The duality transformation is concept in computational geometry that maps coordinate points to lines and lines to coordinate points. In this article, we will learn all about the duality transformation of lines and points, and implement C++ code to show this transformation. What is Duality Transformation? The duality transformation is a process that converts 2 dimensional lines and coordinates into dual plane. In this transformation, a point in the 2D plane can be represented as a line in the dual space, and a line in the 2D plane can be represented as a point in the dual space. This ... Read More

Threaded binary tree is a binary tree that provides the facility to traverse the tree in a particular order. In this article, we will learn all about threaded binary trees, their types, and how to implement them in C++. What is a Threaded Binary Tree? A threaded binary tree is a type of binary tree in which NULL pointers are replaced with pointers to the in-order predecessor and successor nodes. This treading will help in faster traversal of the tree without using a stack or recursion. The image below shows a threaded binary tree. There are two types ... Read More

In this article, we will learn how to write an algorithm an C++ code to find all forward edges in a directed graph. What is a Forward Edge? A forward edge is an edge in a directed graph that points from a node to one of it's descendants in the depth-first search (DFS) tree. To understand this concept better, consider the image of a directed graph below: In the above graph, the edge from node 2 to node 5 is a forward edge because to reach node 5 in DFS traversal, we need to move through node ... Read More

In this article, we will learn what is topological sorting, how to use it to detect cycles in a directed graph, and how to implement it in C++. What is Topological Sort? Topological sorting is an operation used to detect cycle in a graph. In this operation we order the vertices in such a way that for every directed edge u -> v, vertex u comes before vertex v in the ordering. If we are able to perform a topological sort in a graph, it means that the graph is a directed acyclic graph (DAG). The image below show ... Read More

In this problem, we are given adjacency lists of a directed graph and we need to check if there is a cycle in the graph using topological sort. If a cycle exists, it is not possible to perform a topological sort. Example: // Input Graph ( as adjacency list ) 0 -> 1 1 -> 2 1 -> 3 2 -> 0 Output: Cycle exists Explanation: The graph has a cycle (0 -> 1 -> 2 -> 0). To solve this problem, we can use Khan's Algorithm, which is a BFS based topological sorting algorithm. To ... Read More

We are given an array of integers containing the degree of each vertex in a graph. Our task is to check if it is possible to construct a graph with the given degree sequence. Example: int degrees[] = {3, 2, 2, 0} Output: Not Possible Explanation: The first vertex has degree 3, which means it must be connected to three other vertices. But the last vertex has degree 0, meaning it cannot be connected to any other vertex. Hence, it is impossible to construct a graph with this degree sequence. To implement this in C++, we can ... Read More

In this article, we will learn how to print a diamond shape with 2n rows and n columns for a given size n using C++. For example, if n = 4, the diamond shape will look like this: Algorithm to Print Diamond Shape To print the diamond shape, we can follow these steps: Take an integer input n from the user. To print upper half of the diamond, use a loop that runs from i = 1 to i

In this article, we will explain the Vizing's theorem and implement it in C++ to color a graph using the Greedy Coloring Algorithm. What is Vizing's Theorem? Vizing's theorem states that for any graph, the minimum number of colors needed to color the edges (chromatic index) is either equal to the maximum degree G of the graph or one more than maximum degree G + 1. The degree of a vertex is the number of edges connected to it. The maximum degree G refer to highest degree for any vertex in the graph. It is ... Read More

In this article, we will explain the 4 color problem to color a graph and implement the backtracking algorithm to solve it in C++. The 4 Color Problem The 4-color problem states that the maximum number of colors needed to color any planar graph (or a 2D map) is four, such that no two adjacent nodes have the same color. For example, suppose that you want to color a world map, such that no two countries sharing a border have the same color. According to this theorem, the maximum number of colors needed to do this is four. Now, ... Read More

In this article, we will explain the dominating set problem and implement it's solution in C++. First of all, let's understand what a dominating set is. Dominating Set of a Graph A dominating set for a graph is a subset of the set of all the vertices. Every vertex that is not in the dominating set should be adjacent of at least one vertex in the dominating set. To understand this clearly, consider the following graph: In the above graph, the set of vertices {B, D} is one of the dominating sets, because: ... Read More