Recitation 9 Graph Algorithms Solution

Objectives




What is a Graph?



Depth First Search



Breadth First Search



Exercise



Challenge 1 (Silver Badge)



Challenge 2 (Gold Badge)















In real world, many problems are represented in terms of objects and connections between them. For example, in an airline route map, we might be interested in questions like: “What’s the fastest way to go from Denver to San Francisco” or “What is the cheapest way to go from Denver to San Francisco” To answer these questions we need information about connections between objects. Graphs are data structures used for solving these kinds of problems.







Graph: A graph is a pair (V, E), where V is a set of nodes, called vertices and E is a collection of pairs of vertices, called edges.




Directed Edge




Ordered pair of vertices (u, v)



First vertex u is the origin



Second vertex v is the destination



Example: one way road traffic

























































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Undirected Edge:




Unordered pair of vertices (u, v)



Example: Railway lines






























Applications of Graphs




Representing relationships between components in electronic circuits.



Transportation networks: Highway network, Flight network



Computer networks: Local area network, Internet web









Graph Representation




As in other Abstract Data types, to manipulate graphs we need to represent them in some useful form. Basically, there are two ways of doing this:




Adjacency Matrix



Adjacency Lists



Graph Traversals




To solve problems on graphs, we need a mechanism for traversing the graphs. Graph traversal algorithms are also called a graph search algorithms. Like tree traversal algorithms (Inorder, Preorder, Postorder traversals), graph search algorithms can be thought of as starting at some source vertex in a graph, and ‘search’ the graph by going through the edges and making the vertices. Now, we will discuss two such algorithms for traversing the graphs.




Depth First Search (DFS)



Breadth First Search (BFS)









Depth First Search (DFS)




The strategy followed by depth-first search is, as its name implies, to search “deeper” in the graph whenever possible.










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The DFS algorithm is a recursive algorithm that uses the idea of backtracking. The idea is to travel as deep as possible from neighbour to neighbour before backtracking. What determines how deep is possible is that you must follow edges, and you don't visit any vertex twice. To do this properly we need to keep track of which vertices have already been visited, plus how we got to where we currently are, so that we can backtrack.




Here, the word backtrack means that when you are moving forward and there are no more nodes along the current path, you move backwards on the same path to find nodes to traverse.




All the nodes on the current path are visited, after which the next path will be selected. We would be using the Adjacency lists to get the neighbours of any vertex.




Here’s a pseudocode of this recursive approach:







DFS(G, u)

u.visited = true​

​for​ each v ∈ G.Adj[u]

if​​ v.visited == false​

DFS(G,v)




init() {

​for​ each u ∈ G

u.visited = false​

​for​ each u ∈ G

If u.visited==​false

DFS(G, u)




}










Make sure that the nodes that are visited are marked. This will prevent you from visiting the same node more than once. ​If you do not mark the nodes that are visited and you visit the same node more than once, you may end up in an infinite loop.







This recursive behaviour (which uses a system stack) can be simulated by an iterative algorithm explicitly using a stack. So, DFS uses a stack to push nodes we mean to visit onto that.




We gave a pseudocode for DFS traversal using recursion and we will be giving a visualization of DFS traversal using stacks to help you understand better. The following will give you the basic idea of Stack implementation.



















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As C does not have any unvisited adjacent node so we keep popping the stack until we find a node that has an unvisited adjacent node. In this case, there's none and we keep popping until the stack is empty.




Applications of DFS




Finding connected components in an undirected graph



(a variant of DFS is used for doing the same in directed graphs - Kosaraju algorithm)




Solving puzzles, such as mazes (DFS helps to reach the goal faster)



Breadth First Search (BFS)




Breadth-first search is one of the simplest algorithms for searching a graph and the archetype for many important graph algorithms.




BFS works level by level. Initially, BFS starts at a given vertex, which is at level 0. In the first stage it visits all vertices at level 1. In the second stage, it visits all vertices at second level. These new vertices are the one which are adjacent to level 1 vertices.




BFS continues this process until all the levels of the graph are completed. Generally queue data structure is used for storing the vertices of a level. As similar to DFS, assume that initially all vertices are marked as unvisited. Vertices that have been processed and removed from the queue are marked visited. We use a queue to represent the visited set as it will keep the vertices in order of when they were first visited.




Breadth-first search is so named because it expands the frontier between discovered and undiscovered vertices uniformly across the breadth of the frontier. That is, the algorithm discovers all vertices at distance k from s before discovering any vertices at distance k + 1.




Similarly we have an example for BFS as well using queues.










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Applications of BFS




Finding all connected components in an undirected graph.



Finding the shortest path between two nodes.




































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Comparison of BFS vs DFS on a simple graph:









































































Exercise







Silver Badge Problem (Mandatory)



Download the Lab9​zipped ​file from Moodle.
Follow the TODOs and in Graph.cpp, complete the ​findShortestPath()​function.



This function finds the length of the shortest path between a source and destination vertex in an undirected and unweighted graph.









Gold Badge Problem






In Graph.cpp, complete the​printPath()​function after modifying the findShortestPath() ​function so as to print the shortest path after finding it’s length.










































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