Data Structures and Algorithms Lecture 2: Analysis of Algorithms, Asymptotic notation

© 2004 Goodrich, Tamassia
Data Structures and Algorithms
Lecture 2: Analysis of Algorithms,
Asymptotic notation
Lilia Georgieva

Analysis of Algorithms 2
Outline
 Pseudocode
 Theoretical Analysis of Running time
 Primitive Operations
 Counting primitive operations
 Asymptotic analysis of running time

Pseudocode
 In this course, we will
mostly use
pseudocode to
describe an algorithm
 Pseudocode is a high-
level description of an
algorithm
 More structured than
English prose
 Less detailed than a
program
 Preferred notation for
describing algorithms
 Hides program design
issues
Algorithm arrayMax(A, n)
Input: array A of n integers
Output: maximum element of A
currentMax ←A[0]
for i ←1 to n −1 do
if A[i] > currentMax then
currentMax ←A[i]
return currentMax
Example: find max
element of an array

Pseudocode Details
 Control flow
 if … then … [else …]
 while … do …
 repeat … until …
 for … do …
 Indentation replaces
braces
 Method declaration
Algorithm method (arg, arg…)
Input …
Output …
currentMax ←A[0]
currentMax ←A[i]
return currentMax

Pseudocode Details
 Method call
var.method (arg [, arg…])
 Return value
return expression
 Expressions
← Assignment
(like = in Java)
= Equality testing
(like = = in Java)
n2
superscripts and
other mathematical
formatting allowed
currentMax ←A[0]
currentMax ←A[i]
return currentMax

Comparing Algorithms
 Given 2 or more algorithms to solve the
same problem, how do we select the best
one?
 Some criteria for selecting an algorithm
1) Is it easy to implement, understand, modify?
2) How long does it take to run it to completion?
3) How much of computer memory does it use?
 Software engineering is primarily
concerned with the first criteria
 In this course we are interested in the
second and third criteria

Comparing Algorithms
 Time complexity
 The amount of time that an algorithm needs to
run to completion
 Space complexity
 The amount of memory an algorithm needs to
run
 We will occasionally look at space
complexity, but we are mostly interested
in time complexity in this course
 Thus in this course the better algorithm is
the one which runs faster (has smaller
time complexity)

How to Calculate Running time
 Most algorithms transform input objects into
output objects
 The running time of an algorithm typically
grows with the input size
 idea: analyze running time as a function of input
size
sorting
algorithm
5 1
3 2 1 3
2 5
input object output object

How to Calculate Running Time
 Even on inputs of the same size, running time
can be very different
 Example: algorithm that finds the first prime
number in an array by scanning it left to right
 Idea: analyze running time in the
 best case
 worst case
 average case

How to Calculate Running Time
 Best case running
time is usually
useless
 Average case time is
very useful but often
difficult to determine
 We focus on the
worst case running
time
 Easier to analyze
 Crucial to applications
such as games,
finance and robotics
0
20
40
60
80
100
120
Running
Time
1000 2000 3000 4000
Input Size
best case
average case
worst case

Experimental Evaluation of Running Time
 Write a program
implementing the
algorithm
 Run the program with
inputs of varying size and
composition
 Use a method like
System.currentTimeMillis(
) to get an accurate
measure of the actual
running time
 Plot the results
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 50 100
Input Size
Time
(ms)

Limitations of Experiments
 Experimental evaluation of running
time is very useful but
 It is necessary to implement the
algorithm, which may be difficult
 Results may not be indicative of the
running time on other inputs not included
in the experiment
 In order to compare two algorithms, the
same hardware and software
environments must be used

Theoretical Analysis of Running Time
 Uses a pseudo-code description of
the algorithm instead of an
implementation
 Characterizes running time as a
function of the input size, n
 Takes into account all possible
inputs
 Allows us to evaluate the speed of
an algorithm independent of the
hardware/software environment

