Algorithms and Discrete Mathematics 20082009

1
Algorithms and Discrete Mathematics 2008/2009

• Lecture 7
• Big-O notation

Ioannis Ivrissimtzis
24-Nov-2008
2
Overview of the lecture

• The Big-O notation

3
The Big-O notation

• Let f and g be functions from the set of integers
  or the set of real
• numbers to the set of real numbers. We say that
  f(x) is O(g(x)) if there
• are constants C and k such that
• whenever x gt k. This is read as f(x) is
  big-oh of g(x).
• Rosen, p.180

4
The Big-O notation

• The definition says that after a certain point,
  namely after k, the
• absolute value of f(x) is at most C times the
  absolute value of g(x).
• The absolute value of a number, is the number
  without its sign.
• Formally
• C is a fixed constant. We are not allowed to
  increase it as x increases.

5
The Big-O notation

• The constants C and k in the definition of big-O
  notation are called
• witnesses to the relationship f(x) is O(g(x)).
• If there is a pair of witnesses to the
  relationship f(x) is O(g(x)), then
• there are infinitely many pairs of witnesses to
  that relationship.
• Indeed, if C and k are one pair of witnesses,
  then any pair C and k,
• where C lt C and k lt k, is also a pair of
  witnesses.
• To establish that f(x) is O(g(x)) we need only
  one pair of witnesses to
• this relationship.

6
Examples

• Example 5.4 Rosen, p.181 Let
• f(x) x2 2x 1
• Prove that f(x) is in O(x2).
• Proof For x gt 1, we have 1 lt x lt x2. That gives
• f(x) x2 2x 1 x2 2x2 x2 4x2
• Because the above equation holds for every
  positive integer x gt 1,
• using k 1 and C 4 as witnesses, we get
• f(x) Cx2, for every x gt k
• showing that f(x) is O(x2).

7
Examples

• Example 5.5 Show that f(x) 3x3 7x2 - 4x
  2 is O(x3).
• Proof For all x gt 1, we have
• So, for the pair of witnesses k 1, C 16, we
  have f(x) Cx3 for
• every x gt k, giving f(x) O(x3).

8
Examples

• Remark The inequality
• can be obtained with repetitive use of the
  triangle inequality
• Exercise 5.1 Using the triangle inequality show
  that

9
Big-O of polynomials

• Proposition 5.1 Rosen, p.184 Any polynomial
  function is Big-O its
• leading degree without leading constant.
• Proof Let f(x) be a polynomial of degree k, that
  is,
• with ak?0.

10
Big-O of polynomials

• We have
• With witnesses
  
• and k 1, we have f(x) Cxk, for every x gt
  k, giving f(x) O(xk).

11
The Big-O notation

• Some orders that often occur in practice are