COT 3100 Spring 2010 Midterm 2
|
|
- Lorena Hamilton
- 6 years ago
- Views:
Transcription
1 COT 3100 Spring 2010 Midterm 2 For the first two questions #1 and #2, do ONLY ONE of them. If you do both, only question #1 will be graded. 1. (20 pts) Give iterative and recursive algorithms for finding the n th term of the sequence {a n defined by a 0 = 2, a 1 = 2 and a n+1 = a n (a n-1 ) 2. Which algorithm is more efficient and why? : + Input: n, a 0 = 2, a 1 = 2, a n+1 = a n (a n-1 ) 2 ; + Output: The n th term of the sequence {a n. Iterative algorithm: for (int i = 2; i <=n; i++) { result = a 1 * a 0 * a 0 ; a 0 = a 1 ; a 1 = result; return result; Recursive algorithm: int a(int k) { if (k <= 1 ) return 2; else return a(k-1)*a(k-2)*a(k-2); return result = a(n); Which algorithm is more efficient and why? Iterative algorithm performs (n-2) iterations and in each iteration, updates values of 3 variables, thus its time complexity is O(n). The recursive algorithm, in order to find a(n), loops into itself 3 times to find a(n-1) and two a(n-2). To find a(n-1), it again loops into itself 3 times to find a(n-2) and two a(n-2), so on and so forth. Thus, its creates a complete 3-leaf tree with height n. Thus, the total time complexity is O(3 n ). Therefore, the iterative algorithm is more efficient. 1
2 2. (20 pts) Give iterative and recursive algorithms for determining whether a string is a palindrome or not. (Note: A string is a palindrome if it can be read the same way in either direction, e.g., ABBA, madam, level ). + Input: A string w of length n; + Output: YES if w is a palindrome, NO otherwise. Iterative algorithm w is a palindrome iff (w[i] == w[n+1-i]) for any i th index from 1 to n. Therefore, we test this property on every i th position and stop when the index i hits the middle of the string w. int middle = n/2; for(int i = 1; i<=middle; i++) { if ( w[i]!= w[n+1-i] ) return NO; return YES; Recursive algorithm A string w of length k is a palindrome iff (w[1] == w[k]) and the substring w[2...k-1] is also a palindrome. This observation suggests the following recursive algorithm boolean ispal(w, k) { if (k==0 k==1) return YES; return (w[1] == w[k]) && ispal(w[2...k-1], k-2); 2
3 3. (10 pts) Prove by induction that (x - y) divides x n - y n for any integer n 1. + Let P(n) be the statement: (x - y) divides x n - y n. + Basis step: n = 1. It is clear that (x - y) divides x 1 - y 1 (= x - y). + Inductive step: - Assume P(n) is true. We will prove that P(n+1) is also true, i.e., (x-y) divides x n+1 - y n+1. - Now: x n+1 - y n+1 = x n+1 - xy n + xy n - y n+1 = x(x n - y n ) + y n (x - y) - Since P(n) is true, we have (x - y) divides x n - y n, which implies (x-y) divides x(x n - y n ). In addition, (x - y) also divides y n (x - y). Therefore, (x - y) divides x(x n - y n ) + y n (x - y), which means P(n+1) is also true. 4. (20 pts) Let S be the subset of the set of ordered pairs of integers defined recursively by: Basis step: (0,0) S. Recursive step: If (a,b) S then (a+4, b+5) S and (a-2, b-7) S. a. Prove by strong induction on the number of applications of recursive steps that a b is even for any (a,b) S. Let n be the number of applications of recursive steps. We will show that for any (a,b) S, a is even which implies a b is even for any (a,b) S Basis step n = 0. It is clear that 0 is even. Thus the basis step is true. Inductive step + Assume that for all 0 < n N, a is even for any (a,b) S. We will show that a' is also an even number for any (a',b') generated in step (N+1). + Consider the two following cases. Case #1: a' is derived by (a+4) for some number a in step N and based on the assumption, we have a is even. Thus a' = a + 4 is also even. Case #2: a' is derived by (a-2) for some number a in step N and based on the assumption, we have a is even. Thus a' = a - 2 is also even. The inductive step is complete. a. Prove by structural induction that 9 a+ b for any (a,b) S Basis step It is clear that The basis step is true. Inductive step + We will prove that, in Recursive Step, any (a', b') obtained from (a,b) S satisfies: 9 (a' + b'). Since (a', b') is derived from (a,b), it must be obtained by either (a+4,b+5) or (a-2,b-7). Thus, (a' + b') equals either (a+b+9) or (a+b-9) (*). + Now, because (a,b) is previously in S, we have 9 (a + b) which implies 9 divides both (a+b+9) and (a+b-9). This fact also implies that 9 (a' + b') due to (*). 3
