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8
Chapter 1
The Role of Algorithms in Computing
orders is exponential in
n
, and so trying all possible orders may take a very
long time. We shall see in Chapter 15 how to use a general technique known as
dynamic programming to solve this problem much more efﬁciently.
•
We are given an equation
ax
≡
b
(
mod
n
)
,whe
re
a
,
b
,and
n
are integers,
and we wish to ﬁnd all the integers
x
, modulo
n
, that satisfy the equation.
There may be zero, one, or more than one such solution. We can simply try
x
=
0
,
1
,...,
n
−
1 in order, but Chapter 31 shows a more efﬁcient method.
•
We are given
n
points in the plane, and we wish to ﬁnd the convex hull of
these points. The convex hull is the smallest convex polygon containing the
points. Intuitively, we can think of each point as being represented by a nail
sticking out from a board. The convex hull would be represented by a tight
rubber band that surrounds all the nails. Each nail around which the rubber

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