CSE 246 Data Structures and Algorithms Graphs Graph data structure  Graph is an ADT.  A graph is a set of vertices and edges 𝐺 = {𝑉, 𝐸} 𝑉={v1,v2,v3,…,vn}, 𝐸={e1,e2,e3,…,en}  An Edge connects exactly two vertices.  A vertex can have any number of edges connected to it. 2 Quratulain Graphs  Edge might be directed or undirected.  Weighted graphs. Additional information on edges. 3 Quratulain Terms to remember  graph  path  vertex (pl vertices)‫‏‬  cycle  edge  Hamiltonian  edge cost  Eulerian  undirected graph  planar graph  directed graph  acyclic directed graph  simple graph  forest  loop  tree Graph Applications  Modelling a road network with vertexes as towns and edge costs as     distances. Modelling a water supply network. A cost might relate to current or a function of capacity and length. As water flows in only 1 direction, from higher to lower pressure connections or downhill, such a network is inherently an acyclic directed graph. Internet traffic modelling. Dynamically modelling the status of a set of routes by which traffic might be directed over the Internet. Flight scheduling. Minimising the cost and time taken for air travel when direct flights don't exist between starting and ending airports. Using a directed graph to map the links between pages within a website and to analyse ease of navigation between different parts of the site. Graph Applications  The Seven Bridges of Königsberg is a problem inspired by an actual place and situation.  The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point. In 1736, Leonhard Euler proved that it was not possible." Source: http://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg 6 Quratulain Three cottage problem Source: http://en.wikipedia.org/wiki/Three_cottage_problem "The three cottage problem is a well-known mathematical puzzle. It can be stated like this: Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. Is there a way to do so without any of the lines crossing each other?" Three cottage problem "The three cottage problem is a well-known mathematical puzzle. It can be stated like this: Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the gas, water, and electric companies. Is there a way to do so without any of the lines crossing each other?"  Source: http://en.wikipedia.org/wiki/Three_cottage_problem 8 Quratulain Data structure to store Graphs  To select data structure for graph:  First programmer need to understand the application data.  Then construct a graph.  The choice and design of data structure(s) will influence and be influenced by program performance and memory usage requirements. 9 Quratulain Data structures to store graph  Adjacency list  Adjacency matrix.  Incident list Source: http://interactivepython.org/courselib/static/pythonds/Graphs/graphintro.html 10 Quratulain Adjacency matrix  implement a graph is to use a two-dimensional matrix. 11 Quratulain Adjacency list  A more space-efficient way to implement a sparsely connected graph is to use an adjacency list 12 Quratulain Graph search algorithms  Breadth first search  Depth first search 13 Quratulain Depth first search  The depth-first search uses a stack to remember where it should go when it reaches a dead end. 14 Quratulain Depth first search Example  Knight’s tour graph 15 Quratulain Breadth first search  the algorithm likes to stay as close as possible to the starting point. It visits all the vertices adjacent to the starting vertex, and only then goes further. This kind of search is implemented using a queue 16 Quratulain Breadth first search Example  Word ladder graph 17 Quratulain Graph Analysis 18 Quratulain