4 5. (10 pts) Give the recursive definition of the sequence {a n, n = 1,2,3, if a. a n = 4n - 9 a 1 = -5; a n+1 = 4(n+1) - 9 = 4n = a n + 4; b. a n = n 2 2n a 1 = -1; a n+1 = (n+1) 2-2(n+1) = n 2 + 2n + 1-2n - 2 = n 2-2n + 2n - 1 = a n + 2n - 1 For questions #6 and #7, please justify your answers. Answers with only numbers will result in no point. 6. (10 pts) How many positive integers from 1000 to 9999 inclusive are divisible by 9 but not by 4? "A number divisible by 9 but not by 4" means a number divisible by 9 but not 36(=4*9). Therefore, we just need to find the number of integers divisible by 9 and subtract the number of those divisible by Number of integers divisible by 9 + Number of integers divisible by 36 + Thus, the answer is = a. (10 pts) A coin is flipped 20 times where each flip comes up either heads or tails. How many possible outcomes contain at most 5 tails? + Number of outcomes contain 0 tails: C(20,0) + Number of outcomes contain 1 tails: C(20,1) + Number of outcomes contain 2 tails: C(20,2) + Number of outcomes contain 3 tails: C(20,3) + Number of outcomes contain 4 tails: C(20,4) + Number of outcomes contain 5 tails: C(20,5) Total possible outcomes = C(20,0) + C(20,1) + C(20,2) + C(20,3) + C(20,4) + C(20,5) 4
5 b. (10 pts) What is the least number of people needed to form a group in which we can surely conclude that there are at least 3 people having the same birthday? Assume that there are 365 days in a year. Based on Pigeonhole Principle, the least number of people we need to choose is one more than the number of days in (3-1) years. Answer = 365*(3-1) (10 pts) Find the error in this "proof" of the clearly false claim: "Every set of lines in the plane, no two of which are parallel, meet in a common point". "Proof": Let P(n) be the statement that "Every set of n lines in the plane, no two of which are parallel, meet in a common point". We will try to prove that P(n) is true for n Basis step: P(2) is true due to the definition of parallel lines. 2. Inductive step: Assume that P(k) is true, i.e., every set of k lines in the plane, no two of which are parallel, meet in a common point. We will try to prove that P(k+1) is also true. 3. Consider the set S of (k+1) distinct lines in the plane, no two of which are parallel. Since P(k) is true, the first k lines of S must meet in a point p 1, and the last k lines of S must meet in a point p If p 1 is different from p 2, then all lines containing both of them must be exactly the same since two distinct points determine a line. This implies the second line, the third line,..., and the k th lines of S are the same, contradicting to the fact that they are all distinct. 5. Therefore, p 1 and p 2 must be the same point, i.e., (k+1) distinct lines in the plane, no two of which are parallel, meet in a common point, which means P(k+1) is true. The problem is at step #4. We cannot show that P(2) implies P(3). When k = 2, the first two lines must meet in a common point p1 and the last two lines must meet in a common point p2. But in this case, p 1 and p 2 do not have to be the same, because only the second line is common to both sets of lines. 5
6 Bonus questions: 1. (5pts) Show that is Θ( ) for any positive integer k > 1. Choose c 1 = and c 2 =, we have = Thus, = Θ( ) c 1 c (5 pts) Prove that if five points are selected from the interior of a 1 1 square, then there are two points whose distance is less than. We first divide the square into 4 smaller squares (by connecting the corresponding middle points) and then select 5 points from the interior of the "big" square. By Pigeonhole Principle, there must be a "small" square containing at least 2 points. In addition, the maximum distance in this "small" square cannot exceed the length of the diagonal, which is. Thus, any two points inside this "small" square has distance less than. 6
ELEC-270 Solutions to Assignment 5
ELEC-270 Solutions to Assignment 5 1. How many positive integers less than 1000 (a) are divisible by 7? (b) are divisible by 7 but not by 11? (c) are divisible by both 7 and 11? (d) are divisible by 7
More information1KOd17RMoURxjn2 CSE 20 DISCRETE MATH Fall
CSE 20 https://goo.gl/forms/1o 1KOd17RMoURxjn2 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Explain the steps in a proof by mathematical and/or structural
More informationCS525 Winter 2012 \ Class Assignment #2 Preparation
1 CS525 Winter 2012 \ Class Assignment #2 Preparation Ariel Stolerman 2.26) Let be a CFG in Chomsky Normal Form. Following is a proof that for any ( ) of length exactly steps are required for any derivation
More information(a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are rational they can be written as the ratio of integers a 1
CS 70 Discrete Mathematics for CS Fall 2000 Wagner MT1 Sol Solutions to Midterm 1 1. (16 pts.) Theorems and proofs (a) (4 pts) Prove that if a and b are rational, then ab is rational. Since a and b are
More informationTo illustrate what is intended the following are three write ups by students. Diagonalization
General guidelines: You may work with other people, as long as you write up your solution in your own words and understand everything you turn in. Make sure to justify your answers they should be clear
More informationMathematical Induction
Mathematical Induction Victor Adamchik Fall of 2005 Lecture 3 (out of three) Plan 1. Recursive Definitions 2. Recursively Defined Sets 3. Program Correctness Recursive Definitions Sometimes it is easier
More informationMultiple-choice (35 pt.)
