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.
The running time of an algorithm typically grows with the input size.

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 (for fixed size) 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 T(n) 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

Pseudocode is a program code, unrelated to the hardware of a particular computer.
It requires 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 n.

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

effects 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 the algorithm arrayMax.

Growth Rates

Function / n	1	2	10	100	1000
1	1	1	1	1	1
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
n^k	1	2^k	10^k	100^k	1000^k
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 (n)
10⁵n²+10⁸n is a quadratic function (n²)

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 (recommendation and practice).

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.