Midterm CSE 21 Spring 2012

Transcription

1 Signature Name Student ID Midterm CSE 21 Spring 2012 Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 _ (20 points) _ (15 points) _ (13 points) _ (23 points) _ (10 points) _ (8 points) Total _ (89 points) (84 points = 100%) ( 5 points Extra Credit = 6%) This exam is to be taken by yourself with closed books, closed notes, no electronic devices. You are allowed one side of an 8.5 x11 sheet of paper handwritten by you. 0

2 Calculate the first 6 terms for the sum of the first n natural numbers (n = 1, n = 2, n = 3,..., n = 6). Then to the right calculate the sequence of differences between these terms. The first natural number is 1 (not 0). You may not need all the slots on the right. n Sequences of differences 1 = 2 = 3 = 4 = 5 = 6 = Write the recurrence relation for the sum of the first n natural numbers T(n). { T(n) = if n if n What is for the closed-form solution to the recurrence relation above? f(n) = Verify this with a proof by induction. Prove T(n) = f(n) for all n _. Proof (Induction on n): : If n =, the recurrence relation says T( ) =, and the closed-form solution says f( ) = _ =, so T( ) = f( ). : Suppose as inductive hypothesis that T( ) = for some k. : Using the recurrence relation, T(k) =, by 2 nd part of RR =, by IHOP = = So, by induction, T(n) = for all (as ). 1

3 Which general recursive decompositions discussed in class are most appropriate for the following algorithms: Factorial Binary Search _ Palindrome Fractal graphic _ Check output (for example, Koch snowflake or tree fractal) List/String reversal moving last element to front, recurse on rest (sa) R a(s) R List/String reversal exchanging first and last elements, recurse on rest Which is true about proof by induction on k (IHOP on k and IS on k+1) vs proof by induction on k-1 (IHOP on k-1 and IS on k)? A) Both the same with the same restriction on k relative to base case value B) First (on k) uses weak induction and second (on k-1) uses strong induction C) Completely different proofs (not the same) D) First (on k) uses strong induction and second (on k-1) uses weak induction E) Both the same with different restrictions on k relative to base case value In a typical algorithm that compares different array elements with each other, how many pairs of array elements are there to compare in an array of n elements? (Same as the number of ways to choose elements from the set {0, 1,, n-1}.) Give the answer in combinatorics notation in terms of n. Give your answer in terms of n not in combinatorics and not using factorials. Given the following recurrence relation, write the actual body of code for the function T(n). Just write the code for the base case and recursive case (do not worry about anything else). T(n) = { 5 if n = 0 T(n 1) + n 5 if n > 0 long long int T( int n ) { // Assume n >= 0 } How many different strings can be formed by rearranging the letters in NONSENSELESSNESS, using all the letters? What is the probability of rolling a 4, 5, 6, 7, 8, 9, 10, or 11 (sum of two fair 6-sided dice will be a 4-11)? (give answer as a reduced fraction) 2

4 How many different strings of length 13 can be formed from a set of 26 refrigerator magnets A-Z? How many strings of length 5 can be formed from a 15-symbol alphabet where no 2 adjacent symbols are the same? What is the value of P(6,2)? Your answer should be an actual number for this one. _ What is the value of C(7,3)? Your answer should be an actual number for this one. _ A certain algorithm processes a list of n elements. Suppose that Subroutine a requires 8n n operations and Subroutine b requires 90n operations. Give a big-theta estimate for the number of operations performed by the following pseudocode segment. for i {1, 2,..., n } do { Subroutine a Subroutine b } Big-Oh provides a(n) bound on the growth rate of a function while big-omega provides a(n) bound on the growth rate of a function. Given K 1 f(n) g(n) K 2 f(n), we say that "g is big- of f" or "g is order f." K 1 f(n) represents the big- of f while K 2 f(n) represents the big- of f. With respect to the graph to the right, A the function labeled represents big-omega of f and the function labeled represents big-oh of f. B There are 47 students pulling an all-nighter in the CSE basement labs to finish their CSE assignments due the next day. There are 33 CSE 21 assignments being worked on and 21 CSE 30 assignments being worked on. How many students did not start early enough and are working on both assignments? _ 3