CS 161 Practice Midterm I Summer 2018 Released: 7/21/18 Multiple-choice (35 pt.) 1. (2 pt.) Which of the following asymptotic bounds describe the function f(n) = n 3? The bounds do not necessarily need
More informationVerifying Safety Property of Lustre Programs: Temporal Induction
22c181: Formal Methods in Software Engineering The University of Iowa Spring 2008 Verifying Safety Property of Lustre Programs: Temporal Induction Copyright 2008 Cesare Tinelli. These notes are copyrighted
More informationProof Techniques Alphabets, Strings, and Languages. Foundations of Computer Science Theory
Proof Techniques Alphabets, Strings, and Languages Foundations of Computer Science Theory Proof By Case Enumeration Sometimes the most straightforward way to prove that a property holds for all elements
More informationCS 3512, Spring Instructor: Doug Dunham. Textbook: James L. Hein, Discrete Structures, Logic, and Computability, 3rd Ed. Jones and Barlett, 2010
CS 3512, Spring 2011 Instructor: Doug Dunham Textbook: James L. Hein, Discrete Structures, Logic, and Computability, 3rd Ed. Jones and Barlett, 2010 Prerequisites: Calc I, CS2511 Rough course outline:
More informationCSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Today's learning goals Explain the steps in a proof by (strong) mathematical induction Use (strong) mathematical induction
More informationCSE101: Design and Analysis of Algorithms. Ragesh Jaiswal, CSE, UCSD
Recap. Growth rates: Arrange the following functions in ascending order of growth rate: n 2 log n n log n 2 log n n/ log n n n Introduction Algorithm: A step-by-step way of solving a problem. Design of
More information1. Chapter 1, # 1: Prove that for all sets A, B, C, the formula
Homework 1 MTH 4590 Spring 2018 1. Chapter 1, # 1: Prove that for all sets,, C, the formula ( C) = ( ) ( C) is true. Proof : It suffices to show that ( C) ( ) ( C) and ( ) ( C) ( C). ssume that x ( C),
More informationDiscrete Mathematics Lecture 4. Harper Langston New York University
Discrete Mathematics Lecture 4 Harper Langston New York University Sequences Sequence is a set of (usually infinite number of) ordered elements: a 1, a 2,, a n, Each individual element a k is called a
More informationWrite an algorithm to find the maximum value that can be obtained by an appropriate placement of parentheses in the expression
Chapter 5 Dynamic Programming Exercise 5.1 Write an algorithm to find the maximum value that can be obtained by an appropriate placement of parentheses in the expression x 1 /x /x 3 /... x n 1 /x n, where
More informationMathematical Induction
COMP 182 Algorithmic Thinking Mathematical Induction Luay Nakhleh Computer Science Rice University Chapter 5, Section 1-4 Reading Material [P (1) ^8k(P (k)! P (k + 1))]!8nP (n) Why Is It Valid? The well-ordering
More informationAnnouncements. CS243: Discrete Structures. Strong Induction and Recursively Defined Structures. Review. Example (review) Example (review), cont.
Announcements CS43: Discrete Structures Strong Induction and Recursively Defined Structures Işıl Dillig Homework 4 is due today Homework 5 is out today Covers induction (last lecture, this lecture, and
More informationIndicate the option which most accurately completes the sentence.
