![]() ![]() Here is a visualization of the binary search algorithm where 4 is the target value. If the search ends with the remaining half being empty, the target is not in the array. Binary search compares the target value to the middle element of the array if they are unequal, the half in which the target cannot lie is eliminated and the search continues on the remaining half until the target value is found. Divide and Conquer Example: Binary Searchīinary search algorithm, also known as half-interval search, is a search algorithm that finds the position of a target value within a sorted array. Let’s go and try to solve some problems using DP and DC approaches to make this illustration more clear. Since we’re now familiar with DP prerequisites and its methodologies we’re ready to put all that was mentioned above into one picture.ĭynamic programming and divide and conquer paradigms dependency So What the Difference Between DP and DC After All The main idea you should grasp here is that because our divide and conquer problem has overlapping sub-problems the caching of sub-problem solutions becomes possible and thus memoization/tabulation step up onto the scene. You may read more about memoization and tabulation comparison here. The memoized fib function would thus look like this: Memoization (top-down cache filling) refers to the technique of caching and reusing previously computed results. For example naive recursive implementation of Fibonacci function has time complexity of O(2^n) where DP solution doing the same with only O(n) time. Dynamic Programming Extension for Divide and Conquerĭynamic programming approach extends divide and conquer approach with two techniques ( memoization and tabulation) that both have a purpose of storing and re-using sub-problems solutions that may drastically improve performance. Once these two conditions are met we can say that this divide and conquer problem may be solved using dynamic programming approach. Overlapping sub-problems - problem can be broken down into sub-problems which are reused several times, or a recursive algorithm for the problem solves the same sub-problem over and over rather than always generating new sub-problems.Optimal substructure - optimal solution can be constructed from optimal solutions of its sub-problems.Dynamic Programming Prerequisites/RestrictionsĪs we’ve just discovered there are two key attributes that divide and conquer problem must have in order for dynamic programming to be applicable: After that dynamic programming extends divide and conquer approach with memoization or tabulation technique. It is because dynamic programming approach may be applied to the problem only if the problem has certain restrictions or prerequisites. So why do we still have different paradigm names then and why I called dynamic programming an extension. The solutions to the sub-problems are then combined to give a solution to the original problem. ![]() Because they both work by recursively breaking down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. I would not treat them as something completely different. ![]() Dynamic Programming and Divide-and-Conquer SimilaritiesĪs I see it for now I can say that dynamic programming is an extension of the divide and conquer paradigm. But I hope this article will shed some extra light and help you to do another step of learning such valuable algorithm paradigms as dynamic programming and divide-and-conquer. I’m still in the process of understanding DP and DC difference, and I can’t say that I’ve fully grasped the concepts so far. These detail tells us that each technique serves best for different types of problems. But when we’re trying to solve the same problem using both DP and DC approaches to explain each of them, it feels for me like we may lose valuable detail that might help to catch the difference faster. When it gets to comparing those two paradigms usually Fibonacci function comes to the rescue as great example. When I started to learn algorithms it was hard for me to understand the main idea of dynamic programming ( DP) and how it is different from divide-and-conquer ( DC) approach. In this article I’m trying to explain the difference/similarities between dynamic programing and divide and conquer approaches based on two examples: binary search and minimum edit distance (Levenshtein distance).Īlso, in the Content-aware image resizing in JavaScript article I went through another powerful but yet simple example of dynamic programming for the Seam Carving algorithm. ![]()
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