RAM: The Random Access Machine
 For theoretical analysis, we assume RAM
model for our “theoretical” computer
 Our RAM model consists of:
 a CPU
 a potentially unbounded bank of
memory cells, each of which can hold an
arbitrary number or character
 memory cells are numbered and
accessing any cell in memory takes unit
time.
1 2 3 ……………………………………

Primitive Operations
 For theoretical analysis, we will count
primitive or basic operations, which are
simple computations performed by an
algorithm
 Basic operations are:
 Identifiable in pseudocode
 Largely independent from the programming
language
 Exact definition not important (we will see
why later)
 Assumed to take a constant amount of time
in the RAM model

Primitive Operations
 Examples of primitive operations:
 Evaluating an expression x2
+ey
 Assigning a value to a variable cnt ← cnt+1
 Indexing into an array A[5]
 Calling a method mySort(A,n)
 Returning from a method return(cnt)

Counting Primitive Operations
 By inspecting the pseudocode, we can determine
the maximum number of primitive operations
executed by an algorithm, as a function of the
input size
currentMax ←A[0] 2
for i ←1 to n −1 do 2+n
if A[i] > currentMax then 2(n −1)
currentMax ←A[i] 2(n −1)
{ increment counter i } 2(n −1)
return currentMax 1
Total 7n −1

Estimating Running Time
 Algorithm arrayMax executes 7n −1 primitive
operations in the worst case. Define:
a = Time taken by the fastest primitive operation
b = Time taken by the slowest primitive
operation
 Let T(n) be worst-case time of arrayMax.
Then
a (7n −1) ≤ T(n) ≤ b(7n −1)
 Hence, the running time T(n) is bounded by
two linear functions

Growth Rate of Running Time
 Changing the hardware/ software
environment
 Affects T(n) by a constant factor, but
 Does not alter the growth rate of T(n)
 Thus we focus on the big-picture which is
the growth rate of an algorithm
 The linear growth rate of the running time
T(n) is an intrinsic property of algorithm
arrayMax
 algorithm arrayMax grows proportionally with n,
with its true running time being n times a
constant factor that depends on the specific

Constant Factors
 The growth rate is not affected by
 constant factors or
 lower-order terms
 Examples
 102
n + 105
is a linear function
 105
n2
+ 108
n is a quadratic function
 How do we get rid of the constant factors to
focus on the essential part of the running
time?

Big-Oh Notation Motivation
 The big-Oh notation is used widely to
characterize running times and space
bounds
 The big-Oh notation allows us to ignore
constant factors and lower order terms
and focus on the main components of a
function which affect its growth

Big-Oh Notation Definition
 Given functions f(n)
and g(n), we say that
f(n) is O(g(n)) if there
are positive
constants
c and n0 such that
f(n) ≤ cg(n) for n ≥ n0
 Example: 2n + 10 is
O(n)
 2n + 10 ≤ cn
 (c −2) n ≥ 10
 n ≥ 10/(c −2)
 Pick c = 3 and n0= 10
0 5 10 15 20 25 30
0
10
20
30
40
50
60
70
80
3n
2n+
10
n
n

Big-Oh Example
 Example: the
function n2
is not
O(n)
 n2
≤ cn
 n ≤ c
 The above
inequality cannot be
satisfied since c
must be a constant 0 100 200 300 400 500
0
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
100,000
n^2
100n
10n
n
n

More Big-Oh Examples
 7n-2
7n-2 is O(n)
need c > 0 and n0 ≥ 1 such that 7n-2 ≤ c•n for n ≥ n0
this is true for c = 7 and n0 = 1

3n3
+ 20n2
+ 5
3n3
+ 20n2
+ 5 is O(n3
)
need c > 0 and n0 ≥ 1 s.t. 3n3
+ 20n2
+ 5 ≤ c•n3
for n ≥ n0

3 log n + 5
3 log n + 5 is O(log n)
need c > 0 and n0 ≥ 1 s.t. 3 log n + 5 ≤ c•log n for n ≥ n0

