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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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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)

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Text Books Data Structures and Algorithm Analysis in C Mark Allen Weiss Data Structures Using C Yedidyah Langsam

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Prerequisites Knowledge 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 29 th January

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Cumulative Final: 40% 1 Midterm: 30% (last class before spring break)

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Course Content Course 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

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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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Sequence 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 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

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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 Foundation Series and summation: 1 + 2 + 3 + ……. N = N(N+1)/2 (arithmetic series) 1 + r 2 + r 3 +………r N-1 = (1- r N )/(1-r), (geometric series) 1/(1-r), r < 1, large N Sum of squares: 1 + 2 2 + 3 2 +………N 2 = N(N + 1)(2N + 1)/6

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Properties of a log Function log x a = 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 a b = b log a log b a = log c a/ log c b a log n = n log a

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a mn = (a m ) n = (a n ) m a m+n = a m a n (2 n) 0.5 (n/e) n n (2 n) 0.5 (n/e) n + (1/12n)

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We will deduce the value of the descending geometric series: N + Nr + Nr 2 + Nr 3 +………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 Numbers Sequence of numbers, F 0 F 1, F 2, F 3,……. F 0 = 1, F 1 = 1, F 2 = 2, F 3 = 3,……. F i = F i-1 + F i-2,

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Will prove that F N < N = (1+sqrt(5))/2 i=1 N-2 F i = F N - 2 N > 2 (Correction) Holds for N = 1 Let it hold for 1,2,….N F N+1 = F N + F N-1 < N + N-1 = N-1 (1 + ) = N-1 2 = N+1

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Proof By Counter Example Want to prove something is not true! Give an example to show that it does not hold! Is F N N 2 ? No, F 11 = 144 However, if you were to show that F N N 2 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 P 1, P 2, ……… P k Z = P 1 P 2 ……… P k + 1 Z is not divisible by P 1, P 2, ……… P k Z is greater than P 1, P 2, ……… P k 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.

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