Unit_V__Graphs.pptx - Unit V Graphs Application/System/Case Study Communication networking Road maps(Game path finding system Web graph system Contents

# Unit_V__Graphs.pptx - Unit V Graphs Application/System/Case...

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Unit V : Graphs Application/System/Case Study: Communication networking, Road maps (Game path finding system, Web graph system) Contents: Graph: Introduction, types of graph, representation of graphs: adjacency matrix, adjacency list, BFS, DFS & traversal, spanning trees, topological sorting Self study: Warshall’s algorithm Further reading: Page ranking Subscribe to view the full document.

Graphs Graph G = ( V , E ) V = set of vertices E = set of edges ( V V ) Types of graphs Undirected: edge ( u , v ) = ( v , u ) ; for all v , ( v , v ) E ( No self loops. ) Directed: ( u , v ) is edge from u to v , denoted as u v . Self loops are allowed. Weighted : each edge has an associated weight , given by a weight function w : E R . 1 2 3 4 3 Applications Applications that involve not only a set of items, but also the connections between them Computer networks Circuits Schedules Hypertext Maps Subscribe to view the full document.

4 Terminology Directed vs Undirected graphs 5 Terminology (cont’d) Complete graph A graph with an edge between each pair of vertices Subgraph A graph (V , E ) such that V V and E E Path from v to w A sequence of vertices <v 0 , v 1 , , v k > such that v 0 =v and v k =w Length of a path Number of edges in the path 1 2 3 4 path from v 1 to v 4 <v 1 , v 2 , v 4 > Subscribe to view the full document.

6 Terminology (cont’d) w is reachable from v If there is a path from v to w Simple path All the vertices in the path are distinct Cycles A path <v 0 , v 1 , , v k > forms a cycle if v 0 =v k and k≥2 Acyclic graph A graph without any cycles 1 2 3 4 cycle from v 1 to v 1 <v 1 , v 2 , v 3 ,v 1 > 7 Terminology (cont’d) Connected and Strongly Connected Subscribe to view the full document.

8 Terminology (cont’d) A tree is a connected, acyclic undirected graph Graphs If ( u , v ) E , then vertex v is adjacent to vertex u . Adjacency relationship is : Symmetric if G is undirected. Not necessarily so if G is directed. If G is connected : There is a path between every pair of vertices . | E | | V | – 1. Furthermore, if | E | = | V | – 1, then G is a tree. Subscribe to view the full document.

Representation of Graphs Two standard ways . Adjacency Lists. Adjacency Matrix. a d c b a b c d b a d d c c a b a c a d c b 1 2 3 4 1 2 3 4 1 0 1 1 1 2 1 0 1 0 3 1 1 0 1 4 1 0 1 0 Adjacency matrix (Sequential representation) Adjacency list ( linked representation) Adjacency Lists Consists of an array Adj of | V | lists. One list per vertex. For u V , Adj [ u ] consists of all vertices adjacent to u . a d c b a b c d b c d d c a d c b a b c d b a d d c c a b a c If weighted, store weights also in adjacency lists. Subscribe to view the full document.

Storage Requirement For directed graphs: Sum of lengths of all adj. lists is out-degree( v ) = | E | v V Total storage: ( V + E ) For undirected graphs: Sum of lengths of all adj. lists is degree( v ) = 2| E | v V Total storage: ( V + E ) No. of edges leaving v No. of edges incident on v. Edge ( u , v ) is incident on vertices u and v . Operations performed on graph Insertion of vertex Deletion of vertex Insertion of edge Deletion of edge To find particular vertex Traversing the graph Subscribe to view the full document.

Graph traversal Depth first search(DFS) Breadth first search(BFS)  • Winter '18
• Mr M V Joshi

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