Chapter 14: Recursion

Chapter Goals

• To learn about the method of recursion
• To understand the relationship between recursion and iteration
• To analyze problems that are much easier to solve by recursion than by iteration
• To learn to "think recursively"
• To be able to use recursive helper functions
• To understand when the use of recursion affects the efficiency of an algorithm

Triangle Numbers

• In this example, we will look at triangle shapes such as the one shown here:

  []
  [][]
  [][][]

• The n-th triangle number is the area of a triangle of width n.
• From above, the third triangle number is 6.
• Here is the outline of the class that we will develop:

  class Triangle
  {
  public:
     Triangle(int w);
     int get_area() const;
  private:
     int width;
  };
  Triangle::Triangle(int w)
  
  {
     width = w;
  }

Triangle Numbers

• If the width of the triangle is 1, then the triangle has an area of 1.

  []
  
  int Triangle::get_area()
  {
     if (width = = 1) return 1;
     . . .
  }

• To deal with the general case, consider this picture.

  []
  [][]
  [][][]
  [][][][]

• We think of area of the larger triangle as

  smaller_area + width

• To find the smaller area, we make a smaller triangle!

  int Triangle::get_area()
  {
     if (width == 1) return 1;
     Triangle smaller_triangle(width - 1);
     int smaller_area = smaller_triangle.get_area();
     return smaller_area + width;
  }

Triangle Numbers

• Here's how the area is computed for a triangle of width 4.
  • The get_area function makes a smaller triangle of width 3.
    • It calls get_area on that triangle.
      • That function makes a smaller triangle of width 2.
        • It calls get_area on that triangle.
          • That function make a smaller triangle of width 1.
          • It calls get_area on that triangle.
        • That function returns 1.
      • The function returns smaller_area + width = 1 + 2 = 3.
    • The function returns smaller_area + width = 3 + 3 = 6.
  • The function returns smaller_area + width = 6 + 4 = 10.

Triangle Numbers

• The technique of expressing a solution to a problem in terms of solution to a smaller version of the same problem is called recursion.
• There are two key requirements to make that recursion is successful:
  • Every recursive call must simplify the computation in some way.
  • The must be special case to handle the simplest computations directly.
• Here, get_area calls itself again with smaller and smaller width values, eventually reaching a width of 1.
• Recursion is not really necessary to solve this problem:
  • A simple loop:

    double area = 0;
    for (int i = 1; i <= width; i++)
       area = area + 1;

  • A formula:

    width * (width + 1) / 2

Triangle Numbers (triangle.cpp)

Permutations

• We will design a function that lists all permutations of a string.
• A permutation is simply a rearrangement of the letters:

  "eat"
  "eta"
  "aet"
  "ate"
  "tea"
  "tae"

• If a string has n letters, then the number of permutations is given by the factorial function:

  n! = 1 x 2 x 3 x . . . x n

• To a compute the value of a n! you could use a loop (see chapter 7), but there is also a recursive solution:

  n! = (n - 1)! x n 

• It is customary to define

  1! = 1
  0! = 0

• You can therefore implement a recursive factorial function as follows:

  int factorial(int n)
  {
     if (n == 0) return 1;
     int smaller_factorial = factorial(n - 1);
     int result = smaller_factorial * n;
     return result;
  }

Permutations

• We will develop a function that generates all permutations of a word.

  vector<string> generate_permutations(string word);

• The following code displays all permutations of the string "eat":

  vector<string> v = generate_permutations("eat");
  for(int i = 0; i < v.size(); i++)
     cout << v[i] << "\n";

• To generate the permutations recursively, generate all permutations that state with the letter 'e', the those that start with 'a', then those that start with 't'.
• Use the recursion to generate all the permutations of the shorter (two-letter) strings.
  • "at" "ta"
  • "et" "te"
  • "ae" "ea"
• Add the first letters to find the permutations.
  • "eat" "eta"
  • "aet" "ate"
  • "tae" "tea"

Permutations

• To carry out the ideas in the previous slide, we implement a loop that creates a shorter word that omits the current position.

  vector<string> generate_permutations(string word)
  {
     vector<string> result;
     ...
     for (int i = 0; i < word.length(); i++)
     {
        string shorter_word = word.substr(0, i)
           + word.substr(i + 1, word.length() - i - 1);
        ...
     }
     return result;
  }

• The next step is to compute the permutations of the shorter word.

