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

not. The optimal performance of method areAdjacent is counteracted by an
increase in space usage, however, which is now O(n 2 ), and in the running time of
other methods. For example, method incidentEdges(v) now requires that we
examine an entire row or column of array A and thus runs in O(n) time.
Moreover, any vertex insertions or deletions now require creating a whole new
array A, of larger or smaller size, respectively, which takes O(n 2 ) time.
Table 13.3 summarizes the performance of the adjacency matrix structure
implementation of a graph. From this table, we observe that the adjacency list
structure is superior to the adjacency matrix in space, and is superior in time for
all methods except for the areAdjacent method.
Table 13.3: Running times for a graph implemented
with an adjacency matrix.
Operation
Time
vertices
O(n)
edges
O(m)
endVertices, opposite, areAdjacent
O(1)
incidentEdges(v)
O(n + deg(v))
replace, insertEdge, removeEdge,
O(1)
insert Vertex, remove Vertex
O(n 2 )
Historically, Boolean adjacency matrices were the first representations used for
graphs (so that A[i, j] = true if and only if (i, j) is an edge). We should not find
this fact surprising, however, for the adjacency matrix has a natural appeal as a
mathematical structure (for example, an undirected graph has a symmetric
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