CSE 220 (Data Structures and Analysis of Algorithms) Instructor: Saswati Sarkar ([email protected]) T.A. Dimosthenes Anthomelidis ([email protected])

CSE 220 (Data Structures and Analysis of Algorithms)

Instructor: Saswati Sarkar ([email protected])

T.A. Dimosthenes Anthomelidis ([email protected])

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

Timings

Class 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)

Text Books

Data Structures and Algorithm Analysis in C

Mark Allen Weiss

Data Structures Using C

Yedidyah Langsam

Prerequisites

Knowledge of C

CSE 260?

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

Cumulative Final: 40%

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

Course ContentCourse Motivation

Mathematical Foundation:

Complexity Analysis:

Data Structures: List, Stacks, Queues

Algorithm: 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

2 review lectures

No class on April 25

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

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

Algorithms

Sequence of actions a ``dumb’’ machine can follow

For j=1 to N

print j

Efficient versus Inefficient algorithm

Linear Search Vs Binary Search

Design of Algorithms

You 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

Formal Study of Algorithms

Design of Algorithms

Proving Correctness of Algorithms

Formal study of efficiency of algorithms

Run time

Storage required

Mathematical Foundation

Series 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

Properties of a log Function

logxa = b if xb = 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

amn = (am )n = (an)m

am+n = am an

(2n)0.5 (n/e)n n (2n)0.5 (n/e)n + (1/12n)

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

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.

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.

Fibonacci Numbers

Sequence of numbers, F0 F1 , F2 , F3 ,…….

F0 = 1, F1 = 1, F2 = 2, F3 = 3,…….

Fi = Fi-1 + Fi-2 ,

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

i=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

Proof By Counter Example

Want 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.

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!

Suppose there are finite number of primes, k primes.

(we do not include 1 in primes here)

We know k2.

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.