Abstract Data Types in C

Recall that abstraction is the idea of separating what something is from how it works, by separating interface from implementation. Previously, we saw procedural abstraction, which applies abstraction to computational processes. With procedural abstraction, we use functions based on their signature and documentation without having to know details about their definition.

The concept of abstraction can be applied to data as well. An abstract data type (ADT) separates the interface of a data type from its implementation, and it encompasses both the data itself as well as functionality on the data. An example of an ADT is the string type in C++, used in the following code:

string str1 = "hello";
string str2 = "jello";
cout << str1 << endl;
if (str1.length() == str2.length()) {
  cout << "Same length!" << endl;
}

This code creates two strings and initializes them to represent different values, prints out one of them, and compares the lengths of both – all without needing to any details about the implementation of string. Rather, it relies solely on the interface provided by the string abstraction.

A string is an example of a full-featured C++ ADT, providing customized initialization, overloaded operations such as the stream-insertion operator, member functions, and so on. We will start with the simpler model of C ADTs, deferring C++ ADTs until next time.

The C language only has support for structs with data members (i.e. member variables). While this is sufficient to represent the data of an ADT, the functions that operate on the ADT must be defined separately from the struct. The following is the data definition of an ADT to represent triangles:

// A triangle ADT.
struct Triangle {
  double a;
  double b;
  double c;
};

int main() {
  Triangle t1 = { 3, 4, 5 };
  Triangle t2 = { 2, 2, 2 };
}

The Triangle struct contains three member variables, one for each side of the triangle, each represented by a double. The example in main() creates and initializes two Triangle structs, resulting in the memory layout in Figure 37.

_images/07_triangle_struct.svg

Figure 37 Two local Triangle objects.

An ADT also includes functions that operate on the data. We can define functions to compute the perimeter of a triangle or to modify it by scaling each of the sides by a given factor:

// REQUIRES: tri points to a valid Triangle
// EFFECTS:  Returns the perimeter of the given Triangle.
double Triangle_perimeter(const Triangle *tri) {
  return tri->a + tri->b + tri->c;
}

// REQUIRES: tri points to a valid Triangle; s > 0
// MODIFIES: *tri
// EFFECTS:  Scales the sides of the Triangle by the factor s.
void Triangle_scale(Triangle *tri, double s) {
  tri->a *= s;
  tri->b *= s;
  tri->c *= s;
}

Our naming convention for functions that are part of a C-style ADT is to prepend the function name with the name of the ADT, Triangle in this case. The first parameter is a pointer to the actual Triangle object the function works on. If the object need not be modified, we declare the pointer as a pointer to const.

The following demonstrates how to use the Triangle ADT functions:

int main() {
  Triangle t1 = { 3, 4, 5 };
  Triangle_scale(&t1, 2);                   // sides are now 6, 8, 10
  cout << Triangle_perimeter(&t1) << endl;  // prints 24
}

The code creates a Triangle as a local variable and initializes it with sides 3, 4, and 5. It then scales the sides by a factor of 2 by calling Triangle_scale(). Since that function takes a pointer to the actual triangle, we use the address-of operator to obtain and pass the address of t1, as shown in Figure 38.

_images/07_triangle_scale.svg

Figure 38 Passing a pointer to a Triangle object.

The function scales each side of the triangle, resulting in t1 having sides of 6, 8, and 10. We then call Triangle_perimeter() on the address of t1, which computes the value 24.

In this example, the code in main() need not worry about the implementation of Triangle_scale() or Triangle_perimeter(). Instead, it relies on abstraction, using the functions for what they do rather than how they do it. However, in initializing t1 itself, the code is relying on implementation details – specifically, that a Triangle is implemented as three double members that represent the lengths of the sides. If the implementation were to change to represent a triangle as two sides and the angle between them, for instance, then the behavior of the code in main() would change, and it would no longer print 24. Thus, we need to abstract the initialization of a Triangle, avoiding having to initialize each member directly. We do so by defining a Triangle_init() function:

