Divide and Conquer Algorithm Last Updated : 15 Nov, 2024 Comments Improve Suggest changes Like Article Like Report Divide and Conquer algorithm is a problem-solving strategy that involves. Divide : Break the given problem into smaller non-overlapping problems.Conquer : Solve Smaller ProblemsCombine : Use the Solutions of Smaller Problems to find the overall result.Examples of Divide and Conquer are Merge Sort, Quick Sort, Binary Search and Closest Pair of Points.Basics of Divide and Conquer Algorithm:Introduction to Divide and ConquerDynamic Programming vs Divide-and-ConquerDecrease and ConquerAdvanced master theorem for divide and conquer recurrencesStandard Algorithms on Divide and Conquer Algorithm:Binary SearchMerge SortQuick SortCalculate pow(x, n)Karatsuba algorithm for fast multiplicationStrassen’s Matrix MultiplicationConvex Hull (Simple Divide and Conquer Algorithm)Quickhull Algorithm for Convex HullPractice problems on Divide and Conquer Algorithm:Square root of an integerMaximum and minimum of an array using minimum number of comparisonsFind frequency of each element in a limited range array in less than O(n) timeTiling ProblemCount InversionsThe Skyline ProblemSearch in a Row-wise and Column-wise Sorted 2D ArrayAllocate minimum number of pagesModular Exponentiation (Power in Modular Arithmetic)Problems Based on Standard Divide and Conquer AlgorithmsBinary Search Based ProblemsMerge Sort Based ProblemsQuick Sort Based Problems Quick Links :Learn Data Structure and Algorithms | DSA Tutorial‘Practice Problems’ on Divide and Conquer‘Quizzes’ on Divide and Conquer Comment More infoAdvertise with us Next Article Introduction to Divide and Conquer Algorithm H harendrakumar123 Follow Improve Article Tags : Divide and Conquer DSA Practice Tags : Divide and Conquer Similar Reads Divide and Conquer Algorithm Divide and Conquer algorithm is a problem-solving strategy that involves. Divide : Break the given problem into smaller non-overlapping problems.Conquer : Solve Smaller ProblemsCombine : Use the Solutions of Smaller Problems to find the overall result.Examples of Divide and Conquer are Merge Sort, Q 1 min read Introduction to Divide and Conquer Algorithm Divide and Conquer Algorithm is a problem-solving technique used to solve problems by dividing the main problem into subproblems, solving them individually and then merging them to find solution to the original problem. Divide and Conquer is mainly useful when we divide a problem into independent su 9 min read Dynamic Programming vs Divide-and-Conquer In this article I’m trying to explain the difference/similarities between dynamic programming and divide and conquer approaches based on two examples: binary search and minimum edit distance (Levenshtein distance).The ProblemWhen I started to learn algorithms it was hard for me to understand the mai 12 min read Decrease and Conquer As divide-and-conquer approach is already discussed, which include following steps: Divide the problem into a number of subproblems that are smaller instances of the same problem. Conquer the sub problems by solving them recursively. If the subproblem sizes are small enough, however, just solve the 5 min read Advanced master theorem for divide and conquer recurrences The Master Theorem is a tool used to solve recurrence relations that arise in the analysis of divide-and-conquer algorithms. The Master Theorem provides a systematic way of solving recurrence relations of the form: T(n) = aT(n/b) + f(n) where a, b, and f(n) are positive functions and n is the size o 5 min read Some standard Divide and Conquer AlgorithmsWrite program to calculate pow(b, e)Given two numbers b and e, the task is to implement a function to compute b^e.Examples: Input: b = 3.00000, e = 5Output: 243.00000Input: b = 0.55000, e = 3Output: 0.16638Input: b = -0.67000, e = -7Output: -16.49971Table of Content[Naive Approach 1] Using Iteration - O(e) Time and O(1) Space[Naive Ap 10 min read Karatsuba algorithm for fast multiplication using Divide and Conquer algorithmGiven two binary strings that represent value of two integers, find the product of two strings. For example, if the first bit string is "1100" and second bit string is "1010", output should be 120.For simplicity, let the length of two strings be same and be n.A Naive Approach is to follow the proces 15+ min read Strassen's Matrix MultiplicationGiven two square matrices arr[][] and brr[][] of order n * n. Your task is to multiply both the matrices and find the resultant matrix.Examples:Input: arr[][] = [ [7, 8], [2, 9] ]brr[][] = [ [14, 5], [5, 18] ]Output: [ [138, 179], [73, 172] ]Input: arr[][] = [ [17, 4], [17, 16] ]brr[][] = [ [9, 2], 15+ min read Convex Hull using Divide and Conquer AlgorithmIn computational geometry, a convex hull is the smallest convex polygon that contains a given set of points. It is a fundamental concept with applications in various fields such as computer graphics, robotics, and image processing. Importance of Convex Hull:Convex hulls are important in computationa 15 min read Quickhull Algorithm for Convex HullGiven a set of points, a Convex hull is the smallest convex polygon containing all the given points. Input : points[] = {{0, 3}, {1, 1}, {2, 2}, {4, 4}, {0, 0}, {1, 2}, {3, 1}, {3, 3}};Output : The points in convex hull are: (0, 0) (0, 3) (3, 1) (4, 4)Input : points[] = {{0, 3}, {1, 1}Output : Not P 14 min read Binary Search based problemsPeak Element in ArrayGiven an array arr[] where no two adjacent elements are same, find the index of a peak element. An element is considered to be a peak element if it is strictly greater than