Graph Representations

1
Lecture 14

• Graph Representations

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Graph Representation

• How do we represent a graph internally?
• Two ways
• adjacency matrix
• list
• Adjacency Matrix
• For each edge (v,w) in E, set Avw edge_cost
• Non existent edges with logical infinity
• Cost of implementation
• O(V2) time for initialization
• O(V2) space
• ok for dense graphs
• unacceptable for sparse graphs

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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

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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 ??

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2
2
0
1
1
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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

6
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((EV)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.