// REQUIRES: tri points to a Triangle object
// MODIFIES: *tri
// EFFECTS:  Initializes the triangle with the given side lengths.
void Triangle_init(Triangle *tri, double a_in,
                   double b_in, double c_in) {
  tri->a = a_in;
  tri->b = b_in;
  tri->c = c_in;
}

int main() {
  Triangle t1;
  Triangle_init(&t1, 3, 4, 5);
  Triangle_scale(&t1, 2);
  cout << Triangle_perimeter(&t1) << endl;
}

The user of the Triangle ADT creates an object without an explicit initialization and then calls Triangle_init() on its address to initialize it, providing the side lengths. After that call, the Triangle has been properly initialized and can be used with the other ADT functions. Now if the implementation of Triangle changes, as long as the interface remains the same, the code in main() will work as before. The code within the ADT, in the Triangle_... functions, will need to change, but outside code that uses the ADT will not. The following illustrates an implementation of Triangle that represents a triangle by two sides and an angle:

// A triangle ADT.
struct Triangle {
  double side1;
  double side2;
  double angle;
};

// REQUIRES: tri points to a Triangle object
// MODIFIES: *tri
// EFFECTS:  Initializes the triangle with the given side lengths.
void Triangle_init(Triangle *tri, double a_in,
                   double b_in, double c_in) {
  tri->side1 = a_in;
  tri->side2 = b_in;
  tri->angle = std::acos((std::pow(a_in, 2) + std::pow(b_in, 2) -
                          std::pow(c_in, 2)) /
                         (2 * a_in * b_in));
}

// REQUIRES: tri points to a valid Triangle
// EFFECTS:  Returns the first side of the given Triangle.
double Triangle_side1(const Triangle *tri) {
  return tri->side1;
}

// REQUIRES: tri points to a valid Triangle
// EFFECTS:  Returns the second side of the given Triangle.
double Triangle_side2(const Triangle *tri) {
  return tri->side2;
}

// REQUIRES: tri points to a valid Triangle
// EFFECTS:  Returns the third side of the given Triangle.
double Triangle_side3(const Triangle *tri) {
  return std::sqrt(std::pow(tri->side1, 2) +
                   std::pow(tri->side2, 2) -
                   2 * tri->side1 * tri->side2 * std::acos(tri->angle));
}

// REQUIRES: tri points to a valid Triangle
// EFFECTS:  Returns the perimeter of the given Triangle.
double Triangle_perimeter(const Triangle *tri) {
  return Triangle_side1(tri) + Triangle_side2(tri) + Triangle_side3(tri);
}

// REQUIRES: tri points to a valid Triangle; s > 0
// MODIFIES: *tri
// EFFECTS:  Scales the sides of the Triangle by the factor s.
void Triangle_scale(Triangle *tri, double s) {
  tri->side1 *= s;
  tri->side2 *= s;
}

Here, we have added accessor or getter functions for each of the sides, allowing a user to obtain the side lengths without needing to know implementation details. Even within the ADT itself, we have used Triangle_side3() from within Triangle_perimeter() to avoid code duplication.

The REQUIRES clauses of the ADT functions make a distiction between Triangle objects and valid Triangle objects. The former refers to an object that is of type Triangle but may not have been properly initialized, while the latter refers to a Triangle object that has been initialized by a call to Triangle_init(). Except for Triangle_init(), the ADT functions all work on valid Triangles.

Now that we have a full definition of a C-style ADT, we adhere to the following convention for working with one: the user of a C-style ADT may only interact with the ADT through its interface, meaning the functions defined as part of the ADT’s interface. The user is generally prohibited from accessing struct member variables directly, as those are implementation details of the ADT. This convention also holds in testing an ADT, since tests should only exercise the behavior of an ADT and not its implementation.

Representation Invariants

When designing an abstract data type, we must build a data representation on top of existing types. Usually, there will be cases where the underlying data representation permits combinations of values that do not make sense for our ADT. For example, not every combination of three doubles represents a valid triangle – a double may have a negative value, but a triangle may not have a side with negative length. The space of values that represent valid instances of a triangle abstraction is a subset of the set of values that can be represented by three doubles, as illustrated in Figure 39.

_images/07_representation_invariants.svg

Figure 39 Representation invariants define the valid subset of the values allowed by the data representation of an ADT.

Thus, when designing an ADT, we need to determine the set of values that are valid for the ADT. We do so by specifying representation invariants for our ADT, which describe the conditions that must be met in order to make an object valid. For a triangle represented as a double for each side, the following representation invariants must hold:

  • The length of each side must be positive.

