Graph Theory
- 1. Introduction Basic Graph Theory Saksham Agrawal August 30, 2015 Saksham Agrawal Basic Graph Theory
- 2. Introduction List Of Contents 1- Introduction To Graph 2- Definitions And Terminology 3- Graph Representation 4- Breadth First Search 5- Depth first search Saksham Agrawal Basic Graph Theory
- 3. Introduction Introduction To Graphs In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A ”graph” in this context is made up of ”vertices” or ”nodes” and lines called edges that connect them. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another. Graphs are one of the prime objects of study in discrete mathematics. Saksham Agrawal Basic Graph Theory
- 4. Introduction Figure: A Typical Graph Saksham Agrawal Basic Graph Theory
- 5. Introduction Definitions And Terminology Graphs are much clear when defined in mathematical terms. A Graph G (V, E), consists of two sets, Set of Vertices, V Set of Edges, E As the name suggest, V, is the set of vertices or, the set of all nodes in a given graph and E, is the set of all the edges between these nodes, or the associations between them. So, V denotes the number of nodes in a graph and E denotes the number of edges Saksham Agrawal Basic Graph Theory
- 6. Introduction Figure: Types Of Graph Saksham Agrawal Basic Graph Theory
- 7. Introduction Graph Representations ⇒ Adjacency Matrix Adjacency List Saksham Agrawal Basic Graph Theory
- 8. Introduction Adjacency Matrix Figure: Adjacency Matrix Of Shown Graph Saksham Agrawal Basic Graph Theory
- 9. Introduction Adjacency List Figure: Adjacency List Of Shown Graph Saksham Agrawal Basic Graph Theory
- 10. Introduction Breadth first Search In BFS,we use a queue and we push the current vertex in the queue and the vertices which are adjacent to the current vertex using the adjacency list.We also maintain a visited array in which we record the visited vertices so as to prevent revisiting the same vertex.After we have pushed the adjacent vertices of a vertex we pop it out of the queue. Saksham Agrawal Basic Graph Theory
- 11. Introduction BFS algorithm Saksham Agrawal Basic Graph Theory
- 12. Introduction Application of BFS BFS is mainly used in a connected graph in which we want to find the shortest path between 2 vertices.In this we also maintain a level array.We start from one of the 2 vertices and set its level as 0 and the set level of the adjacent vertices as level of parent+1 and traverse the graph in this until we reach the other vertex.The level of the vertex represents the shortest path between the two vertices. Saksham Agrawal Basic Graph Theory
- 13. Introduction Depth First Search Algorithm:- When we see a maze in a newspaper or a magazine or anywhere else, the way we solve it is we take a path and go through it. If we find any junction or a crossroad, where we have a choice of paths to choose, we mark that junction, take up a path and traverse the whole route in your brain. If we see a dead end, we realize that this leads we now where so we come back to the junction we marked before and take up another path of the junction. See if that is also a dead end, else we continue to explore the, puzzle as this route contributes in reaching our destination.In code, we do this by recursion. Saksham Agrawal Basic Graph Theory
- 14. Introduction DFS step-by-step process For a given set of vertices V = V1, V2,V3. . . , start with V1, and explore all the vertices that can possibly be reached from V1. Then go to the next vertex V2, if it hasn’t been visited, explore all the nodes reachable from V2, if not, go for V3. Similarly, go on picking up all the vertices one-by-one and explore as much as possible if it wasn’t visited at all. Saksham Agrawal Basic Graph Theory
- 15. Introduction Saksham Agrawal Basic Graph Theory
- 16. Introduction Application of DFS DFS is also used for finding the shortest path between two vertices.It is also used to do topological sorting. Topological Sorting:- It represents the logical order of doing something. For instance,we wear socks before shoes and after we tie laces.In the similar way in topological sorting a vertex appears only when all its adjacent vertices have already appeared.It is logical that topological sorting is done in only a directed graph.Obviously, the vertex with no child is one which appears in front.This is done using a stack and DFS on a graph. Saksham Agrawal Basic Graph Theory