psu cs 311 – algorithm design and analysis dr. mohamed tounsi 1 cs 311 design and algorithms...

1PSUCS 311 – Algorithm Design and Analysis Dr. Mohamed Tounsi

CS 311 Design and Algorithms

Analysis

Dr. Mohamed Tounsi

[email protected]


Problems and Algorithms

Abstractly, a problem is just a function

p : Problem instances -> Solutions

Example: Sorting a list of integersProblem instances = lists of integersSolutions = sorted lists of integersp : ` L-> Sorted version of L

An algorithm for p is a program which computes p.There are four related questions which warrant

consideration:


Question 1: Given an algorithm A to solve a problem p, how “good” is A?

Question 2: Given a particular problem p for which several algorithms are known to exist, which is best?

Question 3: Given a problem p, how does one design good algorithms for p?

Problems and Algorithms


Algorithm Analysis

The main focus of algorithm analysis in this course will be upon the “quality” of algorithms already known to be correct.

This analysis proceeds in two dimensions: Time complexity: The amount of time which

execution of the algorithm takes, usually specified as a function of the size of the input

Space complexity: The amount of space (memory) which execution of the algorithm takes, usually specified as a function of the size of the input.

Such analyses may be performed both experimentally and analytically.


Designing Algorithms

It is important to study the process of designing good algorithms in the first place.

There are two principal approaches to algorithm design.

By problem: Study sorting algorithms, then scheduling algorithms, etc.

By strategy: Study algorithms by design strategy. Examples of design strategies include:

Divide-and-conquerThe greedy methodDynamic programmingBacktrackingBranch-and-bound


Algorithm Definition

AlgorithmInput Output

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


Running Time

Most algorithms transform input objects into output objects.

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

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Runnin

g T

ime

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Input Size

best caseaverage caseworst case


Experimental Studies

Write a program implementing the algorithm

Run the program with inputs of varying size and composition

Use a method like times() to get an accurate measure of the actual running time

Plot the results

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Input Size

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e (

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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


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

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 integersOutput maximum element of A

currentMax A[0]for i 1 to n 1 do

if A[i] currentMax thencurrentMax 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 declarationAlgorithm method (arg [, arg…])

Input …

Output …

Method callvar.method (arg [, arg…])

Return valuereturn expression

Expressions Assignment

(like in Java) Equality testing

(like in Java)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

01

2

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)

# operations

currentMax A[0] 2for 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 operationb = Time taken by the slowest primitive operation

Let T(n) be worst-case time of arrayMax. Thena (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

Growth rates of functions:

Linear n Quadratic n2

Cubic n3

In a log-log chart, the slope of the line corresponds to the growth rate of the function

1E+01E+21E+41E+61E+8

1E+101E+121E+141E+161E+181E+201E+221E+241E+261E+281E+30

1E+0 1E+2 1E+4 1E+6 1E+8 1E+10n

T(n

)

Cubic

Quadratic

Linear


Constant Factors

The growth rate is not affected by

constant factors or lower-order terms

Examples 102n 105 is a linear

function 105n2 108n is a

quadratic function

1E+01E+21E+41E+61E+8

1E+101E+121E+141E+161E+181E+201E+221E+241E+26

1E+0 1E+2 1E+4 1E+6 1E+8 1E+10n

T(n

)

Quadratic

Quadratic

Linear

Linear


Big-Oh Notation

Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constantsc 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

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1,000

10,000

1 10 100 1,000n

3n

2n+10

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

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n 2̂

100n

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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 + 53n3 + 20n2 + 5 is O(n3)need c > 0 and n0 1 such that 3n3 + 20n2 + 5 c•n3 for n

n0

this is true for c = 4 and n0 = 21 3 log n + log log n3 log n + log log n is O(log n)need c > 0 and n0 1 such that 3 log n + log log n c•log n

for n n0

this is true for c = 4 and n0 = 2


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 terms2. 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


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

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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 integersOutput array A of prefix averages of X #operations A new array of n integers nfor i 0 to n 1 do n

s X[0] nfor 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 isO(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

