RAIK 283: Data Structures & Algorithms
Analysis of Algorithms
Dr. Ying Lu
ylu@cse.unl.edu
August 28, 2012
http://www.cse.unl.edu/~ylu/raik283/
Design and Analysis of Algorithms Chapter 2.1
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RAIK 283: Data Structures & Algorithms
 Giving
credit where credit is due:
• Most of the lecture notes are based on the slides
from the Textbook’s companion website
http://www.aw-bc.com/info/levitin
• Several slides are from Jeff Edmonds of the
York University
• I have modified them and added new slides
Design and Analysis of Algorithms Chapter 2.1
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The Time Complexity of an Algorithm
The Time Complexity of an Algorithm
Specifies how the running time depends
on the size of the input
Purpose
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Purpose
To estimate how long a program will run.
 To estimate the largest input that can reasonably be
given to the program.
 To compare the efficiency of different algorithms.
 To help focus on the parts of code that are executed
the largest number of times.
 To choose an algorithm for an application.

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Purpose (Example)

Suppose a machine that performs a million floatingpoint operations per second (106 FLOPS), then how
long an algorithm will run for an input of size n=50?
• 1) If the algorithm requires n2 such operations:
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Purpose (Example)

Suppose a machine that performs a million floatingpoint operations per second (106 FLOPS), then how
long an algorithm will run for an input of size n=50?
• 1) If the algorithm requires n2 such operations:
– 0.0025 second
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Purpose (Example)

Suppose a machine that performs a million floatingpoint operations per second (106 FLOPS), then how
long an algorithm will run for an input of size n=50?
• 1) If the algorithm requires n2 such operations:
– 0.0025 second
• 2) If the algorithm requires 2n such operations:
– A) Takes a similar amount of time (t < 1 sec)
– B) Takes a little bit longer time (1 sec < t < 1 year)
– C) Takes a much longer time (1 year < t)
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Purpose (Example)

Suppose a machine that performs a million floatingpoint operations per second (106 FLOPS), then how
long an algorithm will run for an input of size n=50?
• 1) If the algorithm requires n2 such operations:
– 0.0025 second
• 2) If the algorithm requires 2n such operations:
– over 35 years!
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Time Complexity Is a Function
Specifies how the running time depends on the size of the
input.
A function mapping
“size” of input
“time” T(n) executed .
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Definition of Time?
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Definition of Time

# of seconds (machine, implementation dependent).

# lines of code executed.

# of times a specific operation is performed
addition).
Design and Analysis of Algorithms Chapter 2.1
(e.g.,
13
Theoretical analysis of time efficiency
Time efficiency is analyzed by determining the number of
repetitions of the basic operation as a function of input size

Basic operation: the operation that contributes most
towards the running time of the algorithm.
input size
T(n) ≈ copC(n)
running time
execution time
for basic operation
Number of times
basic operation is
executed
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Input size and basic operation examples
Problem
Input size measure
Basic operation
Search for key in a list of
n items
Multiply two matrices of
floating point numbers
Compute an
Graph problem
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Input size and basic operation examples
Problem
Input size measure
Search for key in a list of Number of items in the
n items
list: n
Basic operation
Key comparison
Multiply two matrices of
floating point numbers
Compute an
Graph problem
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Input size and basic operation examples
Problem
Input size measure
Basic operation
Search for key in a list of Number of items in the
n items
list: n
Key comparison
Multiply two matrices of
Dimensions of matrices
floating point numbers
Floating point
multiplication
Compute an
Graph problem
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Input size and basic operation examples
Problem
Input size measure
Basic operation
Search for key in list of n
Number of items in list n Key comparison
items
Multiply two matrices of
Dimensions of matrices
floating point numbers
Floating point
multiplication
Compute an
Floating point
multiplication
n
Graph problem
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Input size and basic operation examples
Problem
Input size measure
Basic operation
Search for key in list of n
Number of items in list n Key comparison
items
Multiply two matrices of
Dimensions of matrices
floating point numbers
Floating point
multiplication
Compute an
n
Floating point
multiplication
Graph problem
#vertices and/or edges
Visiting a vertex or
traversing an edge
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Theoretical analysis of time efficiency
Time efficiency is analyzed by determining the
number of repetitions of the basic operation as a
function of input size
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Which Input of size n?
Efficiency also depends on the particular input
 For
instance: search a key in a list of n letters
• Problem input: a list of n letters
• How many different inputs?
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Which Input of size n?
Efficiency also depends on the particular input
For instance: search a key in a list of n letters
There are 26n inputs of size n.
Which do we consider
for the time efficiency C(n)?
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Best-case, average-case, worst-case

Worst case: W(n) – maximum over inputs of size n

Best case:

Average case: A(n) – “average” over inputs of size n
B(n) – minimum over inputs of size n
• NOT the average of worst and best case
• Under some assumption about the probability distribution of all
possible inputs of size n, calculate the weighted sum of expected C(n)
(numbers of basic operation repetitions) over all possible inputs of size
n.
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Example: Sequential search

