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 ??

1 2

2 0 1

1

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 (V2)

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.