its adjacent elements. If there are multiple peak elements, return the index of any one of them.Note: Consider the element befo 12 min read Check for Majority Element in a sorted arrayGiven an array arr of N elements, A majority element in an array arr of size N is an element that appears more than N/2 times in the array. The task is to write a function say isMajority() that takes an array (arr[] ), array’s size (n) and a number to be searched (x) as parameters and returns true i 15+ min read K-th Element of Merged Two Sorted ArraysGiven two sorted arrays of sizes m and n respectively, the task is to find the element that would be at the k-th position in the final sorted array formed by merging these two arrays.Examples: Input: a[] = [2, 3, 6, 7, 9], b[] = [1, 4, 8, 10], k = 5Output: 6Explanation: The final sorted array is [1, 15+ min read Find the number of zeroesGiven an array of 1s and 0s which has all 1s first followed by all 0s. Find the number of 0s. Count the number of zeroes in the given array.Examples : Input: arr[] = {1, 1, 1, 1, 0, 0} Output: 2 Input: arr[] = {1, 0, 0, 0, 0} Output: 4 Input: arr[] = {0, 0, 0} Output: 3 Input: arr[] = {1, 1, 1, 1} O 12 min read Rotation Count in a Rotated Sorted arrayGiven an array arr[] having distinct numbers sorted in increasing order and the array has been right rotated (i.e, the last element will be cyclically shifted to the starting position of the array) k number of times, the task is to find the value of k.Examples: Input: arr[] = {15, 18, 2, 3, 6, 12}Ou 12 min read Unbounded Binary Search Example (Find the point where a monotonically increasing function becomes positive first time)Given a function 'int f(unsigned int x)' which takes a non-negative integer 'x' as input and returns an integer as output. The function is monotonically increasing with respect to the value of x, i.e., the value of f(x+1) is greater than f(x) for every input x. Find the value 'n' where f() becomes p 11 min read Median of two Sorted Arrays of Different SizesGiven two sorted arrays, a[] and b[], the task is to find the median of these sorted arrays. Assume that the two sorted arrays are merged and then median is selected from the combined array.This is an extension of Median of two sorted arrays of equal size problem. Here we handle arrays of unequal si 15+ min read The Painter's Partition Problem using Binary SearchGiven an array arr[] and k, where the array represents the boards and each element of the given array represents the length of each board. k numbers of painters are available to paint these boards. Consider that each unit of a board takes 1 unit of time to paint.The task is to find the minimum time 10 min read Some practice problems on Divide and Conquer algorithmProgram for Square Root of IntegerGiven a positive integer n, find its square root. If n is not a perfect square, then return floor of √n.Examples : Input: n = 4Output: 2Explanation: The square root of 4 is 2.Input: n = 11Output: 3Explanation: The square root of 11 lies in between 3 and 4 so floor of the square root is 3.Table of Co 11 min read Maximum and minimum of an array using minimum number of comparisonsGiven an array of size N. The task is to find the maximum and the minimum element of the array using the minimum number of comparisons.Examples:Input: arr[] = {3, 5, 4, 1, 9}Output: Minimum element is: 1 Maximum element is: 9Input: arr[] = {22, 14, 8, 17, 35, 3}Output: Minimum element is: 3 Maximum 15+ min read Find frequency of each element in a limited range array in less than O(n) timeGiven a sorted array arr[] of positive integers, the task is to find the frequency for each element in the array. Assume all elements in the array are less than some constant MNote: Do this without traversing the complete array. i.e. expected time complexity is less than O(n)Examples: Input: arr[] = 10 min read Tiling Problem - L ShapedGiven an n×n board (where n = 2k and k≥1), with one missing cell, the task is to fill the remaining cells using L-shaped tiles. An L-shaped tile covers 3 cells in a 2x2 grid, with one cell missing. You need to tile the entire board using the L-shaped tiles, ensuring that the missing cell remains unc 15+ min read Count Inversions of an ArrayGiven an integer array arr[] of size n, find the inversion count in the array. Two array elements arr[i] and arr[j] form an inversion if arr[i] > arr[j] and i < j.Note: Inversion Count for an array indicates that how far (or close) the array is from being sorted. If the array is already sorted 15+ min read The Skyline Problem | Set-1Given n rectangular buildings in a 2-dimensional city, compute the skyline of these buildings, eliminating hidden lines. The main task is to view buildings from a side and remove all sections that are not visible. All buildings share a common bottom and every building is represented by a triplet (le 13 min read Search in a Row-wise and Column-wise Sorted 2D Array using Divide and Conquer algorithmGiven an n x n matrix, where every row and column is sorted in increasing order. Given a key, how to decide whether this key is in the matrix. Input: x = 62, mat[][] = [[3, 30, 38], [20, 52, 54], [35, 60, 69]]Output: falseExplanation: 62 is not present in the matrix.Input: x = 30, mat[][] = [[3, 30] 7 min read Allocate Minimum PagesGiven an array arr[] and an integer k, where arr[i] denotes the number of pages of a book and k denotes total number of students. All the books need to be allocated to k students in contiguous manner, with each student getting at least one book.The task is to minimize the maximum number of pages all 15+ min read Like