# Divide and Conquer

**Data Structure and Algorithms Course****Practice Problems on Divide and Conquer****Recent Articles on Divide and Conquer**

__What is Divide and Conquer?__

Divide and Conquer is an algorithmic paradigm in which the problem is solved using the Divide, Conquer, and Combine strategy.

A typical Divide and Conquer algorithm solves a problem using following three steps:

**Divide:**This involves dividing the problem into smaller sub-problems.**Conquer:**Solve sub-problems by calling recursively until solved.**Combine:**Combine the sub-problems to get the final solution of the whole problem.

__Standard algorithms that follow Divide and Conquer algorithm__

The following are some standard algorithms that follow Divide and Conquer algorithm.

**Quicksort**is a sorting algorithm. The algorithm picks a pivot element and rearranges the array elements so that all elements smaller than the picked pivot element move to the left side of the pivot, and all greater elements move to the right side. Finally, the algorithm recursively sorts the subarrays on the left and right of the pivot element.**Merge Sort**is also a sorting algorithm. The algorithm divides the array into two halves, recursively sorts them, and finally merges the two sorted halves.**Closest Pair of Points**The problem is to find the closest pair of points in a set of points in the x-y plane. The problem can be solved in O(n^2) time by calculating the distances of every pair of points and comparing the distances to find the minimum. The Divide and Conquer algorithm solves the problem in O(N log N) time.**Strassen’s Algorithm**is an efficient algorithm to multiply two matrices. A simple method to multiply two matrices needs 3 nested loops and is O(n^3). Strassen’s algorithm multiplies two matrices in O(n^2.8974) time.**Cooley–Tukey Fast Fourier Transform (FFT) algorithm**is the most common algorithm for FFT. It is a divide and conquer algorithm which works in O(N log N) time.**Karatsuba algorithm for fast multiplication**does the multiplication of two*n*-digit numbers in at most single-digit multiplications in general (and exactly when*n*is a power of 2). It is, therefore, faster than the classical algorithm, which requires*n*^{2}single-digit products. If*n*= 2^{10}= 1024, in particular, the exact counts are 3^{10}= 59, 049 and (2^{10})^{2}= 1, 048, 576, respectively.

__Example of Divide and Conquer algorithm__

A classic example of Divide and Conquer is Merge Sort demonstrated below. In Merge Sort, we divide array into two halves, sort the two halves recursively, and then merge the sorted halves.

**Topics :**

- Intoduction to Divide and Conquer
- Binary Search
- Randomized Binary Search Algorithm
- Merge Sort
- Quick Sort
- Tiling Problem
- Count Inversions
- Calculate pow(x, n)
- Closest Pair of Points
- Closest Pair of Points | O(nlogn) Implementation
- Multiply two polynomials
- Strassen’s Matrix Multiplication
- The Skyline Problem
- Maximum Subarray Sum
- Longest Common Prefix
- Search in a Row-wise and Column-wise Sorted 2D Array
- Karatsuba algorithm for fast multiplication
- Convex Hull (Simple Divide and Conquer Algorithm)
- Quickhull Algorithm for Convex Hull
- Distinct elements in subarray using Mo’s Algorithm

- Median of two sorted arrays
- Median of two sorted arrays of different sizes
- Floor in a Sorted Array
- Find closest number in array
- Find a Fixed Point in a given arrray
- Find a peak element in a given array
- Check for Majority Element in a sorted array
- K-th Element of Two Sorted Arrays
- Find the Rotation Count in Rotated Sorted array
- Find the minimum element in a sorted and rotated array
- Find the only repeating element in a sorted array of size n
- Find index of an extra element present in one sorted array
- Find the element that appears once in a sorted array
- Count number of occurrences (or frequency) in a sorted array
- Find the maximum element in an array which is first increasing and then decreasing
- Decrease and Conquer
- Binary Search using pthread
- Binary Search on Singly Linked List
- The painter’s partition problem
- The painter’s partition problem | Set 2
- Find the number of zeroes
- Numbers whose factorials end with n zeros
- Find the missing number in Arithmetic Progression
- Number of days after which tank will become empty
- Find bitonic point in given bitonic sequence
- Find the point where a monotonically increasing function becomes positive first time

- Iterative Tower of Hanoi
- Program for Tower of Hanoi
- Square root of an integer
- Find cubic root of a number
- Allocate minimum number of pages
- Collect all coins in minimum number of steps
- Modular Exponentiation (Power in Modular Arithmetic)
- Find a peak element in a 2D array
- Program to count number of set bits in an (big) array
- Maximum and minimum of an array using minimum number of comparisons
- Find frequency of each element in a limited range array in less than O(n) time
- Minimum difference between adjacent elements of array which contain elements from each row of a matrix
- Search element in a sorted matrix
- Easy way to remember Strassen’s Matrix Equation
- Largest Rectangular Area in a Histogram | Set 1
- Advanced master theorem for divide and conquer recurrences
- Place k elements such that minimum distance is maximized
- Iterative Fast Fourier Transformation for polynomial multiplication
- Write you own Power without using multiplication(*) and division(/) operators
- Sequences of given length where every element is more than or equal to twice of previous
- Shuffle 2n integers in format {a1, b1, a2, b2, a3, b3, ……, an, bn} without using extra space

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