Big-Oh and Growth Rate
 The big-Oh notation gives an upper bound on
the growth rate of a function
 The statement “f(n) is O(g(n))” means that the
growth rate of f(n) is no more than the growth
rate of g(n)
 We can use the big-Oh notation to rank
functions according to their growth rate
f(n) is O(g(n)) g(n) is O(f(n))
g(n) grows
more
Yes No
f(n) grows more No Yes
Same growth Yes Yes

Big-Oh Rules
 If is f(n) a polynomial of degree d, then
f(n) is O(nd
), i.e.,
1. Drop lower-order terms
2. Drop constant factors
 Use the smallest possible class of
functions
 Say “2n is O(n)” instead of “2n is O(n2
)”
 Use the simplest expression of the class
 Say “3n + 5 is O(n)” instead of “3n + 5 is
O(3n)”

Big-Oh Rules
f(n) is O(nd
), i.e.,
1. Drop lower-order terms
2. Drop constant factors
 Use the smallest possible class of
functions
 Say “2n is O(n)” instead of “2n is O(n2
)”
 Use the simplest expression of the class
 Say “3n + 5 is O(n)” instead of “3n + 5 is
O(3n)”

Asymptotic Algorithm Analysis
 The asymptotic analysis of an algorithm
determines the running time in big-Oh notation
 To perform the asymptotic analysis
 We find the worst-case number of primitive
operations executed as a function of the input size
 We express this function with big-Oh notation
 Example:
 We determine that algorithm arrayMax executes at
most 7n −1 primitive operations
 We say that algorithm arrayMax “runs in O(n) time”
 Since constant factors and lower-order terms
are eventually dropped anyhow, we can
disregard them when counting primitive
operations

Seven Important Functions
 Seven functions
that often appear in
algorithm analysis:
 Constant ≈ 1
 Logarithmic ≈ log n
 Linear ≈ n
 N-Log-N ≈ n log n
 Quadratic ≈ n2
 Cubic ≈ n3
 Exponential ≈ 2n
1E+0 1E+2 1E+4 1E+6 1E+8 1E+10
1E+0
1E+3
1E+6
1E+9
1E+12
1E+15
1E+18
1E+21
1E+24
1E+27
1E+30
Cubic
Qua-
dratic
Linear
n
T(n)
 In a log-log chart,
the slope of the line
corresponds to the
growth rate of the
function

Computing Prefix Averages
 We further illustrate
asymptotic analysis with
two algorithms for prefix
averages
 The i-th prefix average
of an array X is average
of the first (i + 1)
elements of X:
A[i] = (X[0] + X[1] + … + X[i])/
(i+1)
 Computing the array A
of prefix averages of
another array X has
applications to financial
analysis 5
10
15
20
25
30
35
X
A

Prefix Averages (Quadratic)
 The following algorithm computes prefix averages in quadratic time by
applying the definition
Algorithm prefixAverages1(X, n)
Input array X of n integers
Output array A of prefix averages of X
#operations
A ←new array of n integers n
for i ←0 to n −1 do n
s ←X[0] n
for j ←1 to i do 1 + 2 + …+ (n −1)
s ←s + X[j] 1 + 2 + …+ (n −1)
A[i] ←s / (i + 1) n
return A 1

Arithmetic Progression
 The running time of
prefixAverages1 is
O(1 + 2 + …+ n)
 The sum of the first n
integers is n(n + 1) / 2
 There is a simple
visual proof of this fact
 Thus, algorithm
prefixAverages1 runs in
O(n2
) time 0
1
2
3
4
5
6
7
1 2 3 4 5 6

Prefix Averages (Linear)
 The following algorithm computes prefix averages in linear time by
keeping a running sum
Algorithm prefixAverages2(X, n)
Input array X of n integers
Output array A of prefix averages of X
#operations
A ←new array of n integers n
s ←0 1
s ←s + X[i] n
A[i] ←s / (i + 1) n
return A 1
 Algorithm prefixAverages2 runs in O(n) time