Discrete Structures, CSCI 246, Fall 2015 Final, Dec. 10 Indicate the option which most accurately completes the sentence. 1. Say that Discrete(x) means that x is a discrete structures exam and Well(x)
More informationTHE PRINCIPLE OF INDUCTION. MARK FLANAGAN School of Electrical, Electronic and Communications Engineering University College Dublin
THE PRINCIPLE OF INDUCTION MARK FLANAGAN School of Electrical, Electronic and Communications Engineering University College Dublin The Principle of Induction: Let a be an integer, and let P(n) be a statement
More informationSolution : a) C(18, 1)C(325, 1) = 5850 b) C(18, 1) + C(325, 1) = 343
DISCRETE MATHEMATICS HOMEWORK 5 SOL Undergraduate Course College of Computer Science Zhejiang University Fall-Winter 2014 HOMEWORK 5 P344 1. There are 18 mathematics majors and 325 computer science majors
More informationCS134 Spring 2005 Final Exam Mon. June. 20, 2005 Signature: Question # Out Of Marks Marker Total
CS134 Spring 2005 Final Exam Mon. June. 20, 2005 Please check your tutorial (TUT) section from the list below: TUT 101: F 11:30, MC 4042 TUT 102: M 10:30, MC 4042 TUT 103: M 11:30, MC 4058 TUT 104: F 10:30,
More informationLecture Notes on Induction and Recursion
Lecture Notes on Induction and Recursion 15-317: Constructive Logic Frank Pfenning Lecture 7 September 19, 2017 1 Introduction At this point in the course we have developed a good formal understanding
More informationFall Recursion and induction. Stephen Brookes. Lecture 4
15-150 Fall 2018 Stephen Brookes Lecture 4 Recursion and induction Last time Specification format for a function F type assumption guarantee (REQUIRES) (ENSURES) For all (properly typed) x satisfying the
More informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More information4&5 Binary Operations and Relations. The Integers. (part I)
c Oksana Shatalov, Spring 2016 1 4&5 Binary Operations and Relations. The Integers. (part I) 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition,
More informationSolution to Graded Problem Set 4
Graph Theory Applications EPFL, Spring 2014 Solution to Graded Problem Set 4 Date: 13.03.2014 Due by 18:00 20.03.2014 Problem 1. Let V be the set of vertices, x be the number of leaves in the tree and
More informationMa/CS 6b Class 26: Art Galleries and Politicians
Ma/CS 6b Class 26: Art Galleries and Politicians By Adam Sheffer The Art Gallery Problem Problem. We wish to place security cameras at a gallery, such that they cover it completely. Every camera can cover
More informationMidterm Exam Review. CS231 Algorithms Handout # 23 Prof. Lyn Turbak November 5, 2001 Wellesley College
CS231 Algorithms Handout # 23 Prof. Lyn Turbak November 5, 2001 Wellesley College Midterm Exam Review This handout contains some problems that cover material covered on the midterm exam. Reviewing these
More informationCSCE 110 Dr. Amr Goneid Exercise Sheet (7): Exercises on Recursion
CSCE 110 Dr. Amr Goneid Exercise Sheet (7): Exercises on Recursion Consider the following recursive function: int what ( int x, int y) if (x > y) return what (x-y, y); else if (y > x) return what (x, y-x);
More informationOutline. Introduction. 2 Proof of Correctness. 3 Final Notes. Precondition P 1 : Inputs include
Outline Computer Science 331 Correctness of Algorithms Mike Jacobson Department of Computer Science University of Calgary Lectures #2-4 1 What is a? Applications 2 Recursive Algorithms 3 Final Notes Additional
More informationHMMT February 2018 February 10, 2018
HMMT February 2018 February 10, 2018 Combinatorics 1. Consider a 2 3 grid where each entry is one of 0, 1, and 2. For how many such grids is the sum of the numbers in every row and in every column a multiple
More informationRecursive Definitions Structural Induction Recursive Algorithms
Chapter 4 1 4.3-4.4 Recursive Definitions Structural Induction Recursive Algorithms 2 Section 4.1 3 Principle of Mathematical Induction Principle of Mathematical Induction: To prove that P(n) is true for
More informationCmpSci 187: Programming with Data Structures Spring 2015
