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Understanding Graph Representation: Adjacency Matrix and List

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This lecture explores how to represent graphs internally, focusing on two primary methods: the adjacency matrix and the adjacency list. The adjacency matrix sets edge costs while using O(V²) space, making it suitable for dense graphs. In contrast, the adjacency list is the optimal choice for sparse graphs, maintaining a list of adjacent vertices. Both methods have their advantages and complexities, and understanding their implementation is crucial for effective graph algorithms. We will also discuss the implications of graph density on algorithm performance.

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Understanding Graph Representation: Adjacency Matrix and List

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  1. Lecture 14 Graph Representations

  2. 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

  3. 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; };

  4. 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

  5. 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((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.

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