Asymptotic Notation,

Review of Functions &

Summations
 Asymptotic Complexity
 Running time of an algorithm as a function of
input size n for large n.
 Expressed using only the highest-order term
in the expression for the exact running time.
 Instead of exact running time, say (n2).
 Describes behavior of function in the limit.
 Written using Asymptotic Notation.
 Asymptotic Notation
 , O, , o, 
 Defined for functions over the natural numbers.
 Ex: f(n) = (n2).
 Describes how f(n) grows in comparison to n2.
 Define a set of functions; in practice used to compare
two function sizes.
 The notations describe different rate-of-growth
relations between the defining function and the
defined set of functions.
 -notation
For function g(n), we define (g(n)),
big-Theta of n, as the set:
(g(n)) = {f(n) :
 positive constants c1, c2, and n0,
such that n  n0,
we have 0  c1g(n)  f(n) 
c2g(n)
}
Intuitively: Set of all functions that
have the same rate of growth as g(n).

g(n) is an asymptotically tight bound for f(n).

 -notation
For function g(n), we define (g(n)),
big-Theta of n, as the set:
(g(n)) = {f(n) :
 positive constants c1, c2, and n0,
such that n  n0,
we have 0  c1g(n)  f(n) 
c2g(n)
}
Technically, f(n)  (g(n)).
Older usage, f(n) = (g(n)).
I’ll accept either…

f(n) and g(n) are nonnegative, for large n.

 Example
(g(n)) = {f(n) :  positive constants c1, c2, and n0,
such that n  n0, 0  c1g(n)  f(n)  c2g(n)}

 10n2 - 3n = (n2)
 What constants for n0, c1, and c2 will work?
 Make c1 a little smaller than the leading
coefficient, and c2 a little bigger.
 To compare orders of growth, look at the
leading term.
 Exercise: Prove that n2/2-3n= (n2)
 Example
(g(n)) = {f(n) :  positive constants c1, c2, and n0,
such that n  n0, 0  c1g(n)  f(n)  c2g(n)}

 Is 3n3  (n4) ??
 How about 22n (2n)??
 O-notation
For function g(n), we define O(g(n)),
big-O of n, as the set:
O(g(n)) = {f(n) :
 positive constants c and n0,
such that n  n0,
we have 0  f(n)  cg(n) }
Intuitively: Set of all functions
whose rate of growth is the same as
or lower than that of g(n).
g(n) is an asymptotic upper bound for f(n).
f(n) = (g(n))  f(n) = O(g(n)).
(g(n))  O(g(n)).
 Examples
O(g(n)) = {f(n) :  positive constants c and n0,
such that n  n0, we have 0  f(n)  cg(n) }
 Any linear function an + b is in O(n2). How?
 Show that 3n3=O(n4) for appropriate c and n0.
  -notation
For function g(n), we define (g(n)),
big-Omega of n, as the set:
(g(n)) = {f(n) :
 positive constants c and n0,
such that n  n0,
we have 0  cg(n)  f(n)}
Intuitively: Set of all functions
whose rate of growth is the same
as or higher than that of g(n).
g(n) is an asymptotic lower bound for f(n).
f(n) = (g(n))  f(n) = (g(n)).
(g(n))  (g(n)).
 Example
(g(n)) = {f(n) :  positive constants c and n0, such
that n  n0, we have 0  cg(n)  f(n)}

 n = (lg n). Choose c and n0.

Relations Between , O, 
 Relations Between , , O
Theorem : For any two functions g(n) and f(n),
f(n) = (g(n)) iff
f(n) = O(g(n)) and f(n) = (g(n)).

