# Arrays¶

An array is a set of values where each value is identified by an index. You can make an array of ints, doubles, or any other type, but all the values in an array have to have the same type.

Syntactically, array types look like other Java types except they are followed by []. For example, int[] is the type “array of integers” and double[] is the type “array of doubles.”

You can declare variables with these types in the usual ways:

int[] count;
double[] values;


Until you initialize these variables, they are set to null. To create the array itself, use new.

count = new int[4];
values = new double[size];


The first assignment makes count refer to an array of 4 integers; the second makes values refer to an array of doubles. The number of elements in values depends on size. You can use any integer expression as an array size.

The following figure shows how arrays are represented in state diagrams:

The large numbers inside the boxes are the elements of the array. The small numbers outside the boxes are the indices used to identify each box. When you allocate an array if ints, the elements are initialized to zero.

## Accessing elements¶

To store values in the array, use the [] operator. For example count[0] refers to the “zeroeth” element of the array, and count[1] refers to the “oneth” element. You can use the [] operator anywhere in an expression:

count[0] = 7;
count[1] = count[0] * 2;
count[2]++;
count[3] -= 60;


These are all legal assignment statements. Here is the result of this code fragment:

The elements of the array are numbered from 0 to 3, which means that there is no element with the index 4. This should sound familiar, since we saw the same thing with String indices. Nevertheless, it is a common error to go beyond the bounds of an array, which throws an ArrayOutOfBoundsException.

You can use any expression as an index, as long as it has type int. One of the most common ways to index an array is with a loop variable. For example:

int i = 0;
while (i < 4) {
System.out.println(count[i]);
i++;
}


This is a standard while loop that counts from 0 up to 4, and when the loop variable i is 4, the condition fails and the loop terminates. Thus, the body of the loop is only executed when i is 0, 1, 2 and 3.

Each time through the loop we use i as an index into the array, printing the ith element. This type of array traversal is very common.

## Copying arrays¶

When you copy an array variable, remember that you are copying a reference to the array. For example:

double[] a = new double [3];
double[] b = a;


This code creates one array of three doubles, and sets two different variables to refer to it. This situation is a form of aliasing.

Any changes in either array will be reflected in the other. This is not usually the behavior you want; more often you want to allocate a new array and copy elements from one to the other.

 1 2 3 4 5 6 7 double[] b = new double [3]; int i = 0; while (i < 4) { b[i] = a[i]; i++; } 

## Arrays and objects¶

In many ways, arrays behave like objects:

• When you declare an array variable, you get a reference to an array.
• You have to use new to create the array itself.
• When you pass an array as an argument, you pass a reference, which means that the invoked method can change the contents of the array.

Some of the objects we looked at, like Rectangles, are similar to arrays in the sense that they are collections of values. This raises the question, “How is an array of 4 integers different from a Rectangle object?”

If you go back to the definition of “array” at the beginning of the chapter, you see one difference: the elements of an array are identified by indices, and the elements of an object have names.

Another difference is that the elements of an array have to be the same type. Objects can have instance variables with different types.

## for loops¶

The loops we have written have a number of elements in common. All of them start by initializing a variable; they have a test, or condition, that depends on that variable; and inside the loop they do something to that variable, like increment it.

This type of loop is so common that there is another loop statement, called for, that expresses it more concisely. The general syntax looks like this:

for (INITIALIZER; CONDITION; INCREMENTOR) {
BODY
}


This statement is equivalent to

INITIALIZER;
while (CONDITION) {
BODY
INCREMENTOR
}


except that it is more concise and, since it puts all the loop-related statements in one place, it is easier to read. For example:

for (int i = 0; i < 4; i++) {
System.out.println(count[i]);
}


is equivalent to

int i = 0;
while (i < 4) {
System.out.println(count[i]);
i++;
}


## Array length¶

All arrays have one named instance variable: length. Not surprisingly, it contains the length of the array (number of elements). It is a good idea to use this value as the upper bound of a loop, rather than a constant value. That way, if the size of the array changes, you won’t have to go through the program changing all the loops; they will work correctly for any size array.

for (int i = 0; i < a.length; i++) {
b[i] = a[i];
}


The last time the body of the loop gets executed, i is a.length - 1, which is the index of the last element. When i is equal to a.length, the condition fails and the body is not executed, which is a good thing, since it would throw an exception. This code assumes that the array b contains at least as many elements as a.

