There are n cities and there are roads in between some of the cities. Somehow all the roads are damaged simultaneously. We have to repair the roads to connect the cities again. There is a fixed cost to repair a particular road. Input is in the form of edges {u, v, w} where, u and v are city indices.…

Try it on GfG Practice Reverse Delete algorithm is closely related to Kruskal's algorithm. In Kruskal's algorithm what we do is : Sort edges by increasing order of their weights. After sorting, we one by one pick edges in increasing order. We include current picked edge if by including this in…

Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum product spanning tree for a weighted, connected, and undirected graph is a spanning tree with a…

Try it on GfG Practice If a graph is a complete graph with n vertices, then total number of spanning trees is n(n-2) where n is the number of nodes in the graph. In complete graph, the task is equal to counting different labeled trees with n nodes for which have Cayley's formula. What if Graph is…

Try it on GfG Practice Prim’s algorithm is a Greedy algorithm like Kruskal's algorithm. This algorithm always starts with a single node and moves through several adjacent nodes, in order to explore all of the connected edges along the way. The algorithm starts with an empty spanning tree. The idea…

Try it on GfG Practice A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected, and undirected graph is a spanning tree (no cycles and connects all vertices) that has minimum weight. The weight of a spanning tree is the sum of all edges in the tree. In Kruskal's…

Minimum Spanning Tree (MST) is a fundamental concept in graph theory and has various applications in network design, clustering, and optimization problems. Two of the most commonly used algorithms to find the MST of a graph are Prim's and Kruskal's algorithms. Although both algorithms achieve the…

Try it on GfG Practice Given an undirected, weighted graph with V vertices numbered from 0 to V-1, and E edges represented as a 2D array edges[][], where each element edges[i] = [u, v, w] denotes an edge between nodes u and v with weight w, and all edge weights are positive integers, your task is to…

Given an undirected graph where every edge has a weight as either 0 or 1. The task is to find the shortest path from the source vertex to all other vertices in the graph. Example: Input: Source Vertex = 0 and below graph Having vertices like (u, v, w): edges: [[0,1,0], [0, 7, 1], [1,2,1], [1, 7, 1],…

Given a weighted Graph and a source vertex, the task is to find the shortest paths from the source node to all other vertices. Example: Input : n = 9, src = 0 Output : 0 4 12 19 21 11 9 8 14 We have learned about how to find the shortest path from a given source vertex to all other vertex using…

Given a directed and strongly connected graph with non-negative edge weights. We define the mean weight of a cycle as the summation of all the edge weights of the cycle divided by the no. of edges. Our task is to find the minimum mean weight among all the directed cycles of the graph. The input is…

Given an unweighted, undirected graph of V nodes and E edges, a source node S, and a destination node D, we need to find the shortest path from node S to node D in the graph. Input: V = 8, E = 10, S = 0, D = 7, edges[][] = {{0, 1}, {1, 2}, {0, 3}, {3, 4}, {4, 7}, {3, 7}, {6, 7}, {4, 5}, {4, 6}, {5,…

A Multistage graph is a directed, weighted graph in which the nodes can be divided into a set of stages such that all edges are from a stage to next stage only (In other words there is no edge between vertices of same stage and from a vertex of current stage to previous stage). The vertices of a…

Try it on GfG Practice Given a weighted undirected graph represented as an edge list and a source vertex src, find the shortest path distances from the source vertex to all other vertices in the graph. The graph contains V vertices, numbered from 0 to V - 1. Note: The given graph does not contain…

Try it on GfG Practice Given a matrix dist[][] of size n x n, where dist[i][j] represents the weight of the edge from node i to node j. If there is no direct edge, dist[i][j] is set to a large value (e.g., 10⁸) to represent infinity. The diagonal entries dist[i][i] are 0, since the distance from a…

Try it on GfG Practice Given a weighted graph with V vertices and E edges, along with a source vertex src, the task is to compute the shortest distances from the source to all other vertices. If a vertex is unreachable from the source, its distance should be marked as 108. In the presence of a…

Try it on GfG Practice Given a Weighted Directed Acyclic Graph and a source vertex in the graph, find the shortest paths from given source to all other vertices. Recommended Practice Shortest path from 1 to n Try It! For a general weighted graph, we can calculate single source shortest distances in…

The problem is to find the shortest paths between every pair of vertices in a given weighted directed Graph and weights may be negative. We have discussed Floyd Warshall Algorithm for this problem. The time complexity of the Floyd Warshall Algorithm is Θ(V3). Using Johnson's algorithm, we can find…

Try it on GfG Practice Two sets are called disjoint sets if they don't have any element in common. The disjoint set data structure is used to store such sets. It supports following operations: Merging two disjoint sets to a single set using Union operation.Finding representative of a disjoint set…

In the previous post, we introduced the Union-Find algorithm. We employed the union() and find() operations to manage subsets. Code Implementation: C++ // Find the representative (root) of the // set that includes element i int Find(int i) { // If i itself is root or representative if (parent[i] ==…

