Suppose you are choosing between the following three...

Suppose you are choosing between the following three algorithms: • Algorithm A solves problems by dividing them into five subproblems of half the size, recursively solving each subproblem, and then combining the solutions in linear time. • Algorithm B solves problems of size n by recursively solving two subproblems of size n ? 1 and then combining the solutions in constant time. • Algorithm C solves problems of size n by dividing them into nine subproblems of size n/3, recursively solving each subproblem, and then combining the solutions in O(n 2 ) time. What are the running times of each of these algorithms (in big-O notation), and which would you choose?

Suppose you are choosing between the following three algorithms:
• Algorithm A solves problems by dividing them into five subproblems of half the size, recursively solving each subproblem, and then combining the solutions in linear time.
• Algorithm B solves problems of size n by recursively solving two subproblems of size n ? 1
and then combining the solutions in constant time.
• Algorithm C solves problems of size n by dividing them into nine subproblems of size n/3,
recursively solving each subproblem, and then combining the solutions in O(n
2
) time.
What are the running times of each of these algorithms (in big-O notation), and which would you
choose?

Key Concepts

Big-O Notation

Big-O notation is used to describe the upper bound of an algorithm’s running time, ignoring constant factors and lower order terms. It provides a way to compare the efficiency of different algorithms by focusing on the growth rate of their running time as the input size increases.

Divide-and-Conquer Algorithms

Divide-and-conquer is a fundamental algorithm design paradigm that involves dividing a problem into smaller subproblems, solving each subproblem recursively, and then combining their solutions to solve the original problem. The efficiency of these algorithms is often characterized by the number of subproblems and the work required to combine solutions, aspects that are captured by the corresponding recurrence relations.

Master Theorem

The Master Theorem provides a straightforward method for solving recurrences that arise in divide-and-conquer algorithms. It compares the work done at the subproblem level with the work needed to combine the solutions, allowing one to identify the asymptotic behavior of the algorithm's running time. This theorem is particularly useful when the recurrence fits a standard form.

Recurrence Relations

Recurrence relations express the running time of an algorithm in terms of the running time for smaller input sizes. They are often used for divide-and-conquer algorithms in order to model how the problem is broken down into subproblems and then recombined. Solving these recurrences leads to a clearer understanding of the overall time complexity of an algorithm.

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