Algorithm design techniques are essential skills for any programmer who wants to solve complex problems efficiently and elegantly. In this article, you will learn how to apply one of the most common and powerful techniques: divide and conquer. You will see how this technique can help you sort, search, and merge data in less time and space.
Divide and conquer is a technique that breaks down a large problem into smaller and simpler subproblems, solves them recursively or iteratively, and combines the solutions to obtain the final result. This technique can reduce the complexity and the amount of work required to solve a problem, as well as improve the performance and scalability of the algorithm.
One of the most famous examples of divide and conquer is the merge sort algorithm, which can sort an array of data in O(n log n) time and O(n) space. The idea is to divide the array into two halves, sort each half recursively, and then merge the two sorted halves into one sorted array. The merging process involves comparing the first elements of each half and placing the smaller one into the output array, until one of the halves is empty.
In addition to merge sort, there are several other divide and conquer algorithms that can be used for sorting data. For eg. quicksort, which also has an average time complexity of O(n log n) but has a worst-case time complexity of O(n^2). Quicksort works by partitioning the array into two parts based on a pivot element, sorting the two parts recursively, and then combining them into a sorted array. Quick sort can be faster than merge sort in some cases, especially when working with large datasets. Quick sort is often used in real-world applications where speed is a primary concern. For example, quick sort could be used in a web application to sort a large number of user records based on a specific field such as name or date of birth.
Another example of divide and conquer is the binary search algorithm, which can find an element in a sorted array in O(log n) time and O(1) space. The idea is to compare the target element with the middle element of the array, and then eliminate half of the array based on the comparison. If the target is equal to the middle element, the search is done. If the target is smaller, the search continues in the left half. If the target is larger, the search continues in the right half.
An unpopular chain of opinion, however, considers binary search as a "decrease and conquer" instead of "divide and conquer". People argue that it does not involve combining the subproblems' solutions to get the final result. Instead, it only simplifies the original problem by permanently discarding the subproblem is doesn't consider interesting. The book Fundamentals of Algorithmics, G. Brassard, P. Bratley says the following: "It is probably the simplest application of divide-and-conquer, so simple in fact that strictly speaking this is an application of simplification rather than divide-and-conquer: the solution to any sufficiently large instance is reduced to that of a single smaller one, in this case of half size."
A third example of divide and conquer is the merge k sorted lists problem, which can merge k sorted lists of data into one sorted list in O(n log k) time and O(k) space, where n is the total number of elements in all lists. The idea is to use a min-heap to store the first elements of each list, and then extract the smallest element from the heap and append it to the output list, until the heap is empty. The heap can be updated by inserting the next element of the list that the extracted element came from.
Divide and conquer is a useful technique for algorithm design, as it can reduce the complexity and amount of work required to solve a problem. By breaking it down into smaller and simpler subproblems, the performance and scalability of the algorithm can be improved. Additionally, the readability and modularity of the code can be enhanced by separating the logic of the subproblems and the combination of the solutions.
Parallelization is one of the lesser-known & under-recognized advantages of this technique. Since the subproblems can be solved independently, it is possible to distribute the work among multiple processors or threads and speed up the computation significantly. One real-world example where this technique is used is in image processing. The task of applying a filter or effect to a large image can be divided into smaller regions or pixels that can be processed in parallel. This approach is used in many popular image-processing libraries, such as OpenCV and ImageMagick.
Divide and conquer can also pose some challenges for algorithm design. For example, finding the optimal way to divide the problem and combine the solutions may depend on the nature and size of the problem and the data. Additionally, handling the base cases and the edge cases may require special treatment or extra checks to avoid errors or infinite loops. Furthermore, balancing the trade-offs between time and space complexity may vary depending on the algorithm and the data structure.
An example of the challenge of optimally dividing is in the quicksort algorithm. If the input data is already sorted or nearly sorted, the partitioning process may result in one partition being significantly larger than the other, leading to the worst-case performance of O(n^2) instead of the expected O(n log n). To handle this, the algorithm may need to use a randomized pivot selection or a modified partitioning scheme that handles equal elements differently.