1
CMPS 201
Algorithms and Abstract Data Types
Some Common Functions
We present several common functions and estimates which occur frequently in the analysis of
algorithms.
Floors and Ceilings
Given
R
∈
x
, we denote by
x
and
x
the
floor of x
and the
ceiling of x
, respectively.
These are
defined to be the unique integers satisfying
1
1
+
<
≤
≤
<

x
x
x
x
x
Equivalently, if
R
∈
x
and
Z
∈
N
then
(1)
x
N
=
if and only if
1
+
<
≤
N
x
N
, and
(2)
x
N
=
if and only if
N
x
N
≤
<

1
.
In other words:
(1)
x
is the
greatest integer less than or equal to x
, and
(2)
x
is the
least integer greater than or equal to x
.
Lemma 1:
Let
R
∈
x
and
Z
∈
b
a
,
.
Then
(1)
b
x
a
<
≤
if and only if
b
x
a
<
≤
, and
(2)
b
x
a
≤
<
if and only if
b
x
a
≤
<
.
Proof of (1):
(i)
x
a
≤
implies
x
a
≤
, since among all integers that are less than or equal to
x
,
x
is the greatest.
(ii)
b
x
<
implies
b
x
<
, since
x
x
≤
.
(iii)
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 Spring '09
 AgoreBack
 Algorithms, Derivative, Exponentiation, Inverse function, Logarithm

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