  • The triangle inequality must hold: the sum of any two sides must be strictly greater than the remaining side.

Often, we document the representation invariants as part of the ADT’s data definition:

// A triangle ADT.
struct Triangle {
  double a;
  double b;
  double c;
  // INVARIANTS: a > 0 && b > 0 && c > 0 &&
  //             a + b > c && a + c > b && b + c > a
};

We then enforce the invariants when constructing or modifying an ADT object by encoding them into the REQUIRES clauses of our functions. We can use assertions to check for them as well, where possible:

// REQUIRES: tri points to a Triangle object;
//           each side length is positive (a > 0 && b > 0 && c > 0);
//           the sides meet the triangle inequality
//           (a + b > c && a + c > b && b + c > a)
// MODIFIES: *tri
// EFFECTS:  Initializes the triangle with the given side lengths.
void Triangle_init(Triangle *tri, double a, double b, double c) {
  assert(a > 0 && b > 0 && c > 0);              // positive lengths
  assert(a + b > c && a + c > b && b + c > a);  // triangle inequality
  tri->a = a;
  tri->b = b;
  tri->c = c;
}

// REQUIRES: tri points to a valid Triangle; s > 0
// MODIFIES: *tri
// EFFECTS:  Scales the sides of the Triangle by the factor s.
void Triangle_scale(Triangle *tri, double s) {
  assert(s > 0);  // positive lengths
  tri->a *= s;
  tri->b *= s;
  tri->c *= s;
}

Plain Old Data

As mentioned above, we adhere to the convention of only interacting with an ADT through its interface. Usually, this means that we do not access the data members of an ADT in outside code. However, occasionally we have the need for an ADT that provides no more functionality than grouping its members together. Such an ADT is just plain old data (POD) 1, without any functions that operate on that data, and we define its interface to be the same as its implementation.

1

We use the term “plain old data” in the generic sense and not as the specific C++ term. C++ has a generalization of POD types called aggregates. Technically, the Person struct we saw last time is an aggregate but not a POD. What we mention here for POD types generally applies to aggregates as well.

The following is an example of a Pixel struct used as a POD:

// A pixel that represents red, green, and blue color values.
struct Pixel {
  int r; // red
  int g; // green
  int b; // blue
};

int main() {
  Pixel p = { 255, 0, 0 };
  cout << p.r << " " << p.g << " " << p.b << endl;
}

The Pixel ADT consists of just a data representation with no further functionality. Since it is a POD, its interface and implementation are the same, so it is acceptable to access its members directly.

Abstraction Layers

As with procedural abstraction, data abstraction is also defined in layers, with each layer interacting solely with the interface of the layer below and not its implementation. For example, we can represent an image using three matrices, one for each color channel. Any code that uses an image relies on the image interface, without needing to know that it is implemented over three matrices. Each matrix in turn can be represented using a single-dimensional array. Code that uses a matrix relies on the 2D abstraction provided by the interface without needing to know that it is implemented as a 1D array under the hood.

_images/07_abstraction_layers.svg

Figure 40 Abstraction layers for an image.

Testing an ADT

As mentioned previously, code outside of an ADT’s implementation must interact with the ADT solely through its interface, including test code. Modifying an ADT’s implementation should not require modifying its test code – we should be able to immediately run our regression tests in order to determine whether or not the ADT still works.

Adhering to the interface often means that we can’t test each ADT function individually. For instance, we cannot test Triangle_init() in isolation; instead, we can test it in combination with the side accessors (e.g. Triangle_side1()) to determine whether or not the initialization works correctly. Instead of testing individual functions, we test individual behaviors, such as initialization.

As another example, let’s proceed to design and test an ADT to represent a coordinate in two-dimensional space, using the principle of test-driven development that we saw previously. We will use polar coordinates, which represent a coordinate by the radius from the origin and angle from the horizontal axis, and we reflect this in the name of the ADT and its interface.

_images/07_polar_coordinates.svg

Figure 41 Polar representation of a point in two-dimensional space.

