The correct base case is very important for correctness! Before understanding dynamic programming and backtracking, We always suggest to understand this approach. This is a very good algorithm design strategy to learn about recursive problem solving. Several problems can be solved using the idea similar to the merge sort and binary search. But there are few cases where we use more than two subproblems for the solution. Usually, we solve a divide and conquer problems using only 2 subproblems. Think about the base case of the merge sort.Ĭan we solve other problems using a similar approach?Įxplore the divide and conquer algorithm of quick-sort. This is an important two pointer approach. How can we design the algorithm for merging two sorted half? How we implement the merge sort algorithm? Can we think of an Iterative version of it? So, the time complexity of the merge sort is O(nlog n). The recurrence relation for the above is: T(n) = 2T(n/2) + O(n) The time complexity of this part is O(n) because merging two sorted array or lists requires O(n) operation Time complexity of recursively solving the two sub-problem of size n/2 i.e. It is O(1) operation because mid can be calculated in constant time. Suppose, T(n) = Time complexity of searching the value K in N size array. This is the case of a single element (Think!) : Merge the two sorted halves by merge function : Recursively solve the two smaller sub-problemsġ) Sort the first half: mergeSort(A, l, mid)Ģ) Sort the second half: mergeSort(A, mid+1, r) : Calculate the middle index of the array. Merge the two sorted subsequences to produce a single sorted sequence. Sort the two sub-sequences recursively using merge sort. The algorithm works as follows:ĭivide the n elements sequence into two equal size subsequences of n/2 element each Merge Sort is an efficient O(nlog n) sorting algorithm and It uses the divide-and-conquer approach. Why is Binary Search preferred over Ternary Search? Where we apply the divide and conquer approach but we divide the given array into three parts. Prepare a list of the problems where we can apply the ideas similar to the binary search for the solution. Also, compare the space complexity of both? Think about the recursive and iterative implementation of the binary search algorithms. We can solve the above recurrence relation by recursion tree method or master theorem.Ĭan we use some hypothesis to analyze the time complexity of binary search? Time complexity is O(log n), which is much faster than O(n) algorithm of linear search. It is the time complexity of recursively solving the one sub-problem of size n/2 i.e. It is O(1) operation because the middle index can be calculated in constant time. Suppose, T(n) = Time complexity of searching the value K in n size array. In the worst case, Recursion will terminate at the base case which is On the basis of comparison with the middle value, we are reducing the input size by 1/2 at every step of recursion. This is the case of an unsuccessful search (Think!) if (A > k), then call binarySearch(A, l, mid-1, k) Recursively solve the smaller sub-problem Recursive Call : binarySearch(A, l, r, k)Ĭalculate the middle index of the array. Similarly, if A is less than K then we search value K in the right part. If A is greater than K then definitely K will not be present in the right part, so we search value K in the left part. The basic idea of binary search is to divide the array equally and compare the value K with the middle element. But if we use the sorted property of the array, we can apply the divide and conquer approach to solve it efficiently in O(log n) time complexity. The naive solution for the problem do a linear search to check whether element K is present or not. If yes then return true otherwise return false. Important Problems/Real-Life Applicationsĭividing the problem into two or more than two sub-problems that are similar to the original problem but smaller in size.Ĭombine these solutions to subproblems to create a solution to the original problem. We will be exploring the following things: We will be discussing the Divide and Conquer approach in detail in this blog. This method usually allows us to reduce the time complexity to a large extent. Divide and Conquer is a recursive problem-solving approach which break a problem into smaller subproblems, recursively solve the subproblems, and finally combines the solutions to the subproblems to solve the original problem.
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