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Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Final Value Theorem of Z-TransformThe final value theorem of Z-transform enables us to calculate the steady state value of a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{x}\mathrm{\left(\mathit{\infty}\right)}$ directly from its Z-transform, without the need for finding its inverse Z-transform.Statement - If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a causal sequence, then the final value theorem of Z-transform states that if, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$And if the Z-transform X(z) has no poles outside ... Read More
In this problem, we are given two values lValue and hValue. Our task is to find the largest twins in given range.Two numbers are said to be twin numbers if both of them are prime numbers and the difference between them is 2.Let's take an example to understand the problem, Input : lValue = 65, rValue = 100 Output : 71, 73Solution ApproachA simple solution to the problem is by looping from rValue - 2 to lValue and checking each pair of i and (i+2) for twins and print the first occurred twin.Another Approach is by finding all prime numbers ... Read More
In this problem, we are given an arr[] consisting of N unsorted elements. Our task is to find the largest pair sum in an unsorted array.We will find a pair whose sum is the maximum.Let's take an example to understand the problem, Input : arr[] = {7, 3, 9, 12, 1} Output : 21Explanation −Pair with largest sum, (9, 12). Sum = 21 Solution ApproachA simple solution to the problem is by making a pair of maximum and second maximum elements of the array.For this we will initialise the max and secondMax elements of the array with the first and ... Read More
In this problem, we are given two integer values, n and m. Our task is to find the largest number with n set and m unset bits in the binary representation of the number.Let's take an example to understand the problemInput : n = 3, m = 1 Output : 14Explanation −Largest number will have 3 set bits and then 1 unset bit. (1110)2 = 14Solution ApproachA simple solution to the problem is by finding the number consisting of (n+m) set bits. From this number, toggle off the m bits from the end (LSB). To create a number with (n+m) ... Read More
In this problem, we are given two integer values, N denoting the count of digits of a number and sum denoting the sum of digits of the number. Our task is to find the largest number with given number of digits and sum of digits.Let's take an example to understand the problem, Input : N = 3, sum = 15 Output : 960Solution ApproachA simple approach to solve the problem is by traversing all N digit numbers from the largest to smallest. The find the digit sum numbers, if it's equal to sum return the number.ExampleProgram to illustrate the working ... Read More
In this problem, we are given a doubly linked list LL. Our task is to find the largest node in Doubly Linked List.Let's take an example to understand the problem, Input : linked-list = 5 -> 2 -> 9 -> 8 -> 1 -> 3 Output : 9Solution ApproachA simple approach to solve the problem is by traversing the linked-list and if the data value of max is greater than maxVal's data. After traversing the linked-list, we will return the macVal nodes data.ExampleProgram to illustrate the working of our solution#include using namespace std; struct Node{ int data; ... Read More
In this problem, we are given an array arr[] of size N consisting of single digits only. Our task is to find the largest multiple of 2, 3 and 5.Let's take an example to understand the problem, Input : arr[] = {1, 0, 5, 2} Output : 510Explanation −The number 510 is divisible by all 2, 3, 5. Solution ApproachA simple solution to the problem is by checking for basic divisibility of the number created.So, if the number needs to be divisible by 2 and 5 i.e. it is divisible by 10. For creating a number divisible by 10, the ... Read More
In this problem, we are given a number N. Our task is to find the largest good number in the divisors of given number N.A good number is a number in which every digit is larger than the sum of digits of its right (all less significant bits than it). For example, 732 is a good number, 7> 3+2 and 3>2.Let's take an example to understand the problem, Input : N = 15 Output : 15Explanation −Divisors of 15 : 1, 3, 5, 15. Solution ApproachA simple solution to the problem is by finding all the divisors of N. And ... Read More
In this problem, we are given a binary tree BT. Our task is to find the largest BST subtree in a given Binary Tree.Binary Tree is a special data structure used for data storage purposes. A binary tree has a special condition that each node can have a maximum of two children.Binary Search Tree (BST)is a tree in which all the nodes follow the below-mentioned properties −The value of the key of the left sub-tree is less than the value of its parent (root) node's key.The value of the key of the right subtree is greater than or equal to ... Read More
In this problem, we are given an array aar[] of size n and another array del[] of size m. Our task is to find the largest after deleting the given elements. If the deletion of an element with multiple instances is required, delete the first instance of the element.Let's take an example to understand the problem, Input : arr[] = {3, 5, 1, 7, 9, 2}, del[] = {1, 9, 3} Output : 7Explanation −Array arr[] after deleting the elements : {5, 7, 2} Largest element of the array is 7Solution ApproachA simple solution the problem is by deleting all ... Read More