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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 integersOutput array A of prefix averages of X #operationsA new array of n integers ns 0 1for i 0 to n 1 do n

s s X[i] nA[i] s (i 1) n

return A 1

Algorithm prefixAverages2 runs in O(n) time


properties of logarithms:logb(xy) = logbx + logby

logb (x/y) = logbx - logby

logbxa = alogbx

logba = logxa/logxb properties of exponentials:

a(b+c) = aba c

abc = (ab)c

ab /ac = a(b-c)

b = a logab

bc = a c*logab

Summations Logarithms and Exponents

Proof techniques Basic probability

Math you need to Review *


Relatives of Big-Oh

big-Omega f(n) is (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 (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

little-oh f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant n0 0

such that f(n) c•g(n) for n n0

little-omega f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n0 0

such that 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 (g(n)) if f(n) is asymptotically greater than or equal to g(n)

big-Theta f(n) is (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 (g(n)) if is asymptotically strictly greater than g(n)


Example Uses of the Relatives of Big-Oh

f(n) is (g(n)) if, for any constant c > 0, there is an integer constant n0 0 such that f(n) c•g(n) for n n0

need 5n02 c•n0 given c, the n0 that satifies this is n0 c/5 0

5n2 is (n)

f(n) is (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 (n)

f(n) is (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 (n2)


Best Worst and Average Case

The worst case complexity of the algorithm is the function defined by the maximum number of steps taken on any instance of size n

The best case complexity of the algorithm is the function defined by the minimum number of steps taken on any instance of size n

Each of these complexities defines a numerical function time vs size


Best Worst and Average Case (cont.)


Exact Analysis is Hard !!

We have agreed that the best, worst and average case complexity of an algorithm is a numerical function of the size of the instances

However it is difficult to work with exactly because it is typically very complicated

Thus it is usually cleaner and easier to talk about upper and lower bounds of the function

This is where the big O notation comes in


Formalization of the Concept of Order

The next task is to formalize the notion of one function being more difficult to compute than another, subject to the following assumptions:

The argument n defining the instance size is sufficiently large

Positive constant multipliers are ignored.


Formalization of Concepts (cont)

Definition Let f : Complexity functions will never be negative

for usable arguments. In any case, behavior before no is not

significant for the asymptotic mathematical analysis.


Big O Notation

Definition Let f :

It is said that “g is big-oh of f ”.

We write g = O( f ) or g in O( f )

The intuition is that g is “smaller” than f ; i.e., that g represents a lesser complexity.


, and

The value of n0 shown is the minimum possible value (any greater value would also work


Sequential Searchvoid seqsearch(int n, const keytype S [ ], keytype x, index & location)

{location = 1;while (location <= n && S[location] != x) location ++; if (location > n) location=0;}

T(n) = N


Binary Search

void binsearch (int n, const keytype S[], keytype x, index& location){ index low, high, mid;

low = 1; high = n; location = 0; while (low < = high && location = = 0){ mid = ∟(low + high)/2 ;⌋ if (x = = S[mid]) location = mid; else if (x < S[ mid ]) high = mid - 1; else low = mid + 1; }}


Matrix Multiplication

void matrixmult (int n, const number A[][], const number B[][], number C[][])

{index i, j, k;

for (i=1; j<= n; j++) for (j=1; j<= n; j++) {

C[ i ] [ j ] = 0; for (k=1; k<= n; k++)

C[ i ] [ j ] = C[ i ] [ j ] + A [ i ] [ k ] * B [ k ] [ j ];

}}


Exchange Sort

void exchangesort (int n, keytype S[])

{ index i, j; for (i=1;i<= 1; i++) for (j=i+l; j<= n; j++) if (S[j] < S[i]) exchange S[i] and S[j];}


Add Array Members

number sum (int n, const number S[ ])

{ index i; number result; result = 0; for (i = 1; i <= n; i++) result = result + S[i]; return result;}

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