 I.e., (g(n)) = O(g(n))  (g(n))

 In practice, asymptotically tight bounds are

obtained from asymptotic upper and lower bounds.
 Running Times
 “Running time is O(f(n))”  Worst case is O(f(n))
 O(f(n)) bound on the worst-case running time 
O(f(n)) bound on the running time of every input.
 (f(n)) bound on the worst-case running time 
(f(n)) bound on the running time of every input.
 “Running time is (f(n))”  Best case is (f(n))
 Can still say “Worst-case running time is (f(n))”
 Means worst-case running time is given by some
unspecified function g(n)  (f(n)).
 Example
 Insertion sort takes (n2) in the worst case, so
sorting (as a problem) is O(n2). Why?

 Any sort algorithm must look at each item, so

sorting is (n).

 In fact, using (e.g.) merge sort, sorting is (n lg n)

in the worst case.
 Later, we will prove that we cannot hope that any
comparison sort to do better in the worst case.
Asymptotic Notation in Equations
 Can use asymptotic notation in equations to
replace expressions containing lower-order terms.
 For example,
4n3 + 3n2 + 2n + 1 = 4n3 + 3n2 + (n)
= 4n3 + (n2) = (n3). How to interpret?
 In equations, (f(n)) always stands for an
anonymous function g(n)  (f(n))
 In the example above, (n2) stands for
3n2 + 2n + 1.
 o-notation
For a given function g(n), the set little-o:
o(g(n)) = {f(n):  c > 0,  n0 > 0 such that
 n  n0, we have 0  f(n) < cg(n)}.
f(n) becomes insignificant relative to g(n) as n
approaches infinity:
lim [f(n) / g(n)] = 0
n

g(n) is an upper bound for f(n) that is not

asymptotically tight.
Observe the difference in this definition from previous
ones. Why?
  -notation
For a given function g(n), the set little-omega:

(g(n)) = {f(n):  c > 0,  n > 0 such that

0
 n  n0, we have 0  cg(n) < f(n)}.
f(n) becomes arbitrarily large relative to g(n) as n
approaches infinity:
lim [f(n) / g(n)] = .
n

g(n) is a lower bound for f(n) that is not

asymptotically tight.
Comparison of Functions
fg  ab

f (n) = O(g(n))  a  b
f (n) = (g(n))  a  b
f (n) = (g(n))  a = b
f (n) = o(g(n))  a < b
f (n) = (g(n))  a > b
 Limits
 lim [f(n) / g(n)] = 0  f(n)  (g(n))
n

 lim [f(n) / g(n)] <   f(n)  (g(n))

n

 0 < lim [f(n) / g(n)] <   f(n)  (g(n))

n

 0 < lim [f(n) / g(n)]  f(n) (g(n))

n

 lim [f(n) / g(n)] =   f(n)  (g(n))

n

 lim [f(n) / g(n)] undefined can’t say

n
 Properties
 Transitivity
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
f(n) = O(g(n)) & g(n) = O(h(n))  f(n) = O(h(n))
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))
f(n) = o (g(n)) & g(n) = o (h(n))  f(n) = o (h(n))
f(n) = (g(n)) & g(n) = (h(n))  f(n) = (h(n))

 Reflexivity
f(n) = (f(n))
f(n) = O(f(n))
f(n) = (f(n))
 Properties
 Symmetry
f(n) = (g(n)) iff g(n) = (f(n))

 Complementarity
f(n) = O(g(n)) iff g(n) = (f(n))
f(n) = o(g(n)) iff g(n) = ((f(n))
Common Functions
 Monotonicity
 f(n) is
 monotonically increasing if m  n  f(m)  f(n).
 monotonically decreasing if m  n  f(m)  f(n).
 strictly increasing if m < n  f(m) < f(n).
 strictly decreasing if m > n  f(m) > f(n).
 Exponentials
 Useful Identities:
1 1
a 
a
(a m ) n a mn
a m a n a mn