Problem: Given a list of n elements and a search key K, find
an element equal to K, if any.
Algorithm: Scan the list and compare its successive
elements with K until either a matching element is found
(successful search) or the list is exhausted (unsuccessful
search)
Worst case

Best case

Average case


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An example





Compute gcd(m, n) by applying the algorithm based on
checking consecutive integers from min(m, n) down to
gcd(m, n)
Input size?
Best case?
Worst case?
Average case?
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Types of formulas for basic operation count

Exact formula
e.g., C(n) = n(n-1)/2

Formula indicating order of growth with specific
multiplicative constant
e.g., C(n) ≈ 0.5 n2

Formula indicating order of growth with unknown
multiplicative constant
e.g., C(n) ≈ cn2
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Types of formulas for basic operation count




Exact formula
e.g., C(n) = n(n-1)/2
Formula indicating order of growth with specific
multiplicative constant
e.g., C(n) ≈ 0.5 n2
Formula indicating order of growth with unknown
multiplicative constant
e.g., C(n) ≈ cn2
Most important: Order of growth within a constant
multiple as n→∞
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Asymptotic growth rate

A way of comparing functions that ignores constant factors
and small input sizes

O(g(n)):

Θ (g(n)):

Ω(g(n)):
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Asymptotic growth rate

A way of comparing functions that ignores constant factors
and small input sizes

O(g(n)): class of functions f(n) that grow no faster than g(n)

Θ (g(n)): class of functions f(n) that grow at same rate as g(n)

Ω(g(n)): class of functions f(n) that grow at least as fast as g(n)
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Table 2.1
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Classifying Functions
Giving an idea of how fast a function
grows without going into too much detail.
Which are more alike?
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Which are more alike?
Mammals
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Which are more alike?
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Which are more alike?
Dogs
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Classifying Animals
Vertebrates
Fish
Reptiles
Mammals
Birds
Giraffe
Dogs
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Which are more alike?
n1000
n2
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2n
37
Which are more alike?
n1000
n2
2n
Polynomials
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Which are more alike?
1000n2
3n2
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2n3
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Which are more alike?
1000n2
3n2
2n3
Quadratic
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Classifying Functions?
Functions
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Classifying Functions
Functions
Polynomial
Exponential
Factorial
5 log n
Poly Logarithmic
Logarithmic
Constant
5
(log n)5
n5
25n
n!
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Classifying Functions?
Polynomial
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Classifying Functions
Polynomial
Linear
Quadratic
Cubic
?
5n
5n2
5n3
5n4
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Logarithmic
 log10n
= # digits to write n
 log2n = # bits to write n
= 3.32 log10n
 log(n1000) = 1000 log(n)
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Differ only by a
multiplicative
constant
45
Poly Logarithmic
(log n)5 = log5 n
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Which grows faster?
log1000 n
n0.001
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Poly Logarithmic << Polynomial
log1000 n << n0.001
For sufficiently large n
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Which grows faster?
10000 n
0.0001 n2
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Linear << Quadratic
10000 n << 0.0001 n2
For sufficiently large n
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Which grows faster?
n1000
20.001 n
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Polynomial << Exponential
n1000 << 20.001 n
For sufficiently large n
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Ordering Functions
Functions
<<
<<
25n <<
Design and Analysis of Algorithms Chapter 2.1
Factorial
5 << 5 log n << (log n)5 << n5
<<
Exponential
<<
Polynomial
<<
Poly Logarithmic
Logarithmic
Constant
<<
n!
53
Which Functions are Constant?
•5
• 1,000,000,000,000
• 0.0000000000001
• -5
• 0
• 8 + sin(n)
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Which Functions are Constant?
Yes • 5
Yes • 1,000,000,000,000
Yes • 0.0000000000001
Yes • -5
Yes • 0
No • 8 + sin(n)
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Which Functions are “Constant”?
The running time of the algorithm is a “Constant”
if it does not depend significantly
on the size of the input.
•5
• 1,000,000,000,000
• 0.0000000000001
• -5
• 0
• 8 + sin(n)
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Which Functions are “Constant”?
The running time of the algorithm is a “Constant”
It does not depend significantly
on the size of the input.
Yes
Yes
Yes
No
No
Yes
•5
• 1,000,000,000,000
9
• 0.0000000000001
7
• -5
• 0
Lie in between
• 8 + sin(n)
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Which Functions are Quadratic?
• n2
• 0.001 n2
• 1000 n2
• 5n2 + 3n + 2log n
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Which Functions are Quadratic?
• n2
• 0.001 n2
Lie in between
• 1000 n2
• 5n2 + 3n + 2log n
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Which Functions are Quadratic?
• n2
• 0.001 n2
• 1000 n2
• 5n2 + 3n + 2log n
Ignore low-order terms
Ignore multiplicative constants.
Ignore "small" values of n.
Write θ(n2).
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Examples
f(n)
g(n)
1)
ln2n
ne
2)
nk
cn
3)
n
nsinn
4)
2n
2n/2
5)
nlgc
clgn
6)
O(g(n))?
Ω(g(n))? Θ(g(n))?
lg(n!) lg(nn)
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