What is a common characteristic of divide-and-conquer algorithms?

Divide-and-conquer algorithms are characterized by their approach of breaking down a problem into smaller subproblems, solving those subproblems independently, and then combining their solutions to form the answer to the original problem. A common characteristic that highlights this method is that they can often be expressed using recurrence relations for time complexities.

In this context, a recurrence relation mathematically captures the relationship between the size of the problem and the time it takes to solve it as the problem size scales down. For instance, in merge sort, the time complexity can be expressed as T(n) = 2T(n/2) + Θ(n), where T(n/2) refers to solving two subproblems of size n/2 and Θ(n) refers to the time taken to merge the results. This type of expression succinctly encapsulates the recursive nature of the algorithm as it breaks the initial problem into smaller components.

Understanding this characteristic provides insight into analyzing the efficiency and performance of divide-and-conquer algorithms, often leading to well-known results such as O(n log n) for sorting algorithms like quicksort and mergesort.