## Random numbers¶

Most computer programs do the same thing every time they are executed, so they are said to be deterministic. Usually, determinism is a good thing, since we expect the same calculation to yield the same result. But for some applications we want the computer to be unpredictable. Games are an obvious example, but there are more.

Making a program truly nondeterministic turns out to be not so easy, but there are ways to make it at least seem nondeterministic. One of them is to generate random numbers and use them to determine the outcome of the program. Java provides a method that generates pseudorandom numbers, which may not be truly random, but for our purposes, they will do.

Check out the documentation of the random method in the Math class. The return value is a double between 0.0 and 1.0. To be precise, it is greater than or equal to 0.0 and strictly less than 1.0. Each time you invoke random you get the next number in a pseudorandom sequence. To see a sample, run this loop:

for (int i = 0; i < 10; i++) {
double x = Math.random();
System.out.println(x);
}


To generate a random double between 0.0 and an upper bound like high, you can multiply x by high.

## Array of random numbers¶

How would you generate a random integer between low and high? If your implementation of randomInt is correct, then every value in the range from low to high-1 should have the same probability. If you generate a long series of numbers, every value should appear, at least approximately, the same number of times.

One way to test your method is to generate a large number of random values, store them in an array, and count the number of times each value occurs.

The following method takes a single argument, the size of the array. It allocates a new array of integers, fills it with random values, and returns a reference to the new array.

 1 2 3 4 5 6 7 public static int[] randomArray(int n) { int[] a = new int[n]; for (int i = 0; i

The return type is int[], which means that this method returns an array of integers. To test this method, it is convenient to have a method that prints the contents of an array.

public static void printArray(int[] a) {
for (int i = 0; i<a.length; i++) {
System.out.println(a[i]);
}
}


The following code generates an array and prints it:

int numValues = 8;
int[] array = randomArray(numValues);
printArray(array);


On my machine the output is

 1 2 3 4 5 6 7 8 27 6 54 62 54 2 44 81 

which is pretty random-looking. Your results may differ.

If these were exam scores (and they would be pretty bad exam scores) the teacher might present the results to the class in the form of a histogram, which is a set of counters that keeps track of the number of times each value appears.

For exam scores, we might have ten counters to keep track of how many students scored in the 90s, the 80s, etc. The next few sections develop code to generate a histogram.

## Counting¶

A good approach to problems like this is to think of simple methods that are easy to write, then combine them into a solution. This process is called bottom-up development. See Wikipedia.

It is not always obvious where to start, but a good approach is to look for subproblems that fit a pattern you have seen before.

In Section Looping and counting we saw a loop that traversed a string and counted the number of times a given letter appeared. You can think of this program as an example of a pattern called “traverse and count.” The elements of this pattern are:

• A set or container that can be traversed, like an array or a string.
• A test that you can apply to each element in the container.
• A counter that keeps track of how many elements pass the test.

In this case, the container is an array of integers. The test is whether or not a given score falls in a given range of values.

Here is a method called inRange that counts the number of elements in an array that fall in a given range. The parameters are the array and two integers that specify the lower and upper bounds of the range.

 1 2 3 4 5 6 7 public static int inRange(int[] a, int low, int high) { int count = 0; for (int i = 0; i < a.length; i++) { if (a[i] >= low && a[i] < high) count++; } return count; } 

I wasn’t specific about whether something equal to low or high falls in the range, but you can see from the code that low is in and high is out. That keeps us from counting any elements twice.

