Algorithm Analysis
CS 122: Data Structures I
Spring 2013
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Cs122 algorithmanalysis, Lecture notes of Data Structures and Algorithms
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Algorithm Analysis^ CS 122: Data Structures I
Spring 2013
Reading
•^
Chapter 4 [Goodrich 2011].
-^
Chapter 2 [Weiss 2005]. Spring 2013
Data Structures (1)
Introduction
•^
What to analyze?^ –
Running time.
Space.
-^
For an input of size N, we define:
-^
For an input of size N, we define:^ –
T
avg
(N): the average running time.
T
worst
(N): the worst running time.
T
avg
(N) ≤ T
worst
(N).
•^
For more than one input, T
avg
(N) and T
worst
(N) will
have more than one argument. Spring 2013
Data Structures (1)
Outline
•^
Introduction
•^
Common Functions to Represent Growth Rates.
-^
Asymptotic Notation.
-^
Runtime calculation.
-^
Runtime calculation. Spring 2013
Data Structures (1)
The Constant Function
f(n) = c
•^
c is a fixed constant.^ –
Examples:
c = 5, c = 27, or c = 2
10
•^
Independent of the value of n.
-^
Independent of the value of n.
-^
Use Cases (the number of steps needed to do a basicoperation)^ –
adding two numbers.
assigning a value to some variable.
comparing two numbers.
-^
Appears as: f(n) = 1 and f(n)= c g(n). Spring 2013
Data Structures (1)
The Logarithmic Function
f (n) = log
b^
n, for some constant b > 1
x = log
b^
n if and only if b
x^
= n.
•^
Approximate calculation:
log
a^
n
equals the number of
times
n
is divided by
a
to get a number less than or
equal to
. Examples: -
log
27 = 3, 27 / 3 / 3 / 3 = 1.
log
27 = 3, 27 / 3 / 3 / 3 = 1. 3
log
64 = 3, 64 / 4 / 4 / 4 = 1. 4
log
12 = 4, 12 / 2 / 2 / 2 / 2 = 0.75 ≤ 1. 2
•^
Algorithms: base-2 logarithm appears in algorithmsthat keeps dividing input by 2 in every iteration.
-^
log
c n
is used to denote the function (log
)n c^
(Log
squared, c=2) Spring 2013
Data Structures (1)
The Linear Function
f(n) = n
•^
For an input of size n, a simple basic operation ispreformed for every input value.
-^
Example, comparing a number x to each element of
-^
Example, comparing a number x to each element of an array of size n requires n comparisons.
linear
search.
-^
It represents the best running time that can beachieve for any algorithm that processes a collectionof n objects. Spring 2013
Data Structures (1)
The N-Log-N Function
f(n) = n log n
•^
For an input of size n, the operations performed bythe algorithm are equal to the value of n times the logarithm base 2 of n.logarithm base 2 of n.
-^
A linear algorithm is better than an algorithm thatruns in n log n time.
-^
An n-log-n algorithm is better than a quadraticalgorithm.
-^
Example: merge sort. Spring 2013
Data Structures (1)
Nested Loop and the Quadratic
Function
•^
Consider the following nested loop:^ –
the first iteration of a loop uses one operation,
the second uses two operations,
the third uses three operations,
and so on.
-^
Number of operations: 1+2+3+…+(n-1)+n = n(n+1)/2.
-^
n(n+1)/2 = n
2
/2 + n/
n
2
, quadratic.
Spring 2013
Data Structures (1)
The Cubic Function
f(n) = n
3
•^
For an input value n, the function f assigns theproduct of n with itself three times (“n cubed”).
-^
Polynomials
: f (n) = a
+a 0
n+a 1
n 2
2 +a
n 3
3
ad
n
d
•^
Polynomials
: f (n) = a
+a 0
n+a 1
n 2
+a
n 3
ad
n
a
, a 0
, a 1
, a 2
, ···, a 3
are the coefficients.d
a
d^
≠ 0.
d is the highest power in the polynomial
degree of the polynomial.
Spring 2013
Data Structures (1)
Comparing Growth Rates of
Functions
Spring 2013
Data Structures (1)
The Ceiling and Floor Functions
•^
The value of a logarithm is typically not an integer.
-^
The running time of an algorithm is usually expressedby means of an integer quantity, such as the number of operations performed.of operations performed.
-^
The analysis of an algorithm involves using thefollowing functions:^ –
Floor:
the largest integer less than or equal to x (≤ x).
Ceiling:
the smallest integer greater than or equal to x
(≥ x).
Spring 2013
Data Structures (1)
^ x x^ ^
Asymptotic Complexity
-^
In algorithm analysis, we focus on the growth rate of therunning time as a function of the input size n.
-^
We are more concerned about large values of n.
-^
The calculated function of the complexity gives an approximate measure of efficiency. This is why we call itapproximate measure of efficiency. This is why we call it asymptotic complexity.
-^
Example: Spring 2013
Data Structures (1)
f(n) = n
2 + 100n+ log
n^
- 1000
n<
1000: largest contribution.
n=
1000 , 100n: equal contribution.
n=
(^2) n and 100n: equal contribution..
n>10 0
(^2) n : highest contribution.
Big-Oh
•^
Given: functions f(n) and g(n), a constant c > 0, andn
0
•^
f(n) = O(g(n))
f(n) ≤ c g(n) for all n ≥
n
0
f(n) ≤ c g(n) for all n ≥
n
0
•^
Example: f(n) = 1000n, g(n) = n
1000n ≤ n
2 , n ≥ 1000.
Spring 2013
Data Structures (1)