  vector<string> shorter_permutations
     = generate_permutations(shorter_word);

• For each of the short permutations, add the omitted letter.

  for(int j = 0; j < shorter_permutations.size(); j++)
  {
     string longer_word = word[i] + shorter_permutations[j];
     result.push_back(longer_word);
  }

• You must build in a stopping point for the recursion, as a special case to handle words of length 1.

  if (word.length() == 1)
  {
     result.push_back(word);
     return result;
  }

Permutations (permute.cpp)

Thinking Recursively

• The algorithms for computing the triangle area, the factorial function and the word permutations all work on the same principle.
  • When working on a complex input, first solve the problem with a simpler input.
  • Turn the simpler result into the result for the more complex unit.
• One function calls another that works on an even simpler input until one function's input is so simple that it can compute the results without further help.
• What's important is that you can focus on putting a solution together from the slightly simpler problem.

Thinking Recursively

• This section gives you a step-by-step guide to the method of recursion.
• To illustrate the steps, we will test whether a sentence is a palindrome - a string that is equal to itself when you reverse all characters:
• Examples:
  • rotor
  • A man, a plan, a canal - Panama!
  • Go hand a salami, I'm a lasagna hog
  • Madam, I'm Adam
• Our goal is to implement a predicate function.

  bool is_palindrome(string s)

Thinking Recursively

• Step 1: Consider various ways for simplifying inputs.
• How can you simplify the inputs in such a way that the same problem can be applied to simpler input.
• Here are several possibilities for the palindrome test problem.
  • Remove the first character.
  • Remove the last character.
  • Remove both the first and last character.
  • Remove a character from the middle.
  • Cut the string into two halves.

Thinking Recursively

• Step 2: Combine solutions with simpler inputs to a solution of the original problem.
• Don't worry how those solutions are obtained. These are simpler inputs, so someone else will solve the problem for you.
• Removing the first and last characters seems promising:

  "rotor"

  becomes

  "oto"

• A word is a palindrome if
  • the first and last letters match
  • the word obtained by removing the first and last letters is a palindrome.

Thinking Recursively

• Step 3: Find solutions to the simplest inputs.
• To make sure that the recursion comes to a stop, you must deal with the simplest inputs separately.
• Sometimes you get into philosophical questions dealing with degenerate inputs: empty strings, shapes with no area, and so on.
• You may want to investigate a slightly larger input that gets reduces to a degenerate input and see what value you should attach to the degenerate input yields the correct answer.
• The simplest strings for the palindrome test are:
  • strings with two characters
  • strings with a single character
  • the empty string.
• We don't need a special case for strings with two characters - by removing both the first and last characters, it becomes a string of length 0.
• A single character string is a palindrome.
• An empty string is a palindrome.

Thinking Recursively (palindrome.cpp)

Recursive Helper Functions

• Sometimes it is easier to find a recursive solution if you change the original problem slightly, then solve the problem using a recursive helper function.
• In our palindrome example, it is inefficient to construct new string objects in every step.
• Rather than creating a new string, this function simply skips over matching letter pairs by adjusting the start and end parameters with each recursive call.

  bool substring_is_palindrome(string s, int start, int end)
  {
     // separate case for substrings of length 0 and 1
     if (start >= end) return true;
  
     if (s[start] == s[end])
        // test substring that doesn't contain the first and last letters
        return substring_is_palindrome(s, start + 1, end - 1);
     else
        return false;
  }

• The new function is a helper function that the user of the is_palindrome function doesn't know about.

  bool is_palindrome(string s)
  {
     return substring_is_palidrome(s, 0, s.length() - 1);
  }

Mutual Recursion

• Sometimes, a set of cooperating functions calls each other in a recursive fashion.
• For our example, we will develop a program that can compute the values of arithmetic expressions such as

  3 + 4 * 5
  (3 + 4) * 5
  1 - (2 - (3 - (4 - 5)))

Mutual Recursion

• The follow syntax diagrams describe the syntax of these expressions.
  • An expressions is either a term, or a sum of different terms.

    

  • A term is either a factor, or a product or quotient of factors.

    

  • A factor is either a number or an expression closed in parentheses.

    

Mutual Recursion

• The syntax diagrams accurately represent which operations should be carried out first.
• First, 4 and 5 should be multiplied, and then the result should be added to 3.

  

Mutual Recursion

• Here, 3 and 4 should be added, and the result should be multiplied by 5.

