nextEntry = table[index]; } // constructor public Pair get () { return nextEntry; } // get... } // class HashTableIterator

public void next () { nextEntry = nextEntry.next; if (nextEntry == null) // Look for next non-empty bucket while (++index < table.length) { if (table[index] != null) // Found more data { nextEntry = table[index]; break; } } // next public boolean atEnd () { return index >= table.length; } // atEnd

Uses of the Hash Table Classes SSame interface ( Dictionary ) as the ListDictionary class Similar applications –more efficient –But: unordered

Dictionary ADT Summary ADTAdvantagesDisadvantages List- Dictionary Ordered Flexible size Slow access Internal- HashTable Fast accessUnordered Fixed size External- HashTable Fast access Flexible size Unordered

O Section 3 Algorithm Analysis Big-O

Chapter 8: Big-O Objectives –Introduce algorithm analysis –Study methods for measuring algorithm efficiency and thus comparing algorithms –Consider some common results –Show some simple applications of the techniques of algorithm analysis

Analysing Algorithms Many algorithms may appear simple and efficient This may not be true in fact!

Example Solving simultaneous equations Cramer’s Rule Calculate the determinant For n equations, it takes n! operations

Example (cont.) If n = 30 (a very small set of equations) n! = 30! = 3 × Computers are very fast Assume 10 9 operations per second (1GHz) Time taken = years Longer than the estimated life of the universe!

Example (cont.) Another approach is needed! The tri-diagonal method Needs about 10n operations If n = 30, time taken = seconds

Implications Need to choose algorithms very carefully This example focussed on time –other resources (memory, disk space, etc.) may also be important

Algorithmic Complexity Not how difficult it is to understand! How it behaves with respect to time (or other resources) Example: we say that Cramer’s Rule has a complexity of n!

Algorithmic Complexity Need to measure complexity in a way that is independent of external factors –compiler optimisations, speed of computers, etc. Find some important operation(s) –Often comparisons, or data exchanges –Count them

Use to Compare Algorithms Algorithm 1 –20 comparisons –30 exchanges Algorithm 2 –100 comparisons –150 exchanges  On Apple II (1MHz) –30s On Pentium 4 (3GHz) –3s

The Impact of the Input Data Vitally important Must compare “apples with apples” But we don’t want to get into specific details! Use some abstract measure of data Example: –Cramer’s Rule: n equations

Input Data (cont.) Often the number of data items But not always! Example: text searching algorithms –length of search string x length of document

Input Data (cont.) Often three cases we need to consider: –average case –best case –worst case

Big-O Notation Or order notation Assume we can find a function giving the number of operations: f(n) = 3.5n n + 45 Dominant term Order n 2 or more simply: O(n 2 )