More Examples
Algorithm SumTripleArray(X, n)
Input triple array X[][][] of n by n by n integers
Output sum of elements of X #operations
s ←0 1
for j ←0 to n −1 do n+n+…+n=n2
for k ←0 to n −1 do n2
+n2
+…+n2
= n3
s ←s + X[i][j][k] n2
+n2
+…+n2
= n3
return s 1
 Algorithm SumTripleArray runs in O(n3
) time

Useful Big-Oh Rules
f(n) is O(nd
)
f (n)=a0+a1 n+a2 n
2
+...+ad n
d
 If d(n) is O(f(n)) and e(n) is O(g(n)) then
 d(n)+e(n) is O(f(n)+g(n))
 d(n)e(n) is O(f(n) g(n))
 If d(n) is O(f(n)) and f(n) is O(g(n)) then d(n)
is O(g(n))
 If p(n) is a polynomial in n then log p(n) is
O(log(n))

Relatives of Big-Oh
 big-Omega
 f(n) is Ω(g(n)) if there is a constant c > 0
and an integer constant n0 ≥ 1 such that
f(n) ≥ c•g(n) for n ≥ n0
 big-Theta
 f(n) is Θ(g(n)) if there are constants c’ > 0
and c’’ > 0 and an integer constant n0 ≥ 1
such that c’•g(n) ≤ f(n) ≤ c’’•g(n) for n ≥ n0

Intuition for Asymptotic Notation
Big-Oh
 f(n) is O(g(n)) if f(n) is asymptotically less than or
equal to g(n)
big-Omega
 f(n) is Ω(g(n)) if f(n) is asymptotically greater than
or equal to g(n)
 Note that f(n) is Ω(g(n)) if and only if g(n) is O(f(n))
big-Theta
 f(n) is Θ(g(n)) if f(n) is asymptotically equal to g(n)
 Note that f(n) is Θ(g(n)) if and only if if g(n) is O(f(n))
and if f(n) is O(g(n))

Example Uses of the Relatives of Big-Oh
f(n) is Θ(g(n)) if it is Ω(n2
) and O(n2
). We have already seen the former,
for the latter recall that f(n) is O(g(n)) if there is a constant c > 0 and
an integer constant n0 ≥ 1 such that f(n) < c•g(n) for n ≥ n0
Let c = 5 and n0 = 1
 5n2
is Θ(n2
)
f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant n0 ≥ 1
such that f(n) ≥ c•g(n) for n ≥ n0
let c = 1 and n0 = 1
 5n2
is Ω(n)
f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant n0 ≥ 1
such that f(n) ≥ c•g(n) for n ≥ n0
let c = 5 and n0 = 1
 5n2
is Ω(n2
)

 properties of
logarithms:
logb(xy) = logbx + logby
logb (x/y) = logbx - logby
logbxa = alogbx
logba = logxa/logxb
 properties of
exponentials:
a(b+c)
= ab
a c
abc
= (ab
)c
ab
/ac
= a(b-c)
b = a log
a
b
bc
= a c*log
a
b
 Summations
 Logarithms and Exponents
Math you need to Review

Final Notes
 Even though in this course we focus
on the asymptotic growth using big-Oh
notation, practitioners do care about
constant factors occasionally
 Suppose we have 2 algorithms
 Algorithm A has running time 30000n
 Algorithm B has running time 3n2
 Asymptotically, algorithm A is better
than algorithm B
 However, if the problem size you deal
with is always less than 10000, then
the quadratic one is faster
B
A
problem size
Running time

Data Structures and Algorithms Lecture 2: Analysis of Algorithms, Asymptotic notation

Recommended

More Related Content

Similar to Data Structures and Algorithms Lecture 2: Analysis of Algorithms, Asymptotic notation (20)

Recently uploaded (20)

Data Structures and Algorithms Lecture 2: Analysis of Algorithms, Asymptotic notation