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

insertVertex, insertEdge, removeEdge,
O(1)
removeVertex
O(deg(v))
In contrast to the edge-list way of doing things, the adjacency list structure
provides improved running times for the following methods:
• Method incidentEdges(v) takes time proportional to the number of
incident vertices of v, that is, O(deg(v)) time.
• Method areAdjacent(u,v) can be performed by inspecting either the
incidence collection of u or that of v. By choosing the smaller of the two, we get
O(min(deg(u),deg(v))) running time.
• Method removeVertex(v) takes O(deg(v)) time.
13.2.3 The Adjacency Matrix Structure
Like the adjacency list structure, the adjacency matrix structure of a graph also
extends the edge list structure with an additional component. In this case, we
augment the edge list with a matrix (a two-dimensional array) A that allows us to
determine adjacencies between pairs of vertices in constant time. In the adjacency
matrix representation, we think of the vertices as being the integers in the set
{0,1,..., n − 1} and the edges as being pairs of such integers. This allows us to store
references to edges in the cells of a two-dimensional n × n array A. Specifically, the
adjacency matrix representation extends the edge list structure as follows (see
Figure 13.5):
• A vertex object v stores a distinct integer i in the range 0,1,..., n − 1, called
the index of v.
• We keep a two-dimensional n × n array A such that the cell A[i,j] holds a
reference to the edge (v, w), if it exists, where v is the vertex with index i and w is
the vertex with index j. If there is no such edge, then A[i,j] = null.
Figure 13.5: (a) A graph G without parallel edges; (b)
schematic representation of the simplified adjacency
matrix structure for G.
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