Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

The main feature of the edge list structure is that it provides direct access from
edges to the vertices they are incident upon. This allows us to define simple
algorithms for methods endVertices(e) and opposite(v, e).
Performance of the Edge List Structure
One method that is inefficient for the edge list structure, however, is that of
accessing the edges that are incident upon a vertex. Determining this set of
vertices requires an exhaustive inspection of all the edge objects in the collection
E. That is, in order to determine which edges are incident to a vertex v, we must
examine all the edges in the edge list and check, for each one, if it happens to be
incident to v. Thus, method incidentEdges(v) runs in time proportional to the
number of edges in the graph, not in time proportional to the degree of vertex v.
In fact, even to check if two vertices v and w are adjacent by the
areAdjacent(v,w) method, requires that we search the entire edge collection
looking for an edge with end vertices v and w. Moreover, since removing a vertex
involves removing all of its incident edges, method removeVertex also
requires a complete search of the edge collection E.
Table 13.1 summarizes the performance of the edge list structure implementation
of a graph under the assumption that collections V and E are realized with doubly
linked lists (Section 6.4.2).
Table 13.1: Running times of the methods of a
graph implemented with the edge list structure. The
space used is O(n + m), where n is the number of
vertices and m is the number of edges.
Operation
Time
vertices
O(n)
edges
O(m)
endVertices, opposite
O(1)
incidentEdges, areAdjacent
801