ITCS6114 Midterm Review (Chapter 1- 8, 10, 12)
1. Time complexity analysis
Learning objectives & assessment examples:
• Given an algorithm, no matter whether the algorithm is recursive or not, we are able to define its
growth rate function. [Measure frequency or number of iterations associated with input instances]
• Compare the asymptotic behaviors of functions or describe the relative merits of worst-, average-,
and best-case analysis
Given two functions of input instance size N, we are able to specify their asymptotic order.
The asymptotic behavior of a function f(N) refers to the growth of f(N) as N gets large. We are
interested in how slow the program will be on large inputs. A good rule of thumb is: the slower
the asymptotic growth rate, the better the algorithm.
• Given a recurrence, we are able to analyze its time complexity through different methods:
• Master method
• Substitution and proof
• Recursive tree
Example:
1. For each pair of functions f and g, circle the statements that are true. No justification is
needed.
f(n) = 5 log n, g(n) = log(n 2 ).
Circle all that apply:
(a) f(n) is O(g(n)) (b) f(n) is Ω(g(n))
Solution: (a) and (b) are true.
Recall that log(n 2 ) = 2 log n.
2. For each pair of functions f and g, circle the statements that are true. No justification is
needed.
f(n) = 5 log n, g(n) = (log n) 2 .
Circle all that apply:
(a) f(n) is O(g(n)) (b) f(n) is Ω(g(n))
Solution: Only (a) is true.
