IEOR 269, Spring 2010 Integer Programming and Combinatorial ...

IEOR269 notes, Prof. Hochbaum, 2010 27
15.2 Growth of functions
n ⌈log n⌉ n − 1 2 n n!
1 0 0 2 1
3 2 2 8 6
5 3 4 32 120
10 4 9 1024 3628800
70 7 69 2 70 ≈ 2 332
We are interested in the asymptotic behavior of the running time.
15.3 Definitions for asymptotic comparisons of functions
We define for functions f and g,
f, g : Z + → [0, ∞) .
1. f(n) ∈ O(g(n)) if ∃ a constant c > 0 such that f(n) ≤ cg(n), for all n sufficiently large.
2. f(n) ∈ o(g(n)) if, for any constant c > 0, f(n) < cg(n), for all n sufficiently large.
3. f(n) ∈ Ω(g(n)) if ∃ a constant c > 0 such that f(n) ≥ cg(n), for all n sufficiently large.
4. f(n) ∈ ω(g(n)) if, for any constant c > 0, f(n) > cg(n), for all n sufficiently large.
5. f(n) ∈ Θ(g(n)) if f(n) ∈ O(g(n)) and f(n) ∈ Ω(g(n)).
Examples:
• Bubble sort has complexity O(n 2 )
• Matrix multiplication has complexity O(n 3 )
• Gaussian elimination O(n 3 )
• 2 n /∈ O(n 3 )
• n 3 ∈ 0(2 n )
• n 3 ∈ o(2 n )
• n 4 ∈ Ω(n 3 .5)
• n 4 ∈ Ω(n 4 )
15.4 Properties of asymptotic notation
We mention a few properties that can be useful when analyzing the complexity of algorithms.
Proposition 15.2. f(n) ∈ Ω(g(n)) if and only if g(n) ∈ O(f(n)).
The next property is often used in conjunction with L’hôpital’s Rule.
Proposition 15.3. Suppose that lim n→∞
f(n)
g(n) exists:
f(n)
lim
n→∞ g(n)
= c .
Then,
1. c < ∞ implies that f(n) ∈ O(g(n)).
2. c > 0 implies that f(n) ∈ Ω(g(n)).
3. c = 0 implies that f(n) ∈ o(g(n)).
4. 0 < c < ∞ implies that f(n) ∈ Θ(g(n)).