How do you find the complexity of an algorithm? What is the relation between the time and space complexities of an algorithm? Justify your answer with an example.

Introduction

The complexity of an algorithm is a measure of the amount of resources required by the algorithm to solve a given problem. The two most common types of complexity are time complexity and space complexity.

• Time complexity measures the amount of time required by the algorithm to solve a problem.
• Space complexity measures the amount of memory required by the algorithm to solve a problem.

Time Complexity

The time complexity of an algorithm is typically expressed using Big O notation. Big O notation provides an upper bound on the worst-case time complexity of an algorithm.

• Best-case time complexity: This is the time required for the algorithm to complete when it is given the most favorable input.

• Average-case time complexity: This is the time required for the algorithm to complete when it is given a randomly chosen input.

• Worst-case time complexity: This is the time required for the algorithm to complete when it is given the least favorable input.

For example, consider the following algorithm for finding the maximum element in an array of n integers:

def max_element(array):
  max_value = array[0]
  for i in range(1, n):
    if array[i] > max_value:
      max_value = array[i]
  return max_value

The best-case time complexity of this algorithm is O(1), which means that it will always take a constant amount of time to find the maximum element in the array. The average-case time complexity of this algorithm is O(n), which means that it will take an average of n/2 comparisons to find the maximum element in the array. The worst-case time complexity of this algorithm is also O(n), which means that it will take a maximum of n comparisons to find the maximum element in the array.

Space Complexity

The space complexity of an algorithm is typically expressed using Big O notation. Big O notation provides an upper bound on the worst-case space complexity of an algorithm.

• Best-case space complexity: This is the space required for the algorithm to complete when it is given the most favorable input.

• Average-case space complexity: This is the space required for the algorithm to complete when it is given a randomly chosen input.

• Worst-case space complexity: This is the space required for the algorithm to complete when it is given the least favorable input.

For example, consider the following algorithm for sorting an array of n integers using the quicksort algorithm:

def quicksort(array):
  if len(array) <= 1:
    return array

  pivot = array[len(array) // 2]
  left = []
  right = []
  for i in range(len(array)):
    if array[i] < pivot:
      left.append(array[i])
    elif array[i] > pivot:
      right.append(array[i])

  return quicksort(left) + [pivot] + quicksort(right)

The best-case space complexity of this algorithm is O(log n), which means that it will require a maximum of log n extra space to sort the array. The average-case space complexity of this algorithm is also O(log n), which means that it will require an average of log n extra space to sort the array. The worst-case space complexity of this algorithm is O(n), which means that it will require a maximum of n extra space to sort the array.

Relation between Time and Space Complexity

In general, there is a trade-off between the time complexity and space complexity of an algorithm. Algorithms with lower time complexity typically require more space, and vice versa. This is because algorithms with lower time complexity typically use more efficient data structures, which require more space.

For example, the quicksort algorithm has a lower time complexity than the bubble sort algorithm, but it also has a higher space complexity. This is because the quicksort algorithm uses a more efficient data structure, the quicksort tree, which requires more space than the bubble sort algorithm's data structure, the array.

Conclusion

The complexity of an algorithm is a measure of the amount of resources required by the algorithm to solve a given problem. The two most common types of complexity are time complexity and space complexity. Time complexity measures the amount of time required by the algorithm to solve a problem, while space complexity measures the amount of memory required by the algorithm to solve a problem. In general, there is a trade-off between the time complexity and space complexity of an algorithm.