10. Vectors¶

A vector is a set of values where each value is identified by a number (called an index). A string is similar to a vector, since it is made up of an indexed set of characters. The nice thing about vectors is that they can be made up of any type of element, including basic types like int and double, and user-defined types like Point and Time.

In order to use vector/s, you need to include the header file at the top of your program:

#include <vector>

You can create a vector similarly to how you create other variable types:

vector<int> count;
vector<double> double_vector;

The type that makes up the vector appears in angle brackets (< and >). The first line creates a vector of integers named count; the second creates a vector of doubles. Although these statements are legal, they are not very useful because they create vectors that have no elments (their size is zero). It is more common to specify the size of the vector in parentheses:

vector<int> count(4);

The syntax here is a little odd. It looks like a combination of a variable declaration and a function call. In fact, that’s exactly what it is. The function we are invoking is a vector constructor. A constructor is a special function that creates new objects and initializes their instance variables. In this case, the constructor takes a single argument, which is the size of the new vector.

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

The large circles inside the boxes are the elements of the vector. Each element is accessed using the address of its box, beginning with 0. When you allocate a new vector, the elements are not initialized. They could contain any values.

There is another constructor for vectors that takes two parameters. The second is a fill value, the value that will be assigned to each of the elements.

vector<int> count(4, 0);

This statement creates a vector of four elements and initializes all of them to zero.

10.1. Accessing elements¶

The [] operator reads and writes elements of a vector in much the same way it accesses the characters in a string. As with strings, the indices start at zero, so count[0] refers to the “zeroth” element of the vector, 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;

All of these are legal assignment statements. Here is the effect of this code fragment:

Since elements of this vector are numbered from 0 to 3, there is no element with the index 4.

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

int i = 0;
while (i < 4) {
    cout << count[i] << endl;
    i++;
}

This while loop counts from 0 to 4. 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 vector, outputting the ith element. This type of vector traversal is very common. Vectors and loops go together like fava beans and a nice Chianti.

10.2. Copying vectors¶

There is one more constructor for vectors, which is called a copy constructor because it takes one vector as an argument and creates a new vector that is the same size, with the same elements.

vector<int> copy(count);

Although this syntax is legal, it is almost never used for vectors because there is a better alternative:

vector<int> copy = count;

The = operator works on vectors pretty much the way you would expect.

10.3. `for` loops¶

The loops we have written so far 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 an alternative loop statement, called for, that expresses it more concisely. The general syntax looks like this:

for (INITIALIZER; CONDITION; INCREMENTOR) {
    BODY
}

This statement is exactly equivalent to

INITIALIZER
while (CONDITION) {
    BODY
    INCREMENTOR
}

except that is 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++) {
    cout << count[i] << endl;
}

is equivalent to

int i = 0;
while (i < 4) {
    cout << count[i] << endl;
    i++;
}

Notice that it is legal to declare a variable inside a for statement. This syntax is convenient, but you should be aware that a variable declared inside a loop only exists inside the loop. If you try to refer to i later, you will get a compiler error.

10.4. Vector size¶

There are only a couple of functions you can invoke on a vector. One of them is very useful though: size. Not surprisingly, it returns the size of the vector (the number of elements).

It is a good idea to use this value as the upper bound of a loop, rather than a constant. That way, if the size of the vector changes, you won’t have to go through the program changing all the loops; they will work correctly for any size vector.

for (int i = 0; i < count.size(); i++) {
    cout << count[i] << endl;
}

10.5. 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. For some applications, though, we would like the computer to be upredictable. Games are an obvious example.

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 pseudorandom numbers and use them to determine the outcome of the program. Pseudorandom numbers are not truly random in the mathematical sense, but for our purposes, they will do.

C++ provides a function called rand that generates pseudorandom numbers. It is declared in the header file cstdlib, which contains a variety of standard library functions, hence its name.

