1. Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency
The Design and Analysis of Algorithms
Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency
Asymptotic Notations and Basic Efficiency Classes

2. Section 2.2. Asymptotic Notations and Basic Efficiency Classes
Classifying Functions by Their Asymptotic Growth
Formal Definitions of Theta, Little oh and Little omega
Big Oh and Big Omega
Rules to Manipulate Big-Oh Expressions
Typical Growth Rates

3. Classifying functions by their asymptotic growth
Given a particular function g(n) , the set of all functions can be partitioned into three sets:
Little oh : o(g(n)): the set of functions f(n) that grow slower than g(n)
Theta: Θ (g(n)): the set of functions f(n) that grow at same rate as g(n)
Little omega: ω(g(n)): the set of functions f(n) that grow faster than g(n)

4. Formal Definition of Theta
f(n) and g(n) have the same rate of growth, if
lim( f(n) / g(n) ) = c, 0 < c < ∞, n → ∞
Notation: f(n) = Θ(g(n))
There exist constants c1 and c2 and a nonnegative integer n0 such that
c2g(n) ≤ f(n) ≤ c1g(n) for all n ≥ n0

5. Formal Definition of Little oh
f(n) grows slower than g(n)
( or g(n) grows faster than f(n)) if
lim(f(n)/g(n)) = 0, n → ∞
Notation: f(n) = o(g(n))
There exists a constant c and a nonnegative integer n0 such that
f(n) < c.g(n) for all n ≥ n0

6. Formal Definition of Little omega
f(n) grows faster than g(n)
( or g(n) grows slower than f(n)) if
lim(f(n)/g(n)) = ∞, n → ∞
Notation: f(n) = (g(n))
There exists a constant c and a nonnegative integer n0 such that
c.g(n) < f(n) for all n ≥ n0

7. Big Oh and Big Omega
O(g(n)) = o(g(n)  Θ(g(n))
 (g(n)) = ω (g(n))  Θ(g(n))

8. f(n) =Ω(g(n) f(n) = O(g(n)) f(n) = ω (g(n)) f(n) = Θ (g(n))
f(n) grows faster
than g(n)
f(n) =Ω(g(n)
f(n) = Θ (g(n))
f(n ) grows with
same rate
f(n) = O(g(n))
f(n) =o(g(n))
f(n) grows slower
than g(n)

9. Rules to Manipulate Big-Oh Expressions
Let T1(N) = O(f(N)) and T2(N) = O(g(N))
T1(N) + T2(N) = max(O(f(N)), O(g(N)))
where max(O(f(N)), O(g(N))) =
f(N) if g(N) = O(f(N))
g(N) if f(N) = O(g(N)),
T1(N) * T2(N) = O(f(N) * g(N))
If T(N) is a polynomial of degree k,
T(N) = Θ(Nk) = O(Nk)
logkN = O(N) for any constant k.

10. Typical Growth Rates C constant, we write O(1) logN logarithmic
log2N log-squared
N linear
NlogN
N2 quadratic
N3 cubic
2N exponential
N! factorial