Now we can count the number of scores in the ranges we are interested in:

int[] scores = randomArray(30);
int a = inRange(scores, 90, 100);
int b = inRange(scores, 80, 90);
int c = inRange(scores, 70, 80);
int d = inRange(scores, 60, 70);
int f = inRange(scores, 0, 60);


## The histogram¶

This code is repetitious, but it is acceptable as long as the number of ranges is small. But imagine that we want to keep track of the number of times each score appears, all 100 possible values. Would you want to write this?

int count0 = inRange(scores, 0, 1);
int count1 = inRange(scores, 1, 2);
int count2 = inRange(scores, 2, 3);
...
int count3 = inRange(scores, 99, 100);


I don’t think so. What we really want is a way to store 100 integers, preferably so we can use an index to access each one. Hint: array.

The counting pattern is the same whether we use a single counter or an array of counters. In this case, we initialize the array outside the loop; then, inside the loop, we invoke inRange and store the result:

int[] counts = new int[100];

for (int i = 0; i < counts.length; i++) {
counts[i] = inRange(scores, i, i+1);
}


The only tricky thing here is that we are using the loop variable in two roles: as in index into the array, and as the parameter to inRange.

## A single-pass solution¶

This code works, but it is not as efficient as it could be. Every time it invokes inRange, it traverses the entire array. As the number of ranges increases, that gets to be a lot of traversals.

It would be better to make a single pass through the array, and for each value, compute which range it falls in. Then we could increment the appropriate counter. In this example, that computation is trivial, because we can use the value itself as an index into the array of counters.

Here is code that traverses an array of scores once and generates a histogram.

int[] counts = new int[100];

for (int i = 0; i < scores.length; i++) {
int index = scores[i];
counts[index]++;
}


## Glossary¶

array:
A collection of values, where all the values have the same type, and each value is identified by an index.
element:
One of the values in an array. The [] operator selects elements.
index:
An integer variable or value used to indicate an element of an array.
deterministic:
A program that does the same thing every time it is invoked.
pseudorandom:
A sequence of numbers that appear to be random, but which are actually the product of a deterministic computation.
histogram:
An array of integers where each integer counts the number of values that fall into a certain range.

## Exercises¶

### Exercise¶

Write a method called cloneArray that takes an array of integers as a parameter, creates a new array that is the same size, copies the elements from the first array into the new one, and then returns a reference to the new array.

### Exercise¶

Write a method called randomDouble that takes two doubles, low and high, and that returns a random double $$x$$ so that $$low \le x < high$$.

### Exercise¶

Write a method called randomInt that takes two arguments, low and high, and that returns a random integer between low and high, not including high.

### Exercise¶

Encapsulate the code in Section A single-pass solution in a method called makeHist that takes an array of scores and returns a histogram of the values in the array.

### Exercise¶

Write a method named areFactors that takes an integer n and an array of integers, and that returns true if the numbers in the array are all factors of n (which is to say that n is divisible by all of them). HINT: See this Exercise.

### Exercise¶

Write a method that takes an array of integers and an integer named target as arguments, and that returns the first index where target appears in the array, if it does, and -1 otherwise.

### Exercise¶

Some programmers disagree with the general rule that variables and methods should be given meaningful names. Instead, they think variables and methods should be named after fruit.

For each of the following methods, write one sentence that describes abstractly what the method does. For each variable, identify the role it plays.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 public static int banana(int[] a) { int grape = 0; int i = 0; while (i < a.length) { grape = grape + a[i]; i++; } return grape; } public static int apple(int[] a, int p) { int i = 0; int pear = 0; while (i < a.length) { if (a[i] == p) pear++; i++; } return pear; } public static int grapefruit(int[] a, int p) { for (int i = 0; i

The purpose of this exercise is to practice reading code and recognizing the computation patterns we have seen.

