mathematical algorithm - OpenGenus IQ

Finding the Lexicographical Next Permutation in O(N) time complexity

In Lexicographical Permutation Algorithm we will find the immediate next smallest Integer number or sequence permutation. Finding all permutations take O(N!) time complexity but we present an efficient algorithm which can solve this in O(N) time complexity

algorithm

Heap’s algorithm for generating permutations

Heap's Algorithm is used to generate all the possible permutation of n-decimals of a number. It generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed. For N numbers, it takes O(N!) time complexity

algorithm

Number of unique partitions of an integer

Given a positive integer, the problem is to find all the Unique Partitions of the number. We can understand the above problem using some basic examples. Naive approach takes O((N^N)log N) time complexity while Dynamic programming takes O(2^N)

algorithm

Distance between two points in 2D space

Distance between two points is calculated by creating a right-angled triangle using the two points.The line between the two points is the hypotenuse (the longest side, opposite to the 90° angle). We used the Distance formula derived from Pythagorean theorem with a time complexity of O((log N)^2)

algorithm

Area of Polygon: Shoelace formula

Given coordinates of vertices of polygon, Area of Polygon is calculated using Shoelace formula described by Mathematician and Physicist Carl Friedrich Gauss where polygon vertices are described by their Cartesian coordinates in the Cartesian plane. The time complexity is O(N) with O(1) space

algorithm

Extended Euclidean Algorithm

We will demonstrate Extended Euclidean Algorithm. We will see how you can calculate the greatest common divisor in a naive way which takes O(N) time complexity which we can improve to O(log N) time complexity using Euclid's algorithm. Extended Euclidean Algorithm takes O(log N) time complexity

algorithm

Sieve of Atkin

Sieve of Atkin is an algorithm used to find all prime numbers upto a given number (say N) and does so in O(N) time complexity. With a modified version with enumerating lattice points variation, the time complexity goes to O(N / log log N). It is computationally efficient than Sieve of Eratosthenes

algorithm

Area of Triangle: Heron's formula

We take a look at Heron's formula which will enable us to calculate the area of any triangle given the length of the three sides of that triangle. The advantage is that the area is calculated using arithmetic operations and hence, the time taken can be assumed to be constant,

algorithm

Number of arithmetic progression subsequences in a given set of numbers

The problem is that given an array of n positive integers. The task is to count the number of Arithmetic Progression subsequence in the array. This can be solved using dynamic programming in linear time complexity.

algorithm

Find the Longest Arithmetic Progression using Dynamic Programming

The problem we will solve is that given a set of integers in sorted order, find length of longest arithmetic progression in that set. This can be solved by brute force in O(N^3) while a dynamic programming approach with take O(N^2) time complexity.

game theory

Zobrist Hashing

Algorithm Complexity Implementations Applications Questions Reading time: 15 minutes | Coding time: 5 minutes Zobrist hashing (additionally alluded to as Zobrist keys or Zobrist marks ) is a hash function development utilized as a part

game theory

Alpha Beta Pruning in Minimax Algorithm

Algorithm Complexity Implementations Applications Questions Reading time: minutes | Coding time: minutes Like we discussed in earlier article, a hard coded AI can be used to create an intelligent opponent which you can challenge

game theory

Von Neumann's Minimax Theorem/ algorithm

Introduction Algorithm Complexity Implementations Applications Questions Reading time: 25 minutes | Coding time: 10 minutes Ever since the idea of artificial intelligence dawned upon the humanity, the basic concept of playing games against an

bitwise operation

Detect if a number is power of 4 using bitwise operators

Reading time: 15 minutes | Coding time: 2 minutes Algorithm Implementation Complexity Questions Bitwise operators are one of the most under-rated operators in C/C++, despite being quite fast as most of the processors

bitwise operation

Check if Two Numbers are Equal using Bitwise Operators

Bitwise Operators Explanation Implementations Applications Reading time: 15 minutes | Coding time: 2 minutes In this article, we will explore the technique of checking whether two numbers are equal using bitwise operators. Bitwise operator

bitwise operation

Detect if a number is power of 2 using bitwise operators

Reading time: 15 minutes | Coding time: 1 minutes Algorithm Implementation Complexity Questions Bitwise operators are one of the most under-rated operators in C/C++, despite being quite fast as most of the processors

game theory

Sprague-Grundy Theorem and Game of Kayle

Sprague Grundy function returns smallest non negative integer which is not in the given set. Explore the application of Sprague Grundy Theorem using a famous game of Kayle. Find the complexity and implementation of Sprague Grundy theorem and game of kayle.

game theory

Nimber Arithmetic : A deeper dive in Nim

Explore Nimber Arithmetic in Game theory and always win a game. Nimbers have two particular operations. nim-addition and nim-multiplication. the nimbers are know as Grundy Numbers. Explore applications in the Game of Nim and odd Knight of the round table problem.

game theory

Exploring The Game of Nim

Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. The key to the theory of the game is the binary digital sum of the heap sizes "exclusive or" (xor). It is called the nim-sum. Find implementations of winning strategy and applications

mathematical algorithm

Bell Numbers: Number of partitions of a set

Bell numbers are a sequence of numbers that describe the number of ways a set with N elements can be partitioned into disjoint, non-empty subsets. Dobinski's Formula Satifaction Sum of Sterling's number of second kind, The Peirce Triangle Method. First few Bell numbers are 1, 1, 2, 5, 15, 52, 203

catalan number

Catalan Numbers

Catalan Numbers are one of the widest used and evident number patterns. They are named after the Belgian mathematician Eugène Charles Catalan. A sequence of natural numbers that occur in various counting problems. The Catalan numbers are: 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786,

greatest common divisor

Euclidean Algorithm to Calculate Greatest Common Divisor (GCD) of 2 numbers

The Euclid's algorithm (or Euclidean Algorithm) is a method for efficiently finding the greatest common divisor (GCD) of two numbers. The GCD of two integers X and Y is the largest integer that divides both of X and Y (without leaving a remainder).

sieve theory

The Sieve of Eratosthenes

Sieve of Eratosthenes is a simple and ancient algorithm (over 2200 years old) used to find the prime numbers up to any given limit. It is one of the most efficient ways to find small prime numbers. Implementations in C, C++, C Sharp, Java, Go, Haskell, JavaScript, PHP and Python.