Kruskal's algorithm is a greedy algorithm that works as follows −1. It Creates a set of all edges in the graph.2. While the above set is not empty and not all vertices are covered, It removes the minimum weight edge from this setIt checks if this edge is forming a cycle or just connecting 2 trees. If it forms a cycle, we discard this edge, else we add it to our tree.3. When the above processing is complete, we have a minimum spanning tree.In order to implement this algorithm, we need 2 more data structures.First, we need a priority queue ... Read More

Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. It finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex.How does Prim's algorithm work?Let us look at an illustration of how Prim's algorithm works −1. Choose any arbitrary node as the root node: In this ... Read More

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted (un)directed graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible.

Djikstra's algorithm is used to find distance/path of the shortest path from one node to all other nodes. There are cases where we need to find shortest paths from all nodes to all other nodes. This is where the All pairs shortest path algorithms come in handy. The most used all pairs shortest path algorithm is Floyd Warshall algorithm.Floyd Warshall algorithm works as follows −We initialize an N x N matrix of distances to be Infinity.Then for each edge u, v, we update this matrix to be showing the weight of this edge and for edges v, v we update ... Read More

Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a weighted graph. We'll use the new addEdge and addDirectedEdge methods to add weights to the edges when creating a graph. Let us look at how this algorithm works −Create a distance collection and set all vertices distances as infinity except the source node.Enqueue the source node in a min-priority queue with priority 0 as the distance is 0.Start a loop till the priority queue is empty and dequeue the node with the minimum distance from it.Update the distances of the connected nodes to the popped node ... Read More

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. Here we need to modify our add edge and add directed methods to allow adding weights to the edges as well. Let us look at how we can add this −Example/**    * Adds 2 edges with the same weight in either direction    *    *             weight    * node1 node2    *           ... Read More

A topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge UV from vertex u to vertex v, u comes before v in the ordering. This only makes sense in directed graphs.There are many places where topological sort makes a lot of sense. For example, let's say you're following a recipe, in this, there are some steps that are must for going to the next steps. But some of these can be performed in parallel. In a similar fashion, during college when selecting courses, there are some prerequisites ... Read More

DFS visits the child vertices before visiting the sibling vertices; that is, it traverses the depth of any particular path before exploring its breadth. A stack (often the program's call stack via recursion) is generally used when implementing the algorithm.Following is how a DFS works −Visit the adjacent unvisited vertex. Mark it as visited. Display it. Push it in a stack.If no adjacent vertex is found, pop up a vertex from the stack. (It will pop up all the vertices from the stack, which do not have adjacent vertices.)Repeat Rule 1 and Rule 2 until the stack is empty.Let us ... Read More

BFS visits the neighbor vertices before visiting the child vertices, and a queue is used in the search process. Following is how a BFS works −Visit the adjacent unvisited vertex. Mark it as visited. Display it. Insert it in a queue.If no adjacent vertex is found, remove the first vertex from the queue.Repeat Rule 1 and Rule 2 until the queue is empty.Let us look at an illustration of how BFS Traversal works:StepTraversalDescription1Initialize the queue.2We start by visiting S (starting node) and mark it as visited.3We then see an unvisited adjacent node from S. In this example, we have three ... Read More

Graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals are classified by the order in which the vertices are visited.