### Exercise¶

1. What is the output of the following program?
2. Draw a stack diagram that shows the state of the program just before mus returns.
3. Describe in a few words what mus does.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 public static int[] make(int n) { int[] a = new int[n]; for (int i = 0; i < n; i++) { a[i] = i+1; } return a; } public static void dub(int[] jub) { for (int i = 0; i < jub.length; i++) { jub[i] *= 2; } } public static int mus(int[] zoo) { int fus = 0; for (int i = 0; i < zoo.length; i++) { fus = fus + zoo[i]; } return fus; } public static void main(String[] args) { int[] bob = make(5); dub(bob); System.out.println(mus(bob)); } 

### Exercise¶

Many of the patterns we have seen for traversing arrays can also be written recursively. It is not common to do so, but it is a useful exercise.

1. Write a method called maxInRange that takes an array of integers and a range of indices (lowIndex and highIndex), and that finds the maximum value in the array, considering only the elements between lowIndex and highIndex, including both ends.

This method should be recursive. If the length of the range is 1, that is, if lowIndex == highIndex, we know immediately that the sole element in the range must be the maximum. So that’s the base case.

If there is more than one element in the range, we can break the array into two pieces, find the maximum in each of the pieces, and then find the maximum of the maxima.

2. Methods like maxInRange can be awkward to use. To find the largest element in an array, we have to provide a range that includes the entire array.

double max = maxInRange(array, 0, a.length-1);


Write a method called max that takes an array as a parameter and that uses maxInRange to find and return the largest value. Methods like max are sometimes called wrapper methods because they provide a layer of abstraction around an awkward method and make it easier to use. The method that actually performs the computation is called the helper method.

3. Write a recursive version of find using the wrapper-helper pattern. find should take an array of integers and a target integer. It should return the index of the first location where the target integer appears in the array, or -1 if it does not appear.

### Exercise¶

One not-very-efficient way to sort the elements of an array is to find the largest element and swap it with the first element, then find the second-largest element and swap it with the second, and so on. This method is called a selection sort (see Wikipedia).

1. Write a method called indexOfMaxInRange that takes an array of integers, finds the largest element in the given range, and returns its index. You can modify your recursive version of maxInRange or you can write an iterative version from scratch.
2. Write a method called swapElement that takes an array of integers and two indices, and that swaps the elements at the given indices.
3. Write a method called selectionSort that takes an array of integers and that uses indexOfMaxInRange and swapElement to sort the array from largest to smallest.

### Exercise¶

Write a method called letterHist that takes a String as a parameter and that returns a histogram of the letters in the String. The zeroeth element of the histogram should contain the number of a’s in the String (upper and lower case); the 25th element should contain the number of z’s. Your solution should only traverse the String once.

### Exercise¶

A word is said to be a “doubloon” if every letter that appears in the word appears exactly twice. For example, the following are all the doubloons I found in my dictionary.

Abba, Anna, appall, appearer, appeases, arraigning, beriberi, bilabial, boob, Caucasus, coco, Dada, deed, Emmett, Hannah, horseshoer, intestines, Isis, mama, Mimi, murmur, noon, Otto, papa, peep, reappear, redder, sees, Shanghaiings, Toto

Write a method called isDoubloon that returns true if the given word is a doubloon and false otherwise.

### Exercise¶

Two words are anagrams if they contain the same letters (and the same number of each letter). For example, “stop” is an anagram of “pots” and “allen downey” is an anagram of “well annoyed.”

Write a method that takes two Strings and returns true if the Strings are anagrams of each other.

Optional challenge: read the letters of the Strings only once.

### Exercise¶

In Scrabble each player has a set of tiles with letters on them, and the object of the game is to use those letters to spell words. The scoring system is complicated, but longer words are usually worth more than shorter words.

Imagine you are given your set of tiles as a String, like "quijibo" and you are given another String to test, like "jib". Write a method called canSpell that takes two Strings and returns true if the set of tiles can be used to spell the word. You might have more than one tile with the same letter, but you can only use each tile once.

Optional challenge: read the letters of the Strings only once.

### Exercise¶

In real Scrabble, there are some blank tiles that can be used as wild cards; that is, a blank tile can be used to represent any letter.

Think of an algorithm for canSpell that deals with wild cards. Don’t get bogged down in details of implementation like how to represent wild cards. Just describe the algorithm, using English, pseudocode, or Java.