Chapter 3

Chapter 3: Analysis Tools

Analysis.pdf

An algorithm is a step-by-step procedure for solving a problem in a finite amount of time.

Most algorithms transform input objects into output objects.

Experimental Studies

Write a program implementing the algorithm
Run the program with inputs of varying size and composition
Use a function, like the built-in clock() function, to get an accurate measure of the actual running time
Plot the results

Limitations of Experiments

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

Running Time

The running time of an algorithm typically grows with the input size.
Average case time is often difficult to determine.
We focus on the worst case running time.

Easier to analyze
Crucial to applications such as games, finance and robotics

Theoretical Analysis

Uses a high-level 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

Pseudocode

A program code unrelated to the hardware of a particular computer and requiring conversion to the code used by the computer before the program can be used. No strict syntax rules, designed for humans, not for computers.

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

Example: find max element of an array

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

Pseudocode Details

Control flow

if…then…[else…] while…do… repeat…until… for…do…
Indentation replaces braces

Method declaration

Algorithm method(arg[, arg…])
Input…
Output…

Method/Function call

var.method (arg[, arg…])

Return value

return expression

Expressions

<- Assignment (like = in C++)
= Equality testing (like == in C++)
n² Superscripts and other mathematical formatting allowed

The Random Access Machine - (RAM) Model

A CPU
An 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.

Primitive Operations

Basic computations performed by an algorithm
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

Examples:

Evaluating an expression

Assigning a value to a variable

Indexing into an array

Calling a method

Returning from a method

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.

Algorithm arrayMax(A, n)	Number of operations
`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)

The linear growth rate of the running time T(n) is an intrinsic property of algorithm arrayMax

Growth Rates

Function / n	1	2	10	100	1000
5	5	5	5	5	5
log n	0	1	3.32	6.64	9.96
n	1	2	10	100	1000
n log n	0	2	33.2	664	9996
n²	1	4	100	10⁴	10⁶
n³	1	8	1000	10⁶	10⁹
2ⁿ	2	4	1024	10³⁰	10³⁰⁰
n!	1	2	3628800	10¹⁵⁷	10²⁵⁶⁷
nⁿ	1	4	10¹⁰	10²⁰⁰	10³⁰⁰⁰

Constant Factors

The growth rate is not affected by

constant factors or
lower-order terms

Examples:

10²n+10⁵is a linear function
10⁵n²+10⁸n is a quadratic function

Asymptotic Notation

Big-Oh

Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constants c and N > 0 such that f(n) < c g(n) for n > N.

Example 1: 2n +10 is O(n)
2n +10 ≤ cn; (c − 2) n ≥10; n ≥ 10/(c − 2)
Pick c = 3 and N = 10.

Example 2: the function n² is not O(n)
n² ≤ c n; n ≤ c
The above inequality cannot be satisfied since c must be a constant

Example 3: 7n - 2 is O(n)
need c > 0 and N ≥ 1 such that 7n - 2 ≤ c n for n ≥ N
this is true for c = 7 and N = 1

Example 4: 3n³+ 20n²+ 5 is O(n³)
need c > 0 and N ≥ 1 such that 3n³+ 20n²+ 5 ≤ cn³ for n ≥ N
this is true for c = 4 and N = 21

Example 5:
3 log n + log log n is O(log n)
need c > 0 and N ≥ 1 such that 3 log n + log log n ≤ c log n for n ≥ N
this is true for c = 4 and N = 2

Big-Oh Rules

If is f(n) a polynomial of degree d, then f(n) is O(n^d), 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:

1. We find the worst-case number of primitive operations executed as a function of the input size
2. 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

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.

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 integersOutput array A of prefix averages of XA <- new array of n integersfor i <- 0 to n−1 do s <- X[0] for j<- 1 to i do s<- s+X[j] A[i] <- s/(i+1)return A #operations
nnn
1 + 2 + …+(n−1)
1 + 2 + …+(n−1)
n
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.
Thus, algorithm prefixAverages1 runs in O(n²) time.

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 A <- new array of n integers s <- 0 for i <- 0 to n−1 do s <- s + X[i] A[i] <- s/(i + 1) return A #operationsn1nnn
1

Algorithm prefixAverages2 runs in O(n) time

Relatives of Big-Oh

Big-Omega
f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant N > 0 such that f(n) > c g(n) for n > N.

Big-Theta
f(n) is Θ(g(n)) if there are constants c' > 0 and c" > 0 and an integer constant N > 0 such that c'g(n) < f(n) < c"g(n) for n > N.

little-oh
f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant N > 0 such that f(n) < c g(n) for n > N.

little-omega
f(n) is ω(g(n)) if, for any constant c > 0, there is an integer constant N > 0 such that f(n) < c g(n) for n > N.

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)

big-Theta
f(n) is Θ(g(n)) if f(n) is asymptotically equal to g(n)

little-oh
f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n)

little-omega
f(n)) is ω(g(n)) if f(n) is asymptotically strictly greater than g(n)

Example Uses of the Relatives of Big-Oh

Example 1: 5n² is Ω(n²)
f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant N ≥ 1 such that f(n) ≥ c g(n) for n ≥ N
let c = 5 and N = 1

Example 2: 5n²is Ω(n)
f(n) is Ω(g(n)) if there is a constant c > 0 and an integer constant N ≥ 1 such that f(n) ≥c g(n) for n ≥ N
let c = 1 and N = 1, 5n²≥ n

Example 3: 5n²is ω(n)
f(n) is ω(g(n)) if, for any constant c > 0, there is an integer constant N ≥ 0 such that f(n) ≥c g(n) for n ≥ N
need 5N² ≥ cN - given c, then N that satisfies N ≥ c/5 ≥ 0