CmpSci 187: Programming with Data Structures Spring 2015 Lecture #9 John Ridgway February 26, 2015 1 Recursive Definitions, Algorithms, and Programs Recursion in General In mathematics and computer science
More informationAnalyze the obvious algorithm, 5 points Here is the most obvious algorithm for this problem: (LastLargerElement[A[1..n]:
CSE 101 Homework 1 Background (Order and Recurrence Relations), correctness proofs, time analysis, and speeding up algorithms with restructuring, preprocessing and data structures. Due Thursday, April
More informationComputer Science 236 Fall Nov. 11, 2010
Computer Science 26 Fall Nov 11, 2010 St George Campus University of Toronto Assignment Due Date: 2nd December, 2010 1 (10 marks) Assume that you are given a file of arbitrary length that contains student
More informationCS171 Midterm Exam. October 29, Name:
CS171 Midterm Exam October 29, 2012 Name: You are to honor the Emory Honor Code. This is a closed-book and closed-notes exam. You have 50 minutes to complete this exam. Read each problem carefully, and
More informationCSC 284/484 Advanced Algorithms - applied homework 0 due: January 29th, 11:59pm EST
CSC 84/484 Advanced Algorithms - applied homework 0 due: January 9th, 11:59pm EST Grading: 84: 1 problem solved = A 484: problems solved = A, 1 problem solved = B This homework has different rules than
More informationCS103 Spring 2018 Mathematical Vocabulary
CS103 Spring 2018 Mathematical Vocabulary You keep using that word. I do not think it means what you think it means. - Inigo Montoya, from The Princess Bride Consider the humble while loop in most programming
More informationCSE Discrete Structures
CSE 2315 - Discrete Structures Homework 3- Solution - Fall 2010 Due Date: Oct. 28 2010, 3:30 pm Sets 1. Rewrite the following sets as a list of elements. (8 points) a) {x ( y)(y N x = y 3 x < 30)} {0,
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 11 Ana Bove April 26th 2018 Recap: Regular Languages Decision properties of RL: Is it empty? Does it contain this word? Contains
More informationCSE 332, Spring 2010, Midterm Examination 30 April 2010
CSE 332, Spring 2010, Midterm Examination 30 April 2010 Please do not turn the page until the bell rings. Rules: The exam is closed-book, closed-note. You may use a calculator for basic arithmetic only.
More informationLecture 19 Thursday, March 29. Examples of isomorphic, and non-isomorphic graphs will be given in class.
CIS 160 - Spring 2018 (instructor Val Tannen) Lecture 19 Thursday, March 29 GRAPH THEORY Graph isomorphism Definition 19.1 Two graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are isomorphic, write G 1 G
More informationSolutions to In-Class Problems Week 4, Fri
Massachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Solutions to In-Class Problems Week 4, Fri Definition: The
More information1. [5 points each] True or False. If the question is currently open, write O or Open.
University of Nevada, Las Vegas Computer Science 456/656 Spring 2018 Practice for the Final on May 9, 2018 The entire examination is 775 points. The real final will be much shorter. Name: No books, notes,
More informationGreedy algorithms is another useful way for solving optimization problems.
Greedy Algorithms Greedy algorithms is another useful way for solving optimization problems. Optimization Problems For the given input, we are seeking solutions that must satisfy certain conditions. These
More informationDiscrete Mathematics Exam File Fall Exam #1
Discrete Mathematics Exam File Fall 2015 Exam #1 1.) Which of the following quantified predicate statements are true? Justify your answers. a.) n Z, k Z, n + k = 0 b.) n Z, k Z, n + k = 0 2.) Prove that
More informationChapter Summary. Mathematical Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter Summary Mathematical Induction Recursive Definitions Structural Induction Recursive Algorithms Section 5.1 Sec.on Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken
More informationDo not turn this page until you have received the signal to start. In the meantime, please read the instructions below carefully.