A directed acyclic graph (DAG) is a graph which doesn't contain a cycle and has directed edges. We are given a DAG, we need to clone it, i.e., create another graph that has copy of its vertices and edges connecting them. Examples: Input : 0 - - - > 1 - - - -> 4 | / \ ^ | / \ | | / \ | | / \ | | / \…

We are given a directed graph. We need compute whether the graph has negative cycle or not. A negative cycle is one in which the overall sum of the cycle comes negative. Negative weights are found in various applications of graphs. For example, instead of paying cost for a path, we may get some…

Given an undirected and connected graph and a number n, count the total number of simple cycles of length n in the graph. A simple cycle of length n is defined as a cycle that contains exactly n vertices and n edges. Note that for an undirected graph, each cycle should only be counted once,…

Try it on GfG Practice Given a undirected graph represented by an adjacency list adj, where each adj[i] represents the list of vertices connected to vertex i. Perform a Breadth First Search (BFS) traversal starting from vertex 0, visiting vertices from left to right according to the adjacency list,…

Try it on GfG Practice Given an undirected graph, the task is to check if there is a cycle in the given graph. Examples: Input: V = 4, edges[][]= [[0, 1], [0, 2], [1, 2], [2, 3]] Undirected Graph with 4 vertices and 4 edges Output: true Explanation: The diagram clearly shows a cycle 0 → 2 → 1 → 0…

Try it on GfG Practice Given a directed weighted graph, your task is to find whether the given graph contains any negative cycles that are reachable from the source vertex (e.g., node 0). Note: A negative-weight cycle is a cycle in a graph whose edges sum to a negative value. Example: Input: V = 4,…

Given a directed graph represented by the number of vertices V and a list of directed edges, determine whether the graph contains a cycle. Your task is to implement a function that accepts V (number of vertices) and edges (an array of directed edges where each edge is a pair [u, v]), and returns…

A Graph is a non-linear data structure consisting of vertices and edges. The vertices are sometimes also referred to as nodes and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph is composed of a set of vertices( V ) and a set of edges( E ). The graph is…

Try it on GfG Practice Given an n x m grid of 'W' (Water) and 'L' (Land), the task is to count the number of islands. An island is a group of adjacent 'L' cells connected horizontally, vertically, or diagonally, and it is surrounded by water or the grid boundary. The goal is to determine how many…

Try it on GfG Practice Given a matrix of dimension mxn, where each cell in the matrix can have values 0, 1 or 2 which has the following meaning: 0: Empty cell1: Cells have fresh oranges2: Cells have rotten oranges The task is to find the minimum time required so that all the oranges become rotten. A…

Breadth-First Search (BFS) and Depth-First Search (DFS) are two fundamental algorithms used for traversing or searching graphs and trees. This article covers the basic difference between Breadth-First Search and Depth-First Search. Difference between BFS and DFS ParametersBFSDFSStands forBFS stands…

Try it on GfG Practice In Depth First Search (or DFS) for a graph, we traverse all adjacent vertices one by one. When we traverse an adjacent vertex, we completely finish the traversal of all vertices reachable through that adjacent vertex. This is similar to a tree, where we first completely…

Try it on GfG Practice Given an M x N matrix where each element can either be 0 or 1. We need to find the shortest path between a given source cell to a destination cell. The path can only be created out of a cell if its value is 1. Note: You can move into an adjacent cell in one of the four…

Try it on GfG Practice You are given a 2D grid image[][], where each image[i][j] represents the color of a pixel in the image. Also provided is a coordinate(sr, sc) representing the starting pixel (row and column) and a new color value newColor. Your task is to perform a flood fill starting from the…

Try it on GfG Practice Given the number of vertices V and a list of directed edges, determine whether the graph contains a cycle or not. Examples: Input: V = 4, edges[][] = [[0, 1], [0, 2], [1, 2], [2, 0], [2, 3]] Cycle: 0 → 2 → 0 Output: true Explanation: The diagram clearly shows a cycle 0 → 2 → 0…

Given a directed graph, find out if a vertex v is reachable from another vertex u for all vertex pairs (u, v) in the given graph. Here reachable means that there is a path from vertex u to v. The reach-ability matrix is called transitive closure of a graph. For example, consider below graph: Graph…

Try it on GfG Practice There is an N x M rectangular island that borders both the Pacific Ocean and the Atlantic Ocean. The Pacific Ocean touches the island's left and top edges, and the Atlantic Ocean touches the island's right and bottom edges. The island is partitioned into a grid of square…

Try it on GfG Practice Given a graph with V vertices numbered from 0 to V-1 and a list of edges, determine whether the graph is bipartite or not. Note: A bipartite graph is a type of graph where the set of vertices can be divided into two disjoint sets, say U and V, such that every edge connects a…

Given two empty jugs of m and n litres respectively. The jugs don't have markings to allow measuring smaller quantities. You have to use the jugs to measure d litres of water. The task is to find the minimum number of operations to be performed to obtain d litres of water in one of the jugs. In case…

Try it on GfG Practice Given an undirected connected graph with v nodes, and e edges, with adjacency list adj. We need to write a function that returns 2 if the graph contains an eulerian circuit or cycle, else if the graph contains an eulerian path, returns 1, otherwise, returns 0. A graph is said…