Brian Mitchell - Drexel University MCS680-FCS 1 Brian Mitchell

Brian Mitchell ([email protected]) - Drexel University MCS680-FCS

1

Brian Mitchellhttp://www.mcs.drexel.edu/~bmitchel

email: [email protected] University

Fall 1997

MCS680:Foundations Of

Computer Science

int MSTWeight(int graph[][], int size){

int i,j;int weight = 0;

for(i=0; i<size; i++)for(j=0; j<size; j++)

weight+= graph[i][j];

return weight;}

1

1

nn

O(1)

O(1)

O(n) O(n)

Running Time = 2O(1) + O(n2) = O(n2)


2

Introduction

• Instructor Brian Mitchell - “Brian”

[email protected]/~bmitchel (215)895-2668

• Course Information Foundations of Computer Science (FCS)

MCS 680 Section 510Wednesday 6:00 - 9:00 PMRoom Location: Curtis 250AOffice Hours By Appointment

• Online Information www.mcs.drexel.edu/~bmitchel/course/mcs680-fcs

Please check course web page several times per week. The web page will be my primary mechanism for communicating:

Special and Emergency Information Syllabus Assignments Solutions to Problems


3

Introduction

• Course Objective– To provide a solid background in the

theoretical and mathematical aspects of Computer Science

• Provide essential background knowledge that is required for success in other core computer science graduate courses

• Textbook A.V.Aho, J.D.Ullman, Foundations of

Computer Science, W.H. Freeman and Co., 1995.

• Additional References H.R. Lewis, C.H. Papadimitriou, Elements

Of The Theory Of Computation, Prentice Hall, 1981.

T.H. Cormen, C.E. Leiserson & R.L. Rivest, Introduction To Algorithms, McGraw Hill, 1996.


4

Introduction

• Homework, Exams Assignments & Programming Projects:

Homework sets consisting of 4-5 problems will be regularly assigned (consult web page). Based on difficulty, you will have 1-2 weeks to complete each assignment. Some assignments might require you to write a small program or a program fragment.

For this course you may develop programming solutions using the ‘C’, ‘C++’ or Java programming languages

Midterm: A 90 minute midterm exam will be given during the 5th or 6th week of the course.

Final: A 2 hour final exam will be given during our regularly scheduled class time on finals week.


5

Introduction

• Grading– The following distribution will be used to

determine your final grade in this course:• 50%: Homework & Programming Projects• 20%: Midterm Exam• 30%: Final Exam

• Policies– All homework and programming assignments

are individual efforts, unless specifically stated otherwise in the assignment definition.

• You may use your colleagues for advice, however, all assignments must be your original work

– Late assignments will be penalized 10% per week. Any assignment not submitted within 2 weeks of the deadline will not be accepted unless you work out special arrangements with me.


6

Introduction

• Tentative List Of Topics– Summation and Logarithm Review– Ineration, Induction & Recursion– Big-Oh Analysis – Running Time– Recurrence Relations– Trees (Mathematical Aspects & Algorithms)– Sets– Graph Theory & Graph Algorithms– Relations– Automata – Regular Expressions– Context Free Grammars– Propositional Logic– Predicate Logic


7

Summations

• Algorithms often contain an interative control construct– Loops (while, for, do…while)– Algorithm analysis can be performed by examining

the sum of the times spent on each execution of the loop

– Example: What is the value of i?

n

j=1

j = 1+2+…+(n-1)+n

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

}


8

Summations

• Properties of sums

aj = n

j=1aj

j=1

limn

If the limit does not exist, the sum diverges; otherwise, itconverges.

n

j=1caj + dbj =

n

j=1aj c + d bj

n

j=1

Summations obey the linearity property


9

Common Summations

n

j=1j =

n(n+1)2

n

j=0xj =

xn+1 - 1x-1

n

j=1aj - aj-1 = an - a0

n-1

j=0aj - aj+1 = a0 - an

n

j=1c = n • c


10

Summation Example

• Consider:int foo(int n){int i,j,k;for (i=1; i<=n; i++)for(j=1; j<=n; j++)

k += i + j;}

How much time is spent performing the addition operationsin the above code fragment if an add operation takes 2clock cycles on a Pentium 120Mhz microprocessor?

n

i=1

n

j=1

addOps = n • addOps

n

i=1

= n2 • addOps

On a 120Mhz microprocessor each clock cycle takes8.3333 ns. (10-9 seconds). Thus the total amount of timethe code fragement spends adding is:

= n2 • (2 • 8.3333E-9) = 16.667n2 ns.


