The best-case, average-case, and worst-case time complexities of an algorithm are measures of how efficient the algorithm is under different scenarios of input.
The best-case time complexity refers to the minimum amount of time an algorithm can take to complete its execution. This occurs when the input is already sorted, or when the algorithm can take advantage of some special property of the input to finish its execution quickly. The best-case time complexity of an algorithm is typically denoted by the big omega notation (), and it represents the lower bound of the algorithm’s runtime.
For example, the best-case time complexity of a linear search algorithm is O(1), which occurs when the target element is the first element of the array.
The worst-case time complexity refers to the maximum amount of time an algorithm can take to complete its execution. This occurs when the input is arranged in a way that makes the algorithm perform the maximum number of operations. The worst-case time complexity of an algorithm is typically denoted by the big O notation (O), and it represents the upper bound of the algorithm’s runtime.
For example, the worst-case time complexity of a linear search algorithm is O(n), which occurs when the target element is not present in the array, and the algorithm has to traverse the entire array.
The average-case time complexity refers to the expected amount of time an algorithm takes to complete its execution when the input is drawn from a probability distribution. The average-case time complexity is typically denoted by the big theta notation (), and it represents the average or typical runtime of the algorithm for a given input distribution.
For example, the average-case time complexity of a quicksort algorithm is O(n log n), which occurs when the input is drawn from a uniform distribution.
In summary, the best-case time complexity represents the lower bound of an algorithm’s runtime, the worst-case time complexity represents the upper bound, and the average-case time complexity represents the expected runtime under a given input distribution. The choice of which time complexity to optimize for depends on the specific requirements of the problem at hand.