CSC 165 H1 Term Test 2 / L5101 Fall 2011 Duration: Aids Allowed: 60 minutes none Student Number: Family Name(s): Given Name(s): Do not turn this page until you have received the signal to start. In the
More information1 Definition of Reduction
1 Definition of Reduction Problem A is reducible, or more technically Turing reducible, to problem B, denoted A B if there a main program M to solve problem A that lacks only a procedure to solve problem
More informationMC 302 GRAPH THEORY 10/1/13 Solutions to HW #2 50 points + 6 XC points
MC 0 GRAPH THEORY 0// Solutions to HW # 0 points + XC points ) [CH] p.,..7. This problem introduces an important class of graphs called the hypercubes or k-cubes, Q, Q, Q, etc. I suggest that before you
More informationDiscrete Mathematics Course Review 3
21-228 Discrete Mathematics Course Review 3 This document contains a list of the important definitions and theorems that have been covered thus far in the course. It is not a complete listing of what has
More informationSection 6.3: Further Rules for Counting Sets
Section 6.3: Further Rules for Counting Sets Often when we are considering the probability of an event, that event is itself a union of other events. For example, suppose there is a horse race with three
More informationLamé s Theorem. Strings. Recursively Defined Sets and Structures. Recursively Defined Sets and Structures
Lamé s Theorem Gabriel Lamé (1795-1870) Recursively Defined Sets and Structures Lamé s Theorem: Let a and b be positive integers with a b Then the number of divisions used by the Euclidian algorithm to
More informationCMPSCI 250: Introduction to Computation. Lecture #1: Things, Sets and Strings David Mix Barrington 22 January 2014
CMPSCI 250: Introduction to Computation Lecture #1: Things, Sets and Strings David Mix Barrington 22 January 2014 Things, Sets, and Strings The Mathematical Method Administrative Stuff The Objects of Mathematics
More informationCSC 1351: Quiz 6: Sort and Search
CSC 1351: Quiz 6: Sort and Search Name: 0.1 You want to implement combat within a role playing game on a computer. Specifically, the game rules for damage inflicted by a hit are: In order to figure out
More informationTest 1, Spring 2013 ( Solutions): Provided by Jeff Collins and Anil Patel. 1. Axioms for a finite AFFINE plane of order n.
Math 532, 736I: Modern Geometry Test 1, Spring 2013 ( Solutions): Provided by Jeff Collins and Anil Patel Part 1: 1. Axioms for a finite AFFINE plane of order n. AA1: There exist at least 4 points, no
More informationSTUDENT NUMBER: MATH Final Exam. Lakehead University. April 13, Dr. Adam Van Tuyl
Page 1 of 13 NAME: STUDENT NUMBER: MATH 1281 - Final Exam Lakehead University April 13, 2011 Dr. Adam Van Tuyl Instructions: Answer all questions in the space provided. If you need more room, answer on
More informationCS 341 Homework 1 Basic Techniques
CS 341 Homework 1 Basic Techniques 1. What are these sets? Write them using braces, commas, numerals, (for infinite sets), and only. (a) ({1, 3, 5} {3, 1}) {3, 5, 7} (b) {{3}, {3, 5}, {{5, 7}, {7, 9}}}
More informationMatching Theory. Figure 1: Is this graph bipartite?
Matching Theory 1 Introduction A matching M of a graph is a subset of E such that no two edges in M share a vertex; edges which have this property are called independent edges. A matching M is said to
More informationToday. Finish Euclid. Bijection/CRT/Isomorphism. Review for Midterm.
Today Finish Euclid. Bijection/CRT/Isomorphism. Review for Midterm. Finding an inverse? We showed how to efficiently tell if there is an inverse. Extend euclid to find inverse. Euclid s GCD algorithm.
More information1. Find f(1), f(2), f(3), and f(4) if f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2,
Exercises Exercises 1. Find f(1), f(2), f(3), and f(4) if f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, a) f(n + 1) = f(n) + 2. b) f(n + 1) = 3f(n). c) f(n + 1) = 2f(n). d) f(n + 1) = f(n)2
More information8/22/12. Outline. Backtracking. The Eight Queens Problem. Backtracking. Part 1. Recursion as a Problem-Solving Technique
Part 1. Recursion as a Problem-Solving Technique CS 200 Algorithms and Data Structures CS 200 Algorithms and Data Structures [Fall 2011] 2 Outline Backtracking Formal grammars Relationship between recursion
More informationSolutions to the Second Midterm Exam
CS/Math 240: Intro to Discrete Math 3/27/2011 Instructor: Dieter van Melkebeek Solutions to the Second Midterm Exam Problem 1 This question deals with the following implementation of binary search. Function
More informationUniversity of Nevada, Las Vegas Computer Science 456/656 Fall 2016
University of Nevada, Las Vegas Computer Science 456/656 Fall 2016 The entire examination is 925 points. The real final will be much shorter. Name: No books, notes, scratch paper, or calculators. Use pen
More informationRecursion. What is Recursion? Simple Example. Repeatedly Reduce the Problem Into Smaller Problems to Solve the Big Problem
Recursion Repeatedly Reduce the Problem Into Smaller Problems to Solve the Big Problem What is Recursion? A