Floor and Ceiling Functions

1
Floor and Ceiling Functions

• Floorx is the largest integer less than or
  equal to x.
• Ceilingx is the smallest integer greater than
  or equal to x.

2
Logarithms

• For bgt1 and xgt0, logbx (read log to the base b
  of x) that is real number L such that bL x
• logbx is the power to which b must be raised to
  get x.

3
Logarithm properties

• def lg x log2 x ln x loge x
• Let x and y be arbitrary positive real numbers,
  and let bgt1 and cgt1 be real numbers.
• logb is a strictly increasing function, if x gt y
  then logb x gt logb y
• logb is a one-to-one function, if logb x logb
  y then x y
• logb 1 0 logb b 1 logb xa a logb x
• logb(xy) logb x logb y
• xlog y ylog x
• change base logcx (logb x)/(logb c)

4
Series

• A series is the sum of a sequence.
• Arithmetic series
• The sum of consecutive integers
• Polynomial Series
• The sum of squares
• The general case is sum of
• numbers raised to
• the power of k

5
Series

• Powers of 2
• Arithmetic Geometric Series

6
Summations Using Integration

• If f(x) is nondecreasing then
• If f(x) is nonincreasing then

7
Monotonic Functions

• A function f(x) is said to be monotonic, or
  nondecreasing, or increasing in wider sense if x
  ? y always implies that f(x) ? f(y).
• A function f(x) is antimonotonic, or
  nonincreasing, if f(x) is monotonic.

8
Asymptotic Growth Rates

• asymptotic growth rate, asymptotic order, or
  order of functions
• Comparing and classifying functions that ignores
  constant factors and small inputs.
• The Sets big oh O(g), big theta ?(g), big omega
  ?(g)

9
Classifying functions by theirAsymptotic Growth
Rates

?(g) functions that grow at least as fast as g
?(g) functions that grow at the same rate as g
O(g) functions that grow no faster as g
10
The Sets O(g), ?(g), ?(g)

• Let g and f be functions from the nonnegative
  integers into the positive real numbers
• For some real constant c gt 0 and some
  nonnegative integer constant n0
• O(g) is the set of functions f, such
  that f(n) ? c g(n) for all n ? n0
• O(g) f ? ?c ?n0 f(n) ? c g(n) for all n ?
  n0

11
?(g), ?(g)

• ?(g) is the set of functions f, such that
• f(n) ? c g(n) for all n ? n0
• ?(g) O(g) ? ?(g)
• asymptotic order of g
• f ??(g) read as f is asymptotic order g or f
  is order g

12
Comparing asymptotic growth rates

• Comparing f(n) and g(n) as n approaches infinity,
  

13
Limit

• lt ?, including the case in which the limit is 0
  then f ?O(g)
• gt 0, including the case in which the limit is ?
  then f ? ?(g)
• c and 0 lt c lt ? then f ? ?(g)
• 0 then f ? o(g) //read as little oh of g
• ? then f ? ?(g) //read as little omega of g

14
Properties of O(g), ?(g), ?(g)

• Transitive If f ?O(g) and g ?O(h), then f
  ?O(h).O is transitive. Also ?, ?, o, ? are
  transitive.
• Reflexive f ? ?(f)
• Symmetric If f ? ?(g), then g ? ?(f)

15
?

• ? defines an equivalence relation on the
  functions.
• Each set ?(f) is an equivalence class (complexity
  class).
• f ?O(g) ? g ? ?(f)
• O(f g) O(max(f, g)) similar equations hold
  for ? and ?

16
Classification of functions

• lg n ? o(n?) for any ? gt 0, including fractional
  powers
• nk ? o(cn) for any k gt 0 and any c gt 1
• powers of n grow more slowly than any
  exponential function cn

17
Classification of functions

• O(1) denotes the set of functions bounded by a
  constant (for large n)
• f ? ?(n), f is linear
• f ? ?(n2), f is quadratic f ? ?(n3), f is cubic