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Data Structure Articles
Page 34 of 164
Nicomachus’ Theorem
According to Nicomachus’ Theorem, the sum of the cubes of the first n integers is equal to the square of the nth triangular number. Or, we can also say − The sum of cubes of first n natural numbers is equal to square of sum of first natural numbers. Putting it algebraically, $$\mathrm{\displaystyle\sum\limits_{i=0}^n i^3=\lgroup \frac{n^2+n}{2}\rgroup^2}$$ Theorem $$1^3 = 1$$ $$2^3 = 3 + 5$$ $$3^3 = 7 + 9 + 11$$ $$4^3 = 13 + 15 + 17 + 19\vdots$$ Generalizing $$n^3 =\lgroup n^2−n+1\rgroup+\lgroup n^2−n+3\rgroup+⋯+\lgroup n^2+n−1\rgroup$$ Proof By Induction For all n Ε Natural ...
Read MoreJava Program to Implement Unrolled Linked List
In this problem, we will learn to implement the unrolled linked list. The unrolled linked list is a specialized version of the linked list. The normal linked list contains a single element in a single node, but the unrolled linked list contains a group of elements in each node. Also, insertion, deletion, and traversal in the unrolled linked list work the same as the typical linked list. The linear search is faster in the array than in the linked list. So, we can add elements in the array and an array in each node of the linked list. Also, ...
Read MoreJava Program to Implement the Vizing\'s Theorem
In this problem, we need to implement Vizing's Theorem. Vizing's Theorem is used with graphs. Theorem statement - For any undirected graph G, the value of the Chromatic index is equal to the d or d + 1, where d is the maximum degree of the graph. The degree for any vertex is the total number of incoming or outgoing edges. Problem statement - We have given a graph and need to implement Vizing's Theorem to find the Chromatic index of the graph. Note - The chromatic index is a positive integer, requiring a ...
Read MoreJava Program to Implement the Schonhage-Strassen Algorithm for Multiplication of Two Numbers
The Schonhage-Strassen algorithm is useful when we need to multiply large decimal numbers. As Java supports the 1018 size of integers, and if we need to multiply the digits of more than 1018, we need to use the Schonhage-Strassen algorithm, as it is one of the fastest multiplication algorithms. It uses the basic rules for the multiplication of two numbers. It first performs the linear convolution and then performs the carry to get the final result. Problem statement - We have given mul1 and mul2 large decimal numbers and need to implement the Schonhage-Strassen algorithm to multiply both ...
Read MoreEqual sum array partition excluding a given element
Problem Statement For an index and an arr[]. Check if the array[] can be partitioned into two disjoint sets, excluding arr[index] such that the sum of both sets has equal value. Example 1 Input arr[] = {4, 3, 1, 2}, Index = 1 Output No Explanation We have to exclude arr[1] = 3 All possible sets are − Set 1: (4), Set 2: (2, 1), sum = 4≠3 Set 1: (4, 1), Set 2: (2), sum = 5≠2 Set 1: (4, 2), Set 2: (1), sum = 6≠1 No combination satisfies the conditions. Example 2Input arr[] ...
Read MoreSquared Triangular Number (Sum of Cubes)
A square triangular number, also referred to as a triangular square number, is a number that is both a triangular number and a perfect square. Square triangular numbers have an unlimited number of possible values; the first few are − 0, 1, 36, 1225, 41616... A triangular number or triangle number counts objects arranged in an equilateral triangle. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, ...
Read MoreJava Program to Implement the String Search Algorithm for Short Text Sizes
In this problem, we need to find the index of the pattern in the string. Implementing an efficient text search is very important to allow users to search large text databases easily. For example, you are writing a blog in Microsoft Word or code in VSCode, containing 1 lakh+ word. If the search algorithm is inefficient, it can take time to show you search results when searching for any word or sentence. We will learn two different approaches to implementing the string search algorithm. One is the naïve approach, and another is the KMP algorithm. Problem statement - ...
Read MoreSum of all prefixes of given numeric string
In this problem, we need to find the sum of all prefixes of the given string. The best solution approach is that traverse through each prefix of the string and add them to get the answer. Problem statement - We have given a string named num_Str containing N digits. We need to find the sum of all prefixes of the given string. Sample examples Input num_str = "1123" Output 1247 Explanation - All prefixes of the given strings are 1, 11, 112, and 1123. The sum of all prefixes is 1247. Input num_str = ...
Read MoreMinimize the cost to convert given string to either type XYXY… or XXYY…
In this problem, we need to convert the given binary string to abababab or aabbaabb format and find the minimum cost for that. Also, we have given the cost to flip any character in the operations array. Problem statement - We have given a bin_str binary string and operations array containing the positive integers. The size of the string and array is the same and even. The task is to find the minimum costs to convert the string in the ababab… or aabbaabb… format. The cost to flip any character in the given string is the value at the ...
Read MoreMaximum string length after choosing strings from given Array with given conditions
In this problem, we need to find the maximum length of the resultant string by appending the array strings to it such that if we choose a string of length x, we can choose the next x/2 strings. We can solve the programming using the recursive function, memoization, and dynamic programming approach. Problem statement - We have given an array of strings named str_array containing the N strings. We need to add the strings given in the array and find the maximum size of the resultant string. While appending the string to the resultant string, we need to ...
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