Lecture 14 Graph Representations

Graph Representation – How do we represent a graph internally? – Two ways adjacency matrix list – Adjacency Matrix For each edge (v,w) in E, set A[v][w] = edge_cost Non existent edges with logical infinity – Cost of implementation O(|V| 2 ) time for initialization O(|V| 2 ) space – ok for dense graphs – unacceptable for sparse graphs

Graph Class (Matrix representation) Public class Graph public static final double NO_EDGE; Graph() {} Graph(int n) {} public void SetSize(int n); public void AddEdge(int origin, int destination, double value); public void RemoveEdge(int origin, int destination); public boolean HasEdge(int origin, int destination) ; public double EdgeValue(int origin, int destination) ; int NumNodes() {} int NumEdges() {} private int myNumEdges; private int [][] M; };

Graph Representation Adjacency List – Ideal solution for sparse graphs – For each vertex keep a list of all adjacent vertices – Adjacent vertices are the vertices that are connected to the vertex directly by an edge. – Example List 0 List 1 List 2 Draw the graph ??

Graph Representation ctd.. The number of list nodes equals to number of edges – O(|E|) space Space is also required to store the lists – O(|V|) for |V| lists Note that the number of edges is at least round(|V|/2) – assuming each vertex is in some edge – Therefore disregard any O(|V|) term when O(|E|) is present Adjacency list can be constructed in linear time (wrt to edges) More on this when we learn pointers

Graph Algorithms Graphs depend on two parameters – edges (E) – Vertices (V) Graph algorithms can be complicated to analyze – One algorithm might be order (V 2 ) for dense graphs – Another might be order((E+V)log E) for sparse graphs Depth First Search – Is the graph connected? If not what are their connected components? – Does the graph have a cycle? – How do we examine (visit) every node and every edge systematically? – First select a vertex, set all vertices connected to that vertex to non- zero – Find a vertex that has not been visited and repeat the step above – Repeat above until all zero vertices are examined.