Efficiency of Algorithms: Best, Worst, and Average Cases

Algorithm Analysis: Time and Space Complexity

Understanding Algorithm Performance

Algorithm analysis is crucial in computer science for understanding how an algorithm's resource consumption (time and space) scales with input size. This analysis utilizes mathematical frameworks considering various scenarios.

Worst-Case Efficiency

Worst-case efficiency describes the maximum time or space an algorithm might require for any input of size n.

Example: Linear Search

In a linear search of an unsorted array, the worst case occurs when the target element is at the end or not present. The algorithm must examine all n elements, resulting in O(n) time complexity.

Best-Case Efficiency

Best-case efficiency describes the minimum time or space an algorithm might require for any input of size n.

Example: Linear Search

The best case for linear search occurs when the target element is the first element. The algorithm completes in constant time, O(1).

Average-Case Efficiency

Average-case efficiency describes the expected time or space an algorithm will require over all possible inputs of size n.

Example: Linear Search

Assuming the target element is equally likely to be anywhere in the array, linear search examines roughly half the elements on average, leading to O(n) average-case time complexity.

Example Algorithm: Linear Search

Algorithm:

def linear_search(arr, target):
  for i in range(len(arr)):
    if arr[i] == target:
      return i
  return -1

Analysis:

• Worst Case: Target at the end or not present; O(n) time complexity.
• Best Case: Target at the beginning; O(1) time complexity.
• Average Case: Target equally likely at any position; O(n) time complexity.