We start by determining the interface of the ADT:

// A set of polar coordinates in 2D space.
struct Polar;

// REQUIRES: p points to a Polar object
// MODIFIES: *p
// EFFECTS: Initializes the coordinate to have the given radius and
//          angle in degrees.
void Polar_init(Polar* p, double radius, double angle);

// REQUIRES: p points to a valid Polar object
// EFFECTS: Returns the radius portion of the coordinate as a
//          nonnegative value.
double Polar_radius(const Polar* p);

// REQUIRES: p points to a valid Polar object
// EFFECTS:  Returns the angle portion of the coordinate in degrees as
//           a value in [0, 360).
double Polar_angle(const Polar* p);

We then proceed to write some test cases, following the principles of test-driven development:

// Basic test of initializing a Polar object.
TEST(test_init_basic) {
  Polar p;
  Polar_init(&p, 5, 45);

  ASSERT_EQUAL(Polar_radius(&p), 5);
  ASSERT_EQUAL(Polar_angle(&p), 45);
}

We can then proceed to define a data representation. As part of this process, we should consider what representation invariants our ADT should have. For our Polar ADT, a reasonable set of invariants is that the radius is nonnegative, and the angle is in the range \([0, 360)\) (using degrees rather than radians) 2:

struct Polar {
  double r;
  double phi;
  // INVARIANTS: r >= 0 && phi >= 0 && phi < 360
};
2

A complete set of invariants would likely also specify a canonical representation of the origin. For example, it may specify that the if the radius is 0, then so is the angle.

Now that we have a data representation, we can make an initial attempt at implementing the functions as well:

void Polar_init(Polar* p, double radius, double angle) {
  p->r = radius;
  p->phi = angle;
}

double Polar_radius(const Polar* p) {
  return p->r;
}

double Polar_angle(const Polar* p) {
  return p->phi;
}

We can run our existing test cases to get some confidence that our code is working. In addition, the process of coming up with a data representation, representation invariants, and function definitions often suggests new test cases. For instance, the following test cases check that the representation invariants are met when Polar_init() is passed values that don’t directly meet the invariants:

// Tests initialization with a negative radius.
TEST(test_negative_radius) {
  Polar p;
  Polar_init(&p, -5, 225);
  ASSERT_EQUAL(Polar_radius(&p), 5);
  ASSERT_EQUAL(Polar_angle(&p), 45);
}

// Tests initialization with an angle >= 360.
TEST(test_big_angle) {
  Polar p;
  Polar_init(&p, 5, 405);
  ASSERT_EQUAL(Polar_radius(&p), 5);
  ASSERT_EQUAL(Polar_angle(&p), 45);
}

Given our initial implementation, these test cases will fail. We can attempt to fix the problem as follows:

void Polar_init(Polar* p, double radius, double angle) {
  p->r = std::abs(radius);  // set radius to its absolute value
  p->phi = angle;
  if (radius < 0) {         // rotate angle by 180 degrees if radius
    p->phi = p->phi + 180;  // was negative
  }
}

Running our test cases again, we find that both test_negative_radius and test_big_angle still fail: the angle value returned by Polar_angle() is out of the expected range. We can fix this as follows:

void Polar_init(Polar* p, double radius, double angle) {
  p->r = std::abs(radius);  // set radius to its absolute value
  p->phi = angle;
  if (radius < 0) {         // rotate angle by 180 degrees if radius
    p->phi = p->phi + 180;  // was negative
  }
  p->phi = std::fmod(p->phi, 360);  // mod angle by 360
}

Now both test cases succeed. However, we may have thought of another test case through this process:

// Tests initialization with a negative angle.
TEST(test_negative_angle) {
  Polar p;
  Polar_init(&p, 5, -45);
  ASSERT_EQUAL(Polar_radius(&p), 5);
  ASSERT_EQUAL(Polar_angle(&p), 315);
}

Unfortunately, this test case fails. We can try another fix:

void Polar_init(Polar* p, double radius, double angle) {
  p->r = std::abs(radius);  // set radius to its absolute value
  p->phi = angle;
  if (radius < 0) {         // rotate angle by 180 degrees if radius
    p->phi = p->phi + 180;  // was negative
  }
  p->phi = std::fmod(p->phi, 360);  // mod angle by 360
  if (p->phi < 0) {         // rotate negative angle by 360
    p->phi += 360;
  }
}

Our test cases now all pass.

