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CS 253: Algorithms
Chapter 3
Growth of Functions
Credit: Dr. George Bebis
Analysis of Algorithms
Goal:

To analyze and compare algorithms in terms of running time and memory
requirements (i.e. time and space complexity)

In other words, how does the running time and space requirements change
as we increase the input size n ?
(sometimes we are also interested in the coding complexity)

Input size (number of elements in the input)
◦ size of an array or a matrix
◦ # of bits in the binary representation of the input
◦ vertices and/or edges in a graph, etc.
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Types of Analysis

Worst case
◦ Provides an upper bound on running time
◦ An absolute guarantee that the algorithm would not run longer, no
matter what the inputs are

Best case
◦ Provides a lower bound on running time
◦ Input is the one for which the algorithm runs the fastest

Average case
◦ Provides a prediction about the running time
◦ Assumes that the input is random
Lower Bound ≤ Running Time ≤ Upper Bound
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Computing the Running Time

Measure the execution time ?
Not a good idea ! It varies for different microprocessors!

Count the number of statements executed?
Yes, but you need to be very careful!
High-level programming languages have statements which require a large
number of low-level machine language instructions to execute (a function
of the input size n).
For example, a subroutine call can not be counted as one statement; it
needs to be analyzed separately

Associate a "cost" with each statement.
Find the "total cost“ by multiplying the cost with the total number of
times each statement is executed.
(we have seen examples before)
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Example
Algorithm X
sum = 0;
for(i=0; i<N; i++)
for(j=0; j<N; j++)
sum += arrY[i][j];
Cost
c1
c2
c3
c4
------------
Total Cost = c1 + c2 * (N+1) + c3 * N * (N+1) + c4 * N2
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Asymptotic Analysis

To compare two algorithms with running times f(n) and g(n), we need
a rough measure that characterizes how fast each function grows
with respect to n

In other words, we are interested in how they behave asymptotically
(i.e. for large n) (called rate of growth)

Big O notation: asymptotic “less than” or “at most”:
f(n)=O(g(n)) implies: f(n) “≤” g(n)

 notation: asymptotic “greater than” or “at least”:
f(n)=  (g(n)) implies: f(n) “≥” g(n)

 notation: asymptotic “equality” or “exactly”:
f(n)=  (g(n)) implies: f(n) “=” g(n)
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Big-O Notation

We say
fA(n) = 7n+18 is order n, or O (n)
It is, at most, roughly proportional to n.
fB(n) = 3n2+5n +4 is order n2, or O(n2).
It is, at most, roughly proportional to n2.
In general, any O(n2) function is fastergrowing than any O(n) function.
Function value 

fB(n)
fA(n)
Increasing n 
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More Examples …

n4 + 100n2 + 10n + 50 
10n3 + 2n2  O(n3)
n3 - n2  O(n3)

constants
O(n4)
10 is O(1)
1273 is O(1)

what is the rate of growth for Algorithm X studied earlier (in
Big O notation)?
Total Time = c1 + c2*(N+1) + c2 * N*(N+1) + c3*N2
If c1, c2, c3 , and c4 are constants then
Total Time = O(N2)
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Definition of Big O

O-notation
Big-O example, graphically

Note that 30n+8 is O(n).
Can you find a c and n0
which can be used in the
formal definition of Big O ?
You can easily see that
30n+8 isn’t less than n
anywhere (n>0).

But it is less than
31n everywhere to
the right of n=8.

So, one possible (c , n0) pair
that can be used in the
formal definition:
c = 31, n0 = 8
cn =31n
30n+8
Function value 

n
n0=8
n
30n+8  O(n)
Big-O Visualization
O(g(n)) is the set of
functions with smaller
or same order of
growth as g(n)
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No Uniqueness

There is no unique set of values for n0 and c in proving the
asymptotic bounds

Prove that 100n + 5 = O(n2)
(i)
100n + 5 ≤ 100n + n = 101n ≤ 101n2
for all n ≥ 5
You may pick n0 = 5 and c = 101 to complete the proof.
(ii) 100n + 5 ≤ 100n + 5n = 105n ≤ 105n2
for all n ≥ 1
You may pick n0 = 1 and c = 105 to complete the proof.
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Definition of 
(g(n)) is the set of functions
with larger or same order of
growth as g(n)
Examples
◦ 5n2 = (n)
 c, n0 such that: 0  cn  5n2  cn  5n2  c = 1 and n > n0=1
◦ 100n + 5 ≠ (n2)
 c, n0 such that: 0  cn2  100n + 5
since 100n + 5  100n + 5n
n1
cn2  105n  n(cn – 105)  0
Since n is positive  (cn – 105)  0  n  105/c
 contradiction: n cannot be smaller than a constant
◦ n = (2n),
n3 = (n2),
n = (logn)
Definition of 

-notation
(g(n)) is the set of
functions with the same
order of growth as g(n)
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Examples
◦ n2/2 –n/2 = (n2)
 ½ n2 - ½ n ≤ ½ n 2
n ≥ 0  c2= ½
 ¼ n2 ≤ ½ n2 - ½ n
n ≥ 2  c1= ¼
◦ n ≠ (n2): c1 n2 ≤ n ≤ c2 n2
 only holds for: n ≤ 1/c1
◦ 6n3 ≠ (n2): c1 n2 ≤ 6n3 ≤ c2 n2
 only holds for: n ≤ c2 /6
◦ n ≠ (logn): c1 logn ≤ n ≤ c2 logn
 c2 ≥ n/logn,  n≥ n0 – impossible
Relations Between Different Sets

Subset relations between order-of-growth sets.
RR
O( f )
( f )
•f
( f )
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Common orders of magnitude
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Common orders of magnitude
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Logarithms and properties

In algorithm analysis we often use the notation “log n”
without specifying the base
Binary logarithm
lg n  log
Natural logarithm
ln n  log e n
2
n
log x  y log x
y
log xy  log x  log y
log
x
 log x  log y
y
lg n  (lg n )
k
k
lg lg n  lg(lg n )
a
log b x
log
b
 x
x
log b a
log
a
x
log
a
b
More Examples

For each of the following pairs of functions, either f(n) is O(g(n)), f(n) is
Ω(g(n)), or f(n) = Θ(g(n)). Determine which relationship is correct.
◦ f(n) = log n2; g(n) = log n + 5
f(n) = (g(n))
◦ f(n) = n;
f(n) = (g(n))
g(n) = log n2
◦ f(n) = log log n;
◦ f(n) = n;
g(n) = log n
g(n) = log2 n
f(n) = O(g(n))
f(n) = (g(n))
◦ f(n) = n log n + n; g(n) = log n
f(n) = (g(n))
◦ f(n) = 10;
g(n) = log 10
f(n) = (g(n))
◦ f(n) = 2n;
g(n) = 10n2
f(n) = (g(n))
◦ f(n) = 2n;
g(n) = 3n
f(n) = O(g(n))
Properties
Theorem:
f(n) = (g(n))  f = O(g(n)) and f = (g(n))

Transitivity:
◦ f(n) = (g(n)) and g(n) =
◦ Same for O and 

(h(n))  f(n) = (h(n))
Reflexivity:
◦ f(n) = (f(n))
◦ Same for O and 

Symmetry:
◦ f(n) =

(g(n)) if and only if g(n) = (f(n))
Transpose symmetry:
◦ f(n) = O(g(n)) if and only if g(n) = (f(n))
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