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Data Structure Articles
Page 64 of 164
Check if the given two numbers are friendly pairs or not
Friendly Numbers − According to number theory, friendly numbers are two or more numbers having the same abundancy index. Abundancy Index − Abundancy index of a natural number can be defined as the ratio between the sum of all the divisors of the natural number and the natural number itself. The abundancy of a number n can be expressed as $\mathrm{\frac{\sigma(n)}{n}}$ where $\mathrm{\sigma(n)}$ denotes the divisor function equal to all the divisors of n. For example, the abundancy index of the natural number 30 is, $$\mathrm{\frac{\sigma(30)}{30}=\frac{1+2+3+5+6+10+15+30}{30}=\frac{72}{30}=\frac{12}{5}}$$ A number n is said to be a ‘friendly number’ if there exists a ...
Read MoreCheck if any valid sequence is divisible by M
A sequence is a collection of objects, and in our case, it is a collection of integers. The task is to find if a sequence with the usage of the addition and subtraction operator within the elements is divisible by M or not. Problem Statement Given an integer M and an array of integers. Using only addition and subtraction between elements check if there is a valid sequence whose solution is divisible by M. Sample Example 1 Input: M = 2, arr = {1, 2, 5} Output: TRUE Explanation − For the given array a valid sequence {1 ...
Read MoreMax occurring divisor in an interval
Let x and y be two numbers. In this case, x is said to be a divisor of y if when y is divided by x it returns zero remainder. The maximum occurring divisor in an interval is the number that is a divisor of the maximum number of elements of that interval. Problem Statement Given an interval [a, b]. Find the maximum occurring divisor in the range including both a and b, except ‘1’. In case all divisors have equal occurrence, return 1. Sample Example 1 Input [2, 5] Output 2 Explanation − Divisors of 2 = ...
Read MoreRamanujan–Nagell Conjecture
Ramanujan-Nagell Equation is an example of the exponential Diophantine equation. The diophantine equation is a polynomial equation with integer coefficients of two or more unknowns. Only integral solutions are required for the Diophantine equation. Ramanujan-Nagell Equation is an equation between a square number and a number that is seven less than the power of 2, where the power of 2 can only be a natural number. Ramanujan conjectured that the diophantine equation 2y - 7 = x2 has positive integral solutions and was later proved by Nagell. $$\mathrm{2y−7=x^2\:has\:x\epsilon\:Z_+:x=1, 3, 5, 11, 181}$$ Triangular Number − It counts objects arranged in ...
Read MoreFind the Number of \'X\' Total Shapes
In this problem, we need to find the total number of ‘X’ shapes in the given matrix. We can construct the single ‘X’ shape using 1 or more adjacent ‘X’ elements. We can use the DFS (depth−first search) technique to solve the problem. For each ‘X’ element, we can find all adjacent elements using DFS and count it as a single ‘X’ shape. If we find a new ‘X’, we find its adjacent again. Here, we will use the iterative and recursive DFS to find the total number of ‘X” shapes. Problem statement − We have given a matrix[] of ...
Read MoreSum of Minimum and Maximum Elements of all Subarrays of Size K.
In this problem, we need to take the maximum and minimum elements of all sub−array of length K and add them to get the answer. The first solution approach is that traverse through all sub−arrays of size K, find the minimum and maximum element of each sub−array, and add them. The optimized approach to solve the problem is using the deque data structure. We will store the index of the minimum and maximum elements of the subarray in the deque. Problem statement − We have given an array nums[] containing N positive or negative integer values. We have also ...
Read MoreReduce the Array to Atmost one Element by the Given Operations
In this problem, we will reduce the array size to 1 or 0 by performing the given operations in each turn. We can sort the array in each turn to get the maximum elements in each iteration. Also, we can use the head data structure to improve the performance of the code. Problem statement − We have given a nums[] array. We need to decrease the array by performing the below operations. Choose two maximum elements of the array. If both elements are the same, remove both elements from the array. If both elements are not the same, remove ...
Read MoreRearrange and Update Array Elements as Specified by the given Queries
In this problem, we will perform the given queries on the array elements. The queries contain the circular left rotation, right rotation, and updation of the array element. The logical part of solving the problem is array rotation. The naïve approach to rotating the array in the left direction is to replace each element with the next element and the last element with the first element. We can use the deque data structure to rotate the array efficiently. Problem statement − We have given an arr[] array containing the integer values. Also, we have given a queries[] array containing ...
Read MorePrint all Prime Levels of a Binary Tree
In this problem, we will print all prime levels of the given binary tree. We will use the level order traversal technique to traverse each binary tree level and check whether all nodes contain the prime integer for the particular level. Problem statement − We have given a binary tree and need to print all prime levels of the binary tree. It is given that if all nodes of any binary tree contain a prime integer, we can say a particular level is a prime level. Sample examples Input ...
Read MorePrint all Palindromic Levels of a Binary Tree
In this problem, we will print each palindromic level of the given binary tree. We can use the breadth−first search algorithm to traverse each level of the given binary tree and check whether a particular level of the binary tree is palindromic. If yes, we print the level in the output. Problem statement − We have given a binary tree, and we need to print all palindromic levels of the binary tree. If any particular level of the binary tree contains the same numbers from left to right and right to left, we can say that the current level is ...
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