Streams in C++

Previously, we learned about the standard input and output streams, as well as file streams. We examine the relationship between streams more closely now, as well as how to write unit tests using string streams.

A stream is an abstraction over a source of input, from which we can read data, or a sink of output, to which we can write data. Streams support the abstraction of character-based input and output over many underlying resources, including the console, files, the network, strings, and so on.

In C++, input streams generally derive from istream 3. We will see what this means specifically when we look at inheritance and polymorphism in the future. For our purposes right now, this means that we can pass different kinds of input-stream objects to a function that takes in a reference to an istream. Similarly, output streams generally derive from ostream, and we can pass different kinds of output-stream objects to a function that takes in a reference to an ostream.

_images/07_stream_hierarchy.svg

Figure 42 Relationships between different kinds of input and output streams.

3

The istream type is actually an alias for basic_istream<char>, which is an input stream that supports input using the char type. The same goes for ostream and basic_ostream<char>.

To write data into an output stream, we use the insertion operator <<. The actual data written out depends on both the value itself as well as its type. For instance, if we use a string as the right-hand-side operand, the insertion operation will write the characters from the string into the stream:

int i = 123;
cout << i;           // writes the characters 123
double d = 12.3;
cout << d;           // writes the characters 12.3
char c = 'c';
cout << c;           // writes the character c
string s = "Hello";
cout << s;           // writes the characters Hello

Expressions that apply an operator generally evaluate to a value. In the case of stream insertion, the result is the actual stream object itself. This allows us to chain insertion operations:

cout << i << d << endl;
// equivalent to ((cout << i) << d) << endl;
//                 ^^^^^^^^^
//       evaluates back to the cout object

To read data from an input stream, we use the extraction operator >>, with an object on the right-hand side. The characters are interpreted according to the type of the object. For built-in types, whitespace is generally skipped when extracting.

char c;
cin >> c;   // reads a single character; does not skip whitespace
string s;
cout >> s;  // reads in one "word", delimited by whitespace
int i;
cin >> i;   // attempts to parse the next characters as an integer value
double d;
cin >> d;   // attempts to parse the next characters as a floating-point value

As with the insertion operator, an expression that applies the extraction operator evaluates back to the stream itself, allowing extraction operations to be chained:

cin >> c >> s >> i >> d;

String Streams

When writing unit tests, we often want the tests to be standalone without requiring access to external data. For tests that work with streams, we can use string streams rather than standard input/output or file streams. To use a string stream, we #include <sstream>. We can then use an istringstream as an input stream, and an ostringstream as an output stream.

The following is an example of using an istringstream to represent input data for testing a function that takes in an input stream:

TEST(test_image_basic) {
  // A hardcoded PPM image
  string input = "P3\n2 2\n255\n255 0 0 0 255 0 \n";
  input += "0 0 255 255 255 255 \n";

  // Use istringstream for simulated input
  istringstream ss_input(input);
  Image *img = new Image;
  Image_init(img, ss_input);

  ASSERT_EQUAL(Image_width(img), 2);
  Pixel red = { 255, 0, 0 };
  ASSERT_TRUE(Pixel_equal(Image_get_pixel(img, 0, 0), red));
  delete img;
}

We start with a string that contains the actual input data and then construct an istringstream from that. We can then pass that istringstream object to a function that has a parameter of type istream &. When that function extracts data, the result will be the data from the string we used to construct the istringstream.

We can similarly use an ostringstream to test a function that takes an output stream:

TEST(test_matrix_basic) {
  Matrix *mat = new Matrix;
  Matrix_init(mat, 3, 3);
  Matrix_fill(mat, 0);
  Matrix_fill_border(mat, 1);

  // Hardcoded correct output
  string output_correct = "3 3\n1 1 1 \n1 0 1 \n1 1 1 \n";

  // Capture output in ostringstream
  ostringstream ss_output;
  Matrix_print(mat, ss_output);
  ASSERT_EQUAL(ss_output.str(), output_correct);
  delete mat;
}

We default construct an ostringstream object and pass it to a function with a parameter of type ostream &. The ostringstream will internally capture the data that the function inserts into it. We can then call .str() on the ostringstream to obtain a string that contains that data, which we can then compare to another string that contains the expected output.