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Data Structure Articles
Page 62 of 164
Golomb sequence
Golomb Sequence − The Golomb Sequence is a non-decreasing sequence of integers where the value of the nth term is the number of times the integer n appeared in the sequence. Some terms of Golomb sequence are, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, … Here as we can see, the 5th term is 3 and 5 also appears 3 times in the sequence. The 6th term is 4 and 6 also appears 4 times in ...
Read MoreFront and Back Search in unsorted array
Unsorted Array − An array is a data structure consisting of a collection of elements of the same type. An unsorted array is such a structure where the order of elements is random, i.e. on insertion, the element is added to the last irrespective of the order of previous elements and searching in such an array is not helped by any search algorithm because of lack of a pattern of the positioning of elements. Searching − Searching in an array means finding a particular element in the array which can be either returning the position of a desired element or ...
Read MoreFind n-th Fortunate Number
Fortunate Numbers − It is the smallest integer m > 1 such that, for a given positive integer n, pn# + m is a prime number, where pn# is the product of the first n prime numbers. For example, for calculating the third fortunate number, first calculate the product of the first 3 prime numbers (2, 3, 5) i.e. 30. Upon adding 2 we get 32 which is an even number, adding 3 gives 33 which is a multiple of 3. One would similarly rule out integers up to 6. Adding 7 gives, 37 which is a prime number. Thus, ...
Read MoreCheck 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 ...
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