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CSE 220 (Data Structures and

Analysis of Algorithms)

Instructor: Saswati Sarkar (swati@ee.upenn.edu)

T.A. Dimosthenes Anthomelidis

(anthomel@gradient.cis.upenn.edu)

Course Web Page:

http://www.seas.upenn.edu/~swati/cse220.html

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TimingsClass MWF: 12-1, 224 Moore

Instructor Office Hours: 11-12, Friday, 360 Moore (correction)

1-2 Wednesday 360 Moore

T.A. Office Hours: Monday 11-12, Friday 1-2, 329 Pender

Recital: Wednesday 5-6 (Moore 216)

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Text BooksData Structures and Algorithm Analysis in C

Mark Allen Weiss

Data Structures Using C

Yedidyah Langsam

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PrerequisitesKnowledge of C

CSE 260?

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Grading

Weekly Homeworks (30%)

Posted every Monday

Due Next Monday before class (solution posted after

class)

No consultation allowed

No late submission accepted

First Homework will be posted on 29th January

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Cumulative Final: 40%

1 Midterm: 30% (last class before spring break)

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Course Content

Course MotivationMathematical Foundation:

Complexity Analysis:

Data Structures: List, Stacks, QueuesAlgorithm: Searching and Trees

Sorting and Heaps

Graph Algorithms (Depth first Search,

Breadth First Search, Topological sort,

Shortest path algorithm, Spanning Tree

Algorithm)

Computability and Complexity

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2 review lectures

No class on April 25

No class on either April 23 or April 27 (will confirm

later)

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Course Motivattion

Need to run computer programs efficiently!

Computer program:

Accepts Input (Data)Performs a Sequence of action with the

input

Generates Output (Data)

Efficient Management of Data

(Data Structures)

Efficient Sequence of Actions

Algorithms

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AlgorithmsSequence of actions a ``dumb machine can follow

For j=1 to N

print j

Efficient versus Inefficient algorithm

Linear Search Vs Binary Search

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Design of AlgorithmsYou have a problem to solve

Design an efficient algorithm

Use good data structures

Show that your algorithm works!

Prove its correctness

Study the efficiency of your algorithm

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Formal Study of Algorithms

Design of Algorithms

Proving Correctness of Algorithms

Formal study of efficiency of algorithms

Run time

Storage required

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Mathematical FoundationSeries and summation:

1 + 2 + 3 + . N = N(N+1)/2 (arithmetic series)

1 + r2 + r3 +rN-1 = (1- rN)/(1-r), (geometric series)

$ 1/(1-r) , r < 1, large N

Sum of squares:

1 + 22 + 32 +N2 = N(N + 1)(2N + 1)/6

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Properties of a log Functionlogxa = b if x

b = a

(we will use base 2 mostly, but may use

other bases occasionally)

Will encounter log functions again and again!

log n bits needed to encode n messages.

log (ab ) = log a + log b

log (a/b ) = log a - log b

log ab = b log a

logba = logca/ logcb

alog n = nlog a

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We will deduce the value of the descending geometric

series:

N + Nr + Nr2 + Nr3 +1

Example: N + N/2 + N/4 + + 1

The summation equals (N-r)/(1-r)

For r = 2, this is 2N - 1

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Proof By Induction

Prove that a property holds for input size 1

Assume that the property holds for input size

1,2,n. Show that the property holds for input

size n+1.

Then, the property holds for all input sizes, n.

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Prove that the sum of 1+2+..+n = n(n+1)/2

Holds for n = 1

Let it hold for 1,2..n

1+2+.+n = n(n+1)/2

1+2+.+n+(n+1) = n(n+1)/2 + (n+1)

= (n+1)(n+2)/2.

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Fibonacci NumbersSequence of numbers, F0 F1 , F2 , F3 ,.

F0 = 1, F1 = 1, F2 = 2, F3 = 3,.

Fi = Fi-1 + Fi-2 ,

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Will prove that FN < *N * = (1+sqrt(5))/2

7i=1N-2 Fi = FN - 2 N > 2 (Correction)

Holds for N = 1

Let it hold for 1,2,.N

FN+1 = FN + FN-1

< *N + *N-1

= *N-1 (1 + * )

= *N-1*2

= *N+1

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Proof By Counter ExampleWant to prove something is not true!

Give an example to show that it does not hold!

Is FN N2 ?

No, F11 = 144

However, if you were to show that FN N2 then you

need to show for all N, and not just one number.

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Proof By Contradiction

Suppose, you want to prove something.

Assume that what you want to prove does not hold.

Then show that you arrive at an impossiblity.

Example: The number of prime numbers is not finite!

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Suppose there are finite number of primes, k primes.

(we do not include 1 in primes here)

We know k"2.

Let the primes be P1, P2 , Pk

Z = P1 P2 Pk + 1

Z is not divisible by P1, P2 , Pk

Z is greater than P1, P2 , Pk Thus Z is not a prime.

We know that a number is either prime or divisible by primes.

Hence contradiction.

So our assumption that there are finite number of primes is

not true.