Saddle Points

Coursework brief

Consider the following very simple "game". There is a two-dimensional array of numbers, and two players, Maxie and Minnie. At each turn, independently of each other, Max chooses a row and Min chooses a column. Where they intersect is the amount of money Min pays to Max (if it is negative, then Max pays Min).

For example, if Max chooses row 1 and Min chooses column 1, then Max wins £14 from Min. If Max chooses row 3 and Min chooses column 2, then Max pays Min £10. In general, Max wants larger numbers, and Min wants smaller numbers.

How should they play?

Your class should be named SaddlePoints; this class, and a Main class, should be placed inside a package called saddlePoints. Note see how these are capitalised and utilise CamelCase.

For each player, we pick the "best of the worst". In this example, the worst Max can do if he picks row 0 is -9; if row 1, then 5; if row 2, then -8; and if row 3, then -10. Of these possible worst outcomes, Max prefers the largest, which is 5 (row 1).

Similarly, Min wants smaller numbers. The worst she can do if she picks column 0 is 10; if column 1, then 17; and if column 2, then 5. Of these possible worst outcomes, Min prefers the smallest, which is 5 (column 2).

Since the "best of the worst" (the maximum of the minimums) for Max is 5, and the "best of the worst" (the minimum of the maximums) for Min is also 5, then Max should always choose row 1, and Min should always choose column 2.

This entry of 5, at row 1 and column 2, is called a saddle point. A saddle point is defined to be an entry such that it is minimum in its row and maximum in its column. Indeed, if Min does choose column 2, then Max will do worse by choosing any other row; if Max does choose row 1, then Min will do worse by choosing any other column.

It is possible for there to be more than one saddle point, but in this case, it is known that they will have the same value.

Specifications

Here are the computational methods you shall implement. You shall develop a unit test for each of these methods.

• int[][] createRandomArray(int numberOfRows, int numberOfColumns, int minValue, int maxValue) Creates and returns an array of the given size and fills it with random values in the specified range.

• int largest(int[] array)
  Finds the largest value in an array of integers.

• int smallest(int[] array)
  Finds the smallest value in an array of integers.

• int[] largestValues(int[][] array)
  Takes a two-dimensional array of integers and returns a one-dimensional array containing the largest values in each column (such as the array [10, 17, 5] in the above example).

• int[] smallestValues(int[][] array)
  Takes a two-dimensional array of integers and returns a one-dimensional array containing the smallest values in each row (such as the array [-9, 5, -8, -10] in the above example).

• boolean hasSaddlePoint(int[][] array)
  Takes a two-dimensional array of integers and returns true if it has a saddle point and false if it does not.

• int saddlePointRow(int[][] array)
  Takes a two-dimensional array of integers that is known to have a saddle point, and returns the row number of that saddle point. If there is more than one saddle point, returns the number of any row that contains a saddle point.

• int saddlePointColumn(int[][] array)
  Takes a two-dimensional array of integers that is known to have a saddle point, and returns the column number of that saddle point. If there is more than one saddle point, returns the number of any column that contains a saddle point.

Here are the I/O methods for which you do not need unit tests:

• public static void main(String[] args) This is responsible for just calling run().

• void run()
  Creates arrays various sizes (including some 2x2 arrays and some larger), fills them with random values, and prints each array and information about it. Keeps generating arrays until it has printed at least one with and one without a saddle point. (Smaller arrays are more likely to have a saddle point; about half of randomly generated 2 by 2 arrays will have one.)

• void printArray(int[][] array)
  Prints the array.

• void printArrayInfo(int[][] array)
  Prints whether the given array has a saddle point, and if so, where it is (row and column) and what its value is. (If there are multiple saddle points, just print any one.)

To save you some typing, we have provided the file SaddlePoints.java, in which all the methods have been entered as stubs, and the comments have been mostly filled out. Start IntelliJ, create the project and the package, and away you go!

JUnit tests in Java

We also provide the template for your tests in SaddlePointsTest.java. Here are some of the more useful JUnit methods:

• assertEquals(expectedValue, actualValue);
• assertArrayEquals(expectedArray, actualArray);
• assertTrue(condition);
• assertFalse(condition);
• fail();

The method

createRandomArray(numberOfRows, numberOfColumns, minValue, maxValue) 

should return a different value every time, but you can still test it. Testing whether it's "really random" is difficult, but the following tests are pretty easy:

• Test whether the array has the requested number of rows and columns.
• Test if all the numbers in the array are in the specified range.
• Test that the array isn't full of the same number, repeated over and over. (This could happen even if your code is correct; but by choosing a larger array or a larger range of values, you can make it extremely unlikely - say, once in a million times.)

Random Numbers

You will need to import java.util.Random. You will need to create a random number generator (an instance of the class Random); you will only need one of these.

To get numbers in a reasonable range, you will need to call one of the methods defined in Random. Please read this documentation for more information.

Arrays in Java

Arrays in Java are like lists in Python, and are indexed the same way. However, the size of an array is specified when you create the array, and you can't add or delete elements. (You can, however, assign a different array to the variable.) You can declare arrays like this:

• type [] name = new type[size];
• type [][] name = new type[rows][columns];

Declared in this way, all locations in the new array contain null, zero, or false, depending on the type of the array. You can put specific values in an array when you declare it (but not later) like this:

• type [] name = {value1, ..., valueN};
• type [][] name = {{value1_1, ..., value1_N}, ... {valueN_1, ..., valueN_N}};

To use a literal array in a statement, rather than in a declaration, the syntax is a little different:

• name = type [] {value1, ..., valueN};
• name = type [][] {{value1_1, ..., value1_N}, ... {valueN_1, ..., valueN_N}};

For testing purposes, you can start by using the following game,

     int[][] with = {{-9, 12, -6},
                     { 7, 14,  5}, 
                     {10, -8,  3}, 
                     { 6, 17,-10}};

which has a saddle point, and this further game

     int[][] without = {{ 1, -2,  3},
                        {-6,  5, -4}, 
                        { 7, -8,  9}};

which does not.