Algorithms - GraphWhat is Graph- A tree which at least one cycleV(G) = {1,2,3,4} E(G) = {(1,2),(1,3),(3,3,),(3,4),(4,1)} Graph Representation- Adjacency Matrix representation- Adjacency List- Adjacency Multilist representationGraph RepresentationGraph Representation- Adjacency Matrix representationGraph Representation- Adjacency ListQuestion ?What graph topics should I study in order to be adequately prepared for a Google Software Engineer interview?Depth-first Search, Breadth-first Search, and Dijkstra's algorithm are obviously required.Would it be worthwhile to also study algorithms for minimum spanning trees, maximum network flows, bipartite matching, coloring algorithms, topological sorting, etc...? How good should I be at implementing these algorithms from scratch?Answer by Gayle Laakmann McDowell, Ex-Googler, Author of Cracking the Coding InterviewI have never seen anything about Dijkstra's algorithm asked. It'd be a pretty stupid question.First, very few interviewers would know it. It's not like they go home every night and re-read CLRS or whatever their algorithm book is.Second, it'd be way too hard. For candidates who were interviewing with me and finished a difficult question with some time to spare (that is, doing very well), I would ask them how to delete a node from a binary search tree. Only about 20% of candidates remembered this -- and this is of the really good ones. You think they know Dijkstra's algorithm?Third, it wouldn't show anything. Knowing an algorithm says that you have a good memory or that you studied some stuff. We don't care about that stuff. We care about your ability to solve *new* problems.The same goes for all these other fancy algorithm problems. Seriously, this is not what tech interviews are all about. The actual knowledge required is really just basic CS knowledge.But here's what you should do: ignore my advice. Go look at actual googleinterviews on CareerCup and Glassdoor. Go make your own decision about how important it is to know fancy algorithms. (Do not look at blogs on interview questions. Best case, they have a huge selection bias and are intentionally picking the ridiculousquestions. Worst case, and more likely, they're lying or pulling their info from someone who lied.)There's a reason I don't cover advanced graph algorithms in Cracking the Coding Interview: this stuff doesn't make it into interviews.The idea that Google asks such things largely comes from an incorrect assumption that a highly selective company must requireadvanced knowledge. No, not exactly. They just set higher standards about your ability to solve new problems with basic CS knowledge.Disclaimer The previous suggestion is the view of the specific person. Use your own judgement on what to study. You are responsible for your career. If extra knowledge does not confuse you, it does not harm.Depth First Searchvoid search(Node root) {if (root == null) return;visit(root);root.visited = true;foreach (Node n in root.adjacent) {if (n.visited == false) {search(n);}}}Please copy on your notebookExplanation visit(root); Do processing needed root.visited = true; Processing is done if (n.visited == false) { Lets not process the same again Not useful in tree as such but in a graphWhat DS is used in DFS Implicitly a stack , because we are using recursion.Breadth First Search1: void search(Node root) {2: Queue queue = new Queue();5: queue.enqueue(root); // Add to end of queue6:7: while (!queue.isEmpty()) {8: Node r = queue.dequeue(); // Remove from front of queue9:visit(r);10:r.visited = true;11: foreach (Node n in r.adjacent) {12:if (n.visited == false) {13:queue.enqueue(n);14:}15:}16:}17:}Please note in your copy.ExampleBFS Explanation When we pick a node we insert all the children at the rear end of the queue. Lets run the BFS on previous tree and see how the queue gets changed.ActivityWhat will be the DFS order ?What will be the BFS order ?Three jug problemI have three jug whose capacity are 8 lt.,5 lt. and 3 lt. and no one jug is calibrated then how can I divide 8 lt. water in two equal parts. Can we write a program to solve this ? Which algorithm will you choose If there is no solution, can our program tell this.Lets give it a manual tryTime : 3 minutesData Structures for Jug Problem How can we store the state ? We need a place to store whether this state is already visited ?Data Structures for Jug Problem How can we store the state ? Can we use a structure with three integers ? We need a place to store whether this state is already visited ? Can we use a three dimensional array for this ?OperationsWhat are the possible operations we can do on the states of the water jug ?OperationsWhat are the possible operations we can do on the states of the water jug ? When we pour from one jug to another Either we fill the other jug to its capacity Or we empty to first jug of whatever it hadApplication of BFS and DFS BFS : Whenever you need to search the minimal number of steps. DFS : When it is necessary to go till the leaf node, or when we cannot maintain an additional data structure such as queue.Problem 11. Implement a function to check if a binary tree is balanced. For the purposes of this question, a balanced tree is defined to be a tree such that the heights of the two subtrees of any node never differ by more than one.Hint to Problem 1During DFS on each node get the height and also check if each sub left tree and right tree is balanced. If any sub tree is not return -1 to indicate that a subtree is not balanced.Problem 2On a chessboard, a knight is placed at a random position and a rook is placed at another random position. We need to determine whether the knight can move over rook and how many minimum steps will it take to reach there.Hint to Problem 2 Use BFS Use int[8][8] to mark the fact that the position is visitedProblem 33. Design an algorithm and write code to find the first common ancestor of two nodes in a binary tree. Avoid storing additional nodes in a data structure. Hint to Problem 3A node is the LCA if its left child reports it as /ancestor of one node its right child reports it as /ancestor of other node Or If it itself is one node the left or right child contains other nodeDo a post order DFS and let every node determine whether it is a LCA node or not. Pass to the parent as /ancestor information for both first node and second node.Problem 44. You are given a binary tree in which each node contains a value. Design an algorithm to print all paths which sum to a given value. The path does not need to start or end at the root or a leaf.Hint to Problem 4Do DFS and we pass the function the full path from root to node & ask, "Does this node complete a path with the sum?Problem 55. Given a binary tree, design an algorithm which creates a linked list of all the nodes at each depth (e.g., if you have a tree with depth D, you'll have D linked lists).Hints to Problem 5 Use a linked list of linked lists to store the result You can actually use either DFS or BFS for this purposeSpanning Tree (Dijkstras method)Spanning Tree (Dijkstras method)Spanning Tree (Dijkstras method)Spanning Tree (Dijkstras method)Spanning Tree (Dijkstras method)Spanning Tree (Dijkstras method)tree=null; edges=an unsorted sequence of all edges of graph; for j=1 to IEI add ej to tree; If there is cycle in tree Remove an edge with maximum weight from this only cycle.