Outline of Class 19: Algorithm Analysis, Continued

Held: Wednesday, February 18, 1998

Administrivia

• Any questions on assignment 3?
  • Note that many parts of this assignment can be found in the appropriate chapter of Bailey.
  • Note also that there were some typos in the assignment which have now been fixed.
• The focus of today's class will be lab 8.

Composing Functions

• Often, we can express the running time of an algorithm as a composition of other functions.
  • For example, we might have a primary control structure which repeats its contents O(h(n)) times and within that control structure, we have a subalgorithm that takes O(g(n)) time.
  • Similarly, we might break our algorithm up into two independent algorithms, one of which takes O(h(n)) time and one of which takes O(g(n)) time.
  • We might also look at other relationships.
• Here are some questions we might ask ourselves about composition of functions.
  • If f(n) is O(h(n)) and g(n) is O(k(n)), then what can we say about f(n) + g(n)?
  • If f(n) is O(h(n)) and g(n) is O(k(n)), then what can we say about f(n) * g(n)?
• We might also ask more general questions about big-O notation.
  • If f(n) is O(g(n)) and g(n) is O(h(n)) then what can we say about f(n) and h(n)?
  • If f(n) is O(kg(n)) and k is a "constant", then what else can we say about f(n) and g(n)?
  • If f(n) is O(g(n) + h(n)) and g(n) is O(h(n)), then what simpler thing can we say about f(n)?

Related Notations

• There are a number of concepts like big-O notation.
• We say that a function f(n) is Omega(g(n)), which is read "Omega of g of n" if
  • there exists a number n0
  • there exists a number c > 0
  • for all n > n0, abs(f(n)) >= abs(c*g(n))
• Omega provides a lower bound on the running time of an algorithm.
• We say that a function f(n) is theta(g(n)), which is read "theta of g of n" if
  • there exists a number n0
  • there exists a number c > 0
  • there exists a number d > 0
  • for all n > n0, c*g(n) <= abs(f(n)) <= abs(d*g(n))
• Theta notation gives a more precise running time, as it provides both a lower and upper bound on the running time.
• Some computer scientists look at even more abstract issues, such as the minimum running time for any algorithm that solves a particular type of problem.

Lab

We'll spend much of today's class on an algorithm analysis lab.