How to analyze the time complexity of C++ recursive functions?

Time complexity analysis of recursive functions involves: identifying base cases and recursive calls. Calculate the time complexity of the base case and each recursive call. Sum the time complexity of all recursive calls. Consider the relationship between the number of function calls and the size of the problem. For example, the time complexity of the factorial function is O(n) because each recursive call increases the recursion depth by 1, giving a total depth of O(n).

Time complexity analysis of recursive functions in C

In computer science, recursion is a programming technique that allows function calls itself. While recursion allows for writing concise and elegant code, an understanding of time complexity is crucial because it affects the performance of your program.

Time Complexity

Time complexity measures how long an algorithm takes to execute relative to the input size. For recursive functions, the input size is usually the size of the problem, such as the number of elements in an array or the depth of the problem to be solved.

Analyzing recursive functions

The time complexity of analyzing recursive functions needs to be identified:

• Basic situation:function Stop calling.
• Recursive call: The situation when the function calls itself.

Calculating time complexity

1. Determine the time complexity of the base case execution to be O(1).

2. For each recursive call, calculate the time complexity associated with the call, including:

   • Time complexity of the function call
   • After the recursive call The time complexity of execution
3. sums the time complexity of all recursive calls.
4. Consider the relationship between the number of function calls and the size of the problem.

Practical case: Factorial function

The factorial function recursively calculates the factorial of an integer n, that is, n (n-1) (n-2) ... 1.

int factorial(int n) {
  // 基本情况
  if (n == 0) {
    return 1;
  }
  // 递归调用
  return n * factorial(n-1);
}

• Basic situation: When n is 0, the time complexity is O(1).
• Recursive calls: Each recursive call performs a multiplication (O(1)) and then calls factorial(n-1) (recursive call).
• Time complexity: Each recursive call increases the recursion depth by 1, so the total depth is O(n). Since the execution time after function calls and recursive calls is O(1), the time complexity is O(n).

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