Arrays Tonga Institute of Higher Education

Introduction An array is a data structure Definitions  Cell/Element – A box in which you can enter a piece of data.  Index – A number or string identifying the location of the cell Example: 0, 1, 2, 3, 4, 5  Field - A piece of data Example: 23, 11, 2, 35, 4, 43 An array must have cells arranged contiguously (Connecting without a break)  No holes allowed!

Arrays Indexes Index – A number identifying the location of the cell Index starts at 0  If your array size is 6, then the indexes are from 0 to

Demonstration Array Applet

Using Arrays in Java - 1 You need to declare and create them Can declare and create them in 2 steps: Can declare and create them in 1 step: Can declare, create and initialize them in 1 step: Bracket means it’s an array Must define the size when you create them Must state what the array will contain

Accessing Array Data in Java Syntax: [ ] Example  intArray3[0] => 0  intArray3[4] => 4  intArary3[10] => Index out of bounds error intArray3.length returns the size of the array

Example Array Code The size of the array is set here The kind of data stored In the array is set here We must state what cell to put the value in Loops are great for going through every item in an array

Common Array Errors Runtime Errors  ArrayIndexOutOfBoundsException – Occurs when we try to use a cell in an array that does not exist Compile Errors  incompatible types – Occurs when we try to put a value in a cell that is different from what the cell can store

Duplicate Keys Many times, we allow duplicate data Sometimes, we don’t want to allow duplicate data  Example: Rugby Jersey Number, Customer ID

Array Searches Don’t Allow Duplicate Keys  We must check every cell until we find our key.  The average number of steps needed to find an item is: Number of Items / 2 = N/2 Allow Duplicate Keys  We must check every cell to find all the cells that contain the key  The average number of steps need to do this is: Number of Items = N

Array Inserts Don’t Allow Duplicate Keys  If we assume that no one will try to enter duplicate data Then, we insert the item.  The number of steps needed to insert an item is:  1  If we assume that someone might try to enter duplicate data We have to check each field to make sure the new key isn’t a duplicate  The average number of steps needed to find an item is:  Number of Items / 2 = N/2 Then, we insert the item.  The number of steps needed to insert an item is:  1 Therefore, the total number of steps needed to insert an item is:  Steps required to check every cell + Steps required to insert and item  N/2 + 1 Allow Duplicate Keys  We know the location of the next empty cell because we know how many items are in the array.  A new item is inserted in the first empty cell  So the number of steps needed to insert an item is: 1

Array Deletes Don’t Allow Duplicate Keys  We must search for the key we want to delete. The average number of steps needed to find an item is:  Number of Items / 2  N/2  Then, we must shift each cell over The average number of steps needed to shift the remaining elements is:  Number of Items / 2  N/2  Therefore, the total number of steps needed is: Average Number of Steps to Find an Item + Average Number of Steps to Shift Remaining Items N/2 + N/2 N Allow Duplicate Keys  We must check every cell to find all the cells that contain the key The average number of steps need to do this is:  Number of Items = N  Then, we must shift each cell over for each deleted cell The average number of steps needed to shift the remaining elements is:  Number of Items / 2  N/2 Therefore, the average number of steps needed to shift the cells for multiple deletes is:  More than N/2  Therefore, the total number of steps needed is: Average Number of Steps to Find all Items + Average Number of Steps to Shift Remaining Items N + More than N/2

Array Performance Summary No DuplicatesDuplicates OK SearchN/2N Insert1* or N/2 + 1** 1 DeleteNN + More than N/2 *We are assuming that the data inserted will not duplicate any data. **Checks for duplicate data.

Encapsulated Array Object We can encapsulate the complexities of an array behind of object

Demonstration Encapsulation of Array Object Project: EncapsulatedArray

Object Arrays Only a few changes are needed to adapt a program to handle objects  The type of array is changed to an object. Example: Person  The key field is a field of the object rather than the value itself. Note: If the field is a string, we need to use the String.equals(…) method  The insert(…) method creates a new object and inserts it into an array.