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Summation Example-Results

Execution Time For Addition Operations

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

n

Exec

utio

n Ti

me

(sec

onds

)


12

Summation Example

int foo(int n){int i,j,k;for (i=1; i<=n; i++)for(j=1; j<=n; j++)

k += (i + j);}

What is the value of k?

)1(

))1((2

)1(2

2)1(

2)1(

2)1(

2

11 1

nn

nnnnnn

nnnnnn

nnnaaan

ii

n

i

n

jji


13

Summation Example

int foo(int n){int i,j,k;for (i=1; i<=n; i++)for(j=1; j<=i; j++)

k += (i + j);}

What is the value of k?

)1(2

)1(2

)1(2

)1(

2)1(

11 1

nnn

nnnnn

iiiaaan

ii

n

i

i

jji


14

Logarithms

• Logarithms appear often during the analysis of algorithms– General form: logb x

• b is the base of the logarithm• if b is not specified then 10 is assumed

– Definition: The logarithm to the base b of x represents the power to which b must be raised to produce x

– Examples: log2 8 = 3 (since 23 = 8)log3 81 = 4 (since 34 =

81) log10 1 = 0 (since 100 =

1)– During the analysis of algorithms, log2 x

appears often• We will use lg x to represent log2 x


15

Logarithms

• Notation– lg n = log2 n = (log n / log 2)

– ln n = loge n

• Identities– logb 1 = 0

– logbk n = (logb n)k

– logb logb n = logb (logb n)

– logb (a/c) = logb a - logb c

– logb n = (1/ logn b)

– logc n = (logarb n / logarb c) - c is any base

– logb (1/n) = - logb n

– logb nr = r logb n

– logb nj = logb n + logb j– a = blogb a

– alogb n = nlogb a


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Logarithms

• Logarithmic algorithms grow slowly with respect to the amount of input data:– Bubble sort growth n2

– Quick sort growth n lg n

Comarision of Execution Complexity

0

50

100

150

200

250

300

350

1 3 5 7 9 11 13 15 17

Sample Size

Exec

utio

n C

ompl

exity

Sample Size (x)x^2x lg x


17

Data Models, Data Structures & Algorithms

• Data Models– Abstractions used to formulate problems– Any mathematical concept can be treated

as a data model• Values that the data objects can assume• Operations on the data

– Programming languages (‘C/C++’, Pascal, Java...) support primitive data models

• Integers, floating point, strings, chars, ...• Operations: +, -, /, *, %, <, >, <=, >=, == ...• However there are other important data

models– Sets, graphs, trees, expressions– These data models are mathematically

understood, however, most programming languages do not provide intrinsic support


18

Data Models, Data Structures & Algorithms

• Data Structures• Most programming languages do not

directly support important data models– Must represent the needed data model by

using abstractions that are supported by the language

• Data structures are methods for representing a data model in a programming language

• Consider the following weighted and directed graph:

A

B

C

D


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Data Structures

A

B

C

D

This graph can be represented in a table:

ABCD

A0001

B4000

C3200

D0600

1

6

4

3

2

A table can be represented in a computer language as atwo-dimensional array:

int graph[4][4];...

graph[0][1] = 4;graph[0][2] = 3;graph[1][2] = 2;graph[1][3] = 6;graph[3][0] = 1;


20

Algorithms

• Precise an unambiguous specification of a sequence of steps that can be carried out automatically

• In Computer Science, algorithms are expressed using programming languages

• Consider an algorithm for calculating the total weight of a graph.– Our data structure is a two-dimensional graph

int GraphWeight(int graph[][], int size){int i,j,weight;

weight = 0;for(i=0; i<size; i++)for(j=0; j<size; j++)weight+= graph[i][j];

return weight;}


21

Running Times Of Algorithms

• Desire to analyze algorithm running time• Used to determine how “good” an

algorithm performs– Inputs of various sizes– Best Case– Worst Case– Average Case– Bounding Performance

• We use “Big-Oh” analysis to model running time (also known as execution complexity)– O(n)– O(n2)– O(2n)– O(n log n)– O(n lg n)

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Brian Mitchell - Drexel University MCS680-FCS 1 Brian Mitchell