  

Mutual Recursion

• To compute the value of an expression, we implement three functions expression_value, term_value, and factor_value.
• The expression_value function calls term_value, checks to see if the next input is + or -, and if so calls term_value again to add or subtract the next term.
• The term_value function calls factor_value in the same way, multiplying or dividing the factor values.
• The factor_value function checks whether the next input is '(' or a digit, calling either expression_value recursively or returning the value of the digit.
• The termination of the recursion is ensured because the expression_value function consumes some of the input characters, ensuring that the next time it is called on a shorter expression.

Mutual Recursion (eval.cpp)

The Efficiency of Recursion

• Although recursion can be a powerful tool to implement complex algorithms, it can lead to algorithms that perform poorly.
• We will analyze the question of when recursion is beneficial and when it is inefficient.
• For our study we will examine the Fibonacci sequence of numbers defined by the equations:
  • f₁ = 1
  • f₂ = 1
  • f_n = f_n-1 + f_n-2
• The first ten terms of the Fibonacci sequence are

  1, 1, 2, 3, 5, 8, 13, 21, 34, 55

• One can easily write a recursive function to compute Fibonacci numbers.

  int fib(int n)
  {
     if (n <= 2) return 1;
     else return fib(n - 1) + fib(n - 2);
  }

The Efficiency of Recursion

• The function above works correctly, but for moderately large values of n it takes an amazing long time to compute the value.
• Try running a test program with n between 30 and 50 to see this effect.
• To determine the problem, we insert trace messages into the function.

  int fib(int n)
  {
     cout << "Entering fib: n = " << n << "\n";
     int f;
     if (n <= 2) f = 1;
     else f = fib(n - 1) + fib(n - 2);
     cout << "Exiting fib: n = " << n
        << " return value = " << f << "\n";
     return f;
  }

The Efficiency of Recursion

• The output from the function indicates why the computation takes so long.

  Entering fib: n = 6
  Entering fib: n = 5
  Entering fib: n = 4
  Entering fib: n = 3
  Entering fib: n = 2
  Exiting fib: n = 2 return value = 1
  Entering fib: n = 1
  Exiting fib: n = 1 return value = 1
  Exiting fib: n = 3 return value = 2
  Entering fib: n = 2
  Exiting fib: n = 2 return value = 1
  Exiting fib: n = 4 return value = 3
  Entering fib: n = 3
  Entering fib: n = 2
  Exiting fib: n = 2 return value = 1
  Entering fib: n = 1
  Exiting fib: n = 1 return value = 1
  Exiting fib: n = 3 return value = 2
  Exiting fib: n = 5 return value = 5
  Entering fib: n = 4
  Entering fib: n = 3
  Entering fib: n = 2
  Exiting fib: n = 2 return value = 1
  Entering fib: n = 1
  Exiting fib: n = 1 return value = 1
  Exiting fib: n = 3 return value = 2
  Entering fib: n = 2
  Exiting fib: n = 2 return value = 1
  Exiting fib: n = 4 return value = 3
  Exiting fib: n = 6 return value = 8

The Efficiency of Recursion

• The call tree helps illustrate how the functions call one another.

  

• Because the fib(4) is called twice and fib(3) is called three times, the functions spends a lot of time needlessly computing the same values over and over.

The Efficiency of Recursion

• A person would just write down the values as they were computed and add up the last two to get the next one; no sequence value would ever be computed twice.
• To imitate the pencil-and-paper process, you would write a loop to compute Fibonacci numbers.

  int fib(int n)
  {
     if (n <= 2) return 1;
     int fold = 1;
     int fold2 = 1;
     int fnew;
     for (int i = 3; i <= n; i++)
     {
        fnew = fold + fold2;
        fold2 = fold;
        fold = fnew;
     }
     return fnew;
  }

The Efficiency of Recursion

• Can you always speed up a recursive solution by changing it into a loop?
• Frequently, the iterative and recursive solution have essentially the same performance.
• For example:

  bool is_palindrome(string s)
  {
     int start = 0;
     int end = text.length() - 1;
     while (start < end)
     {
        if (s[start] != s[end] return false;
        start++;
        end--;
     }
     return true;
  }

• The iterative and the recursive solutions to the palindrome problem run at about the same speed.
• There are quite a few problems that are dramatically easier to solve recursively that iteratively.
• Solving the permutation problem iteratively yields code that is complex, but no faster.