 Exponentials and polynomials

nb
lim n 0
n  a

 n b o(a n )
 Logarithms
a b logb a
x = logba is the
exponent for a = bx. log c (ab) log c a  log c b
n
log b a n log b a
Natural log: ln a = logea
log c a
Binary log: lg a = log2a
log b a 
log c b
log b (1 / a )  log b a
lg2a = (lg a)2
lg lg a = lg (lg a)
1
log b a 
log a b
a logb c c logb a
Logarithms and exponentials – Bases
 If the base of a logarithm is changed from one
constant to another, the value is altered by a
constant factor.
 Ex: log10 n * log210 = log2 n.
 Base of logarithm is not an issue in asymptotic
notation.
 Exponentials with different bases differ by a
exponential factor (not a constant factor).
 Ex: 2n = (2/3)n*3n.
 Polylogarithms
 For a  0, b > 0, lim n ( lga n / nb ) = 0,
so lga n = o(nb), and nb = (lga n )
 Prove using L’Hopital’s rule repeatedly

 lg(n!) = (n lg n)
 Prove using Stirling’s approximation (in the text) for lg(n!).
 Exercise
Express functions in A in asymptotic notation using functions in B.

A B
5n2 + 100n 3n2 + 2 A  (B)

A  (n2), n2  (B)  A  (B)

log3(n2) log2(n3) A  (B)
logba = logca / logcb; A = 2lgn / lg3, B = 3lgn, A/B =2/(3lg3)
nlg4 3lg n A  (B)
alog b = blog a; B =3lg n=nlg 3; A/B =nlg(4/3)   as n
lg2n n1/2 A  (B)
lim ( lga n / nb ) = 0 (here a = 2 and b = 1/2)  A  (B)
n
Summations – Review
 Review on Summations
 Why do we need summation formulas?
For computing the running times of iterative
constructs (loops). (CLRS – Appendix A)
Example: Maximum Subvector
Given an array A[1…n] of numeric values (can be
positive, zero, and negative) determine the
subvector A[i…j] (1 i  j  n) whose sum of
elements is maximum over all subvectors.

1 -2 2 2
 Review on Summations
MaxSubvector(A, n)
maxsum  0;
for i  1 to n
do for j = i to n
sum  0
for k  i to j
do sum += A[k]
maxsum  max(sum, maxsum)
return maxsum

n n j
T(n) =    1
i=1 j=i k=i

NOTE: This is not a simplified solution. What is the final answer?

 Review on Summations
 Constant Series: For integers a and b, a  b,
b

1 b  a 1
i a

 Linear Series (Arithmetic Series): For n  0,

n
n(n  1)
 i 1  2   n 
i 1 2

 Quadratic Series: For n  0,

n
n(n  1)(2n  1)
 i 2 12  22    n 2 
i 1 6
 Review on Summations
 Cubic Series: For n  0,
n 2 2
3 3 3 3 n ( n  1)
 i 1  2    n 
i 1 4

 Geometric Series: For real x  1,

n n 1
k 2 n x 1
 x 1  x  x    x 
k 0 x 1
For |x| < 1,

k 1
 x 
k 0 1 x
 Review on Summations
 Linear-Geometric Series: For n  0, real c  1,

n n 1 n 2
i 2 n (n  1)c  nc c
 ic c  2c    nc  2
i 1 (c  1)

 Harmonic Series: nth harmonic number, nI+,

1 1 1
H n 1     
2 3 n
n
1
 ln(n)  O (1)
k 1 k
 Review on Summations
 Telescoping Series:
n

a
k 1
k  ak  1 an  a0

 Differentiating Series: For |x| < 1,


k x
 kx  2
k 0 1  x 
 Review on Summations
 Approximation by integrals:
 For monotonically increasing f(n)
n n n 1

 f ( x)dx  f (k )  f ( x)dx
m 1 k m m

 For monotonically decreasing f(n)

n1 n n

 f ( x)dx  f (k )  f ( x)dx
m k m m 1

 How?
 Review on Summations
 nth harmonic number
n n1
1 dx
   ln(n  1)
k 1 k 1
x

n n
1 dx
   ln n
k 2 k 1
x

n
1
  ln n  1
k 1 k
 Reading Assignment
 Chapter 4 of CLRS.