Demonstration Array of Objects Project: ObjectArray

Ordered Arrays The data inside an array is ordered so:  The smallest value is at index 0  Each next cell holds a value larger than the previous cell

Demonstration Ordered Array Applet

Ordered Array Advantages / Disadvantages Advantages  Fast Search  Very fast access to data if the index is known Disadvantages  Slow Insert  Slow Deletion  Fixed Size

Linear Searches with Arrays Linear Searches  In an normal array: Check each cell looking for a match  In an ordered array: Check each cell looking for a match If the key in a cell is larger than the search value, then quit

Binary Searches with Arrays - 1 Array must be ordered Must faster than linear searches It works like the Guess-a-Number Game  Rules: Person A picks a number from 1 to 100. Person B guesses a number from 1 to 100. Person A says whether their number is bigger or smaller than that number. Person B keeps guessing from the small set of choices until they find the number

Binary Searches with Arrays - 2 Each guess allows you to reduce the range of possible choices The fastest way to win is to pick the middle number. This will reduce the possible choices by half. This shows the progression if the number is is the maximum number of guesses it will take to find any number! Much faster than a Linear Search!

Code View: Binary Search Binary Search Method Project: OrderedArray

Analyzing Algorithms To understand how good or bad an algorithm is, we need to know how it works over all cases To determine the complexity of an algorithm we must think about runnining it on all possible cases of a data set that can be used 1. Worst–case complexity : The maximum number of steps taken on any instance of size n. 2. Best–case complexity : The minimum number of steps taken on any instance of size n. 3. Average–case complexity :The maximum number of steps taken on any instance of size n.

Big O Notation Comparing the speed of 2 algorithms is tricky because the speed of an algorithm changes depending on how much data we have.  For example Algorithm A may be 3 times faster than algorithm B if there are only 5 pieces of data. Algorithm B may be 10 times faster than algorithm A if there are 1,000,000 pieces of data We use Big O Notation to tell us how fast an algorithm is.  It gives us a general estimation of the speed  It considers how many pieces of data we have The O stands for Order of Growth

Depending on the number range, the Guess-a-Number game will always require a maximum number of guesses as follows: Depending on the number of guesses, the Guess-a- Number game will work with a range as follows: Comparing Speed in Binary Search

Understanding Logarithms As the range grows exponentially, the steps grows slowly. range = 2 steps The inverse of raising a number to a power is called a logarithm We can use a logarithm to solve for steps: steps = log 2 (range)

Logarithmic Growth Rates Using this logarithmic function, we can see that every time we multiply the Range by 10, the Steps increases by Thus, a logarithmic growth rate means that as your data range grows exponentially, your steps don’t grow nearly as fast. steps = log2(range) Do these look familiar?

Big O Notation - 2 Constant  The number of steps is always the same.  It doesn’t matter how many pieces of data we have  Example: O(1) O(10) Proportional to N  Example: Linear searches  The number of steps is proportional to the number of pieces of data.  Ignore constants  Example: O(N)

Big O Notation - 3 Proportional to log(N)  Example: Binary searches  The number of steps is proportional to the logarithm of the number of pieces of data.  Ignore constants  Example: O(log N) Exponential  The number of steps is proportional to the logarithm of the number of pieces of data.  Ignore constants  Example: O(N 2 )

Arrays vs. Ordered Arrays Array  Advantages Fast Insert*  O(1)  Disadvantages Slow Search  O(N) Slow Deletion  O(N) Fixed Size Ordered Array  Advantages Fast Search  O(log N)  Disadvantages Slow Insert  O(N) Slow Deletion  O(N) Fixed Size *if duplicates are ok or we are assuming that the data inserted will not duplicate any data.

General Array Advantages / Disadvantages Advantages  Very fast access to data if the index is known Disadvantages  Fixed Size *if duplicates are ok or we are assuming that the data inserted will not duplicate any data.

Vectors A vector is an class provided by Java Grow in size when they become too full Usually expand in fixed-sized increments such as doubling in size when it’s about to overflow The process of growing bigger is slow because they are copying data from 1 array to another Useful when the amount of data is not known in advance

Demonstration Vectors Project: Vector