Detailed explanation of C++ function recursion: complexity analysis of recursion

Recursion is a process in which a function calls itself. The time complexity of recursion can be analyzed by counting the number of recursive calls, for example, the factorial function is O(n^2), and the recursive function of the nth term of the Fibonacci sequence is O(φ^n), where φ is the golden ratio.

Detailed explanation of C function recursion: complexity analysis of recursion

What is recursion?

Recursion is the behavior of a function calling itself. Recursion occurs when a function calls itself within itself.

Example of recursion

The following is a recursive function that calculates factorial:

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

Complexity analysis of recursion

The complexity of a recursive function can be analyzed by counting the number of recursive calls.

For the factorial function:

• When n is 0, call it recursively once.
• When n is 1, recursive calls are made 2 times (1 self-call, 1 tail call).
• When n is 2, recursively call 3 times (1 self-call, 2 tail calls).

By analogy, when n is k, the number of recursive calls is k 1.

The number of recursive calls forms an arithmetic sequence: 1, 2, 3, ..., k 1, and its summation formula is:

1 + 2 + 3 + ... + (k + 1) = (k + 1) * (k + 2) / 2

Therefore, the complexity of the factorial function is O (n^2).

Practical case

The following is a recursive function that calculates the nth term of the Fibonacci sequence:

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

The number of recursive calls is related to the golden ratio , its complexity is O(φ^n), where φ ≈ 1.618 is the golden ratio.

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