5 Match the time complexity class names in the box to the right with their big-oh equivalent. O( n ) O( n 2 ) O( 1 ) O( log 2 n ) O( n! ) O( 2 n ) A) polynomial E) linear B) exponential F) logarithmic C) factorial G) n log 2 n D) constant H) super exponential Now rank the different time complexity classes (A-H) from smallest to largest according to how fast they grow as their input n grows large. smallest/slowest largest/fastest growing (faster algorithms) growing (slower algorithms) How many 7-digit phone numbers are possible within an area code? Phone numbers can contain all zeros thru all nines. How many 7-digit phone numbers have all different digits (no duplicates)? How many 7-digit phone numbers contain only odd digits? How many 7-digit phone numbers have at least one even digit? How many 7-digit phone numbers start with an odd digit and end with an even digit? How many 7-digit phone numbers can be formed with at least one duplicate digit (for example, , , and )? The easiest way to solve the previous question is to use the technique called There are 11 customers and 3 cashiers. How many ways can the customers line up to the cashiers, if the order of each line does not matter. Rick has 38 comics to show in the last 7 lectures for CSE 21. If he distributes the comics evenly across the lectures, this guarantees that at least one lecture will have how many comics? 4

6 An urn contains 10 balls numbered 0-9. Six balls are drawn from the urn in sequence, and the numbers on the balls are recorded. How many ways are there to do this, if when each ball is drawn it is not replaced? each ball is replaced before the next one is drawn? all six balls are drawn at once (one handful of six balls)? _ How many four-digit binary strings is there that do not contain 101 or 010? First draw a decision tree. Each slot/line should have a single 0 or 1. 0s to the left and 1s to the right at each split. How many such four-digit binary strings that do not contain 101 or 010? There are 38 possible outcomes in American roulette (1-36 and 0 and 00). The numbers 1-36 alternate between red and black while 0 and 00 are green. What is the probability of a roulette ball landing in a red slot? P(X = red) = _ The payout for a bet on red is 1-to-1 (for example, $1 bet pays $1 + the original $1 bet for a total of $2), the Expected Value of the amount of money you will win (pull off the table) in terms of P(X=x) is E(X) = 2 * P(X = red) + 0 * P(X!= red) Now replace the P(X=x) values with their numeric probabilities keeping your answer in terms of fractions vs. decimals. Reduced fractions are preferred. E(X) = 2 * + 0 * = If your bet is $1 (costs you $1 to play), what is your expected return each time you make this kind of bet? Express your answer as a positive or negative reduced fraction. E(X) - 1 = 5

7 If you walk up to an American roulette table and see there have been 15 consecutive red winners before you got there, and you decide to place a bet on black, what is the probability of the roulette ball landing in a black slot for this bet? Match the person to what the person is famous for. (1/2 point each) Invented Quicksort algorithm. Proved that what is now known as the halting problem is undecidable. Coined the term "artificial intelligence." Main notation for expressing context-free grammars in programming. Helped popularize the term "debugging." null reference self-described as a billion-dollar mistake. Shortest-Path algorithm. Invented Merge sort algorithm. Co-authored Concrete Mathematics (a blend of CONtinuous and discrete math) with Ron Graham. "Software is getting slower more rapidly than hardware becomes faster." First Turing Award winner. A case against the goto statement author. Known as the father of the analysis of algorithms. 1) Edsger Dijkstra 2) Donald Knuth 3) Alan Turing 4) Grace Hopper 5) John von Neumann 6) Alan Perlis 7) John McCarthy 8) John Backus 9) Niklaus Wirth 10) C.A.R. Hoare Conceptualized the idea of machine-independent programming languages, which led to the development of COBOL 6

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