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

cover a big triangle of area n 2 /2 (base n and height n)
plus n small triangles of area 1/2 each (base 1 and
height 1). In (b), which applies only when n is even, the
rectangles are shown to cover a big rectangle of base
n/2 and height n+ 1.
The lesson to be learned from Proposition 4.3 is that if we perform an algorithm
with nested loops such that the operations in the inner loop increase by one each
time, then the total number of operations is quadratic in the number of times, n,
we perform the outer loop. In particular, the number of operations is n 2 /2 + n/2, in
this case, which is a little more than a constant factor (1/2) times the quadratic
function n 2 . In other words, such an algorithm is only slightly better than an
algorithm that uses n operations each time the inner loop is performed. This
observation might at first seem nonintuitive, but it is nevertheless true, as shown
in Figure 4.1.
4.1.6 The Cubic Function and Other
Polynomials
Continuing our discussion of functions that are powers of the input, we consider the
cubic function,
f(n) = n 3 ,
218