Growth of Functions

1
Growth of Functions
2
Introduction

• Order of growth of the running time of an
  algorithm
• Characterization of the algorithms efficiency
• Is used to compare relative performance of
  alternative algorithms
• Asymptotic efficiency of algorithms
• How the running time of an algorithm increase
  with the size of the input in the limit, as the
  size of the input increases without bound
• Usually, an algorithm that is asymptotically more
  efficient will be the best choice for all but
  very small inputs.

3
Asymptotic Notation
4
?-notation

• ?(g(n)) f(n) there exist positive constants
  c1, c2 and n0 such that 0 ? c1g(n) ? f(n) ?
  c2g(n) for all n ? n0
• Domains of functions are the set of natural
  number
• f(n) and g(n) must be asymptotically nonnegative
• ?(g(n)) is a set of functions
• f(n) is a member of ?(g(n)) ? f(n) ? ?(g(n)) ?
  f(n) ?(g(n))
• f(n) is equal to g(n) within a constant factor
• g(n) is an asymptotically tight bound for f(n)
• The lower-order terms of an asymptotically
  positive function can be ignored in determining
  asymptotically tight bounds because they are
  insignificant for large n

5
?-notation (Cont.)
6
?-notation (Cont.)

• Prove that
• Determine c1, c2 and n0 such that
• c11/14, c21/2, n07
• Prove that
• Suppose for the purpose of contraction that c2
  and n0 exist such that

Impossible for arbitrarily large n, since c2 is
constant.
7
O-notation

• O(g(n)) f(n) there exist positive constants c
  and n0 such that 0 ? f(n) ? cg(n) for all n ? n0
• Asymptotic upper bound
• f(n) ?(g(n)) implies f(n) O(g(n)) ? ?(g(n))
  ? O(g(n))
• an2 bn c O(n2)
• an b O(n2) (Why???)
• O-notation is often used to describe the
  worst-case running time of an algorithm
• The running time of an algorithm is O(g(n)) ?
  there is a function f(n) that is O(g(n)) such
  that for any value of n, no matter what
  particular input of size n is chosen, the running
  time on that input is bounded from above by the
  value f(n)

8
O-notation (Cont.)
9
?-notation
an2 bn c ?(n2)an2 bn c ?(n)

• ?(g(n)) f(n) there exist positive constants c
  and n0 such that 0 ? cg(n) ? f(n) for all n ? n0
• Asymptotic lower bound
• f(n) ?(g(n)) implies f(n) ?(g(n)) ? ?(g(n))
  ? ?(g(n))
• ?-notation is often used to describe the
  best-case running time of an algorithm
• The running time of an algorithm is ?(g(n)) ?
  there is a function f(n) that is ?(g(n)) such
  that for any value of n, no matter what
  particular input of size n is chosen, the running
  time on that input is bounded from below by the
  value f(n)
• Theorem 3.1. For any two functions f(n) and g(n),
  we have f(n) ?(g(n)) if and only if f(n)
  O(g(n)) and f(n) ?(g(n))

10
?-Notation (Cont.)
11
o-notation and ?-notation

• o(g(n)) f(n) for any positive constant c,
  there exists a positive constant n0 such that 0 ?
  f(n) lt cg(n) for all n ? n0
• Denotes an upper bound that is not asymptotically
  tight
• 2n2 O(n2) ? asymptotically tight
• 2n O(n2) ? not asymptotically tight
• 2n o(n2) but 2n2 ? o(n2)
• ?(g(n)) f(n) for any positive constant c,
  there exists a positive constant n0 such that 0 ?
  cg(n) lt f(n) for all n ? n0
• Denotes an lower bound that is not asymptotically
  tight
• n2/2 ?(n) but n2/2 ? ?(n2)

12
Self-Study

• Section 3.1. Asymptotic notation in equations and
  inequalities (pp. 46-47)
• Section 3.1. Comparison of functions (pp. 49-50)
• Transitivity, reflexivity, symmetry, transpose
  symmetry, and trichotomy of asymptotically
  positive functions
• Section 3.2. Standard notations and common
  functions (pp. 51-56)