The return value from rand is an integer between 0 and RAND_MAX, where RAND_MAX is a large number (about 2 billion on our system) also defined in the header file. Each time you call rand you get a different pseudorandom number. To rand in action, compile and run this program:

#include <cstdlib>
#include <iostream>
using namespace std;

int main()
{
    cout << "RAND_MAX is: " << RAND_MAX << endl;
    cout << "Let's generate 10 random numbers." << endl;
    for (int i = 1; i < 11; i++) {
        cout << "Random number " << i << ": " << rand() << endl;
    }

    return 0;
}

Now try running the program again several times. What do you notice? You get the same set of 10 random numbers each time!

If we want our numbers to vary on program runs, we can set a random seed for the pseudorandom number function rand by calling srand (for “seed rand”). The srand function takes an integer argument. With a different argument to srand, rand generates a different sequence of numbers.

A common way to give srand a “random” seed is to use the time function, which returns the current time in seconds.

Note

The time function returns Unix time, which is the number of seconds elapsed since January 1, 1970.

To use srand this way, we include the ctime header file and seed it with the current time:

srand((unsigned) time(NULL));

which will cause rand to generate a different sequence of numbers.

Of course, we don’t usually want to work with gigantic integers. More often we want to generate integers between some lower bound and some upper bound. We will leave this as an exercise for you to do this.

10.6. Statistics¶

The numbers generated by rand are supposed to be distributed uniformly. That means that each value in the range should be equally likely. If we count the number of times each value appears, it should be roughly the same for all values, provided that we generate a large number of values.

In the next few sections, we will write programs that generate a sequence of random numbers and check whether this property holds true.

10.7. Vector of random numbers¶

The first step is to generate a large number of random values and store them in a vector. By “large number”, of course, we mean 20. It’s always a good idea to start with a manageable number, to help with debugging, and then increase it later.

The following function takes a single argument, the size of the vector. It allocates a new vector of ints, and fills it with random values between 0 and upper_bound - 1.

vector<int> random_vector(int n, int upper_bound)
{
    vector<int> vec(n);

    for (int i = 0; i < vec.size(); i++) {
        vec[i] = rand() % upper_bound;
    }

    return vec;
}

The return type is vector<int>, which means that this function returns a vector of integers. To test this function, it is convenient to have a function that outputs the contents of a vector.

void print_vector(const vector<int>& vec)
{
    for (int i = 0; i < vec.size(); i++) {
        cout << vec[i] << " ";
    }
}

Notice that it is legal to pass vectors by reference. In fact it is quite common, since it makes it unnecessary to copy the vector. Since print_vector does not modify the vector, we declare the parameter const.

The following code generates a vector and outputs it:

int num_values = 20;
int upper_bound = 10;
vector<int> vector = random_vector(num_values, upper_bound);
print_vector(vector);

On our machine we got this as output:

3 6 7 5 3 5 6 2 9 1 2 7 0 9 3 6 0 6 2 6

which is pretty random-looking.

If these numbers are really random, we expect each digit to appear the same number of times - twice each. In fact, the number 6 appears five times, and the numbers 4 and 8 never appear at all.

Do these results mean the values are not really uniform? It’s hard to tell. With so few values, the chances are slim that we would get exactly what we expect. But as the number of values increases, the outcome should be more predictable.

To test this theory, we’ll write some programs that count the number of times each value appears, and then see what happens when we increase the number of values.

10.8. Counting¶

A good approach to problems like this is to think of simple functions that are easy to write, and that might turn out to be useful. Then you can combine them into a solution. This approach is sometimes called bottom-up design. Of course it is not easy to know ahead of time which functions are likely to be useful, but as you gain experience you will have a better idea.

Also, it is not always obvious what sort of things are easy to write, but a good approach is to look for subproblems that fit a pattern you have seen before.

Back in the Traversal section of the Strings and things chapter we looked 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 be traversed, like a string or a vector.
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, we have a function in mind called how_many that counts the number of elements in a vector that equal a given value. The parameters are the vector and the integer value we are looking for. The return value is the number of times the value appears.

int how_many(const vector<int>& vec, int value)
{
    int count = 0;
    for (int i = 0; i < vec.size(); i++) {
        if (vec[i] == value) count++;
    }
    return count;
}

how_many only counts the occurances of a particular value, and we are interested in seeing how many times each value appears. We can solve this problem with a loop:

int num_values = 20;
int upper_bound = 10;
vector<int> vector = random_vector(num_values, upper_bound);
cout << "value\thow_many" << endl;
for (int i = 0; i < upper_bound; i++) {
    cout << i << '\t' << how_many(vector, i) << endl;
}

This code uses the loop variable as an argument to how_many, in order to check each value between 0 and 9, in order. The result is:

value   how_many
     2
     1
     3
     3
     0
     2
     5
     2
     0
     2

Again, it is hard to tell if the digits are really appearing equally often. If we increase num_values to 100,000 we get the following:

value   how_many
     10130
     10072
     9990
     9842
     10174
     9930
     10059
     9954
     9891
     9958

In each case, the number of appearances is within about 1% of the expected value (10,000), so we conclude that the random numbers are probably uniform.

10.9. A histogram¶

It is often useful to take the data from the previous tables and store them for later access, rather than just print them. What we need is a way to store 10 integers. We could create 10 integer variables with names like how_manyOnes, how_manyTwos, etc. But that would require a lot of typing, and it would be a real pain later if we decided to change the range of values.

A better solution is to use a vector with size 10. That way we can create all ten storage locations at once and we can access them using indices, rather than ten different names. Here’s how:

int num_values = 100000;
int upper_bound = 10;
vector<int> vector = random_vector(num_values, upper_bound);
vector<int> histogram(upper_bound);

for (int i = 0; i < upper_bound; i++) {
    int count = how_many(vector, i);
    histogram[i] = count;
}

We called the vector histogram because that’s a statistical term for a vector of numbers that counts the number of appearances of a range of values.

The trick thing here is that we are using the loop variable in two different ways. First, it is an argument to how_many, specifying which value we are interested in. Second, it is an index into histogram, specifying which location the result should be stored in.

Although this code works, it is not as efficient as it could be. Every time it calls how_many, it traverses the entire vector. In this example we have to traverse the vector ten times!

It would be better to make a single pass through the vector. For each value in the vector we could find the corresponding count and increment it. In other words, we could use the value from the vector as an index into the histogram. Here’s what that looks like:

vector<int> histogram(upper_bound, 0);

for (int i = 0; i < num_values; i++) {
    int index = vector[i];
    histogram[index]++;
}

The first line intializes the elements of the histogram to zeros. That way, when we use the increment operator (++) inside the loop, we know we are starting from zero. Forgetting to initialize counters is a common error.

10.10. Glossary¶

bottom-up design¶: A method of program development that starts by writing small, useful functions and then assembling them into larger solutions.
constructor¶: A special function that creates a new object and initializes its instance variables.
deterministic¶: A program that does the same thing every time it is run.
element¶: One of the values in a vector. The [] operator selects elements of a vector.
histogram¶: A vector of integers where each integer counts the number of values that fall into a certain range.
pseudorandom¶: A sequence of numbers that appear to be random, but which are actually the result of a deterministic computation.
seed¶: A value used to initialize a pseudorandom number sequence. Using the same seed should yield the same sequence of values.
vector¶: A named collection of values, where all the values have the same type, and each value is identified by an index.

10.11. Exercises¶

Chapter 10 Exercise Set 1

10. Vectors¶

10.1. Accessing elements¶

10.2. Copying vectors¶

10.3. for loops¶

10.4. Vector size¶

10.5. Random numbers¶

10.6. Statistics¶

10.7. Vector of random numbers¶

10.8. Counting¶

10.9. A histogram¶

10.10. Glossary¶

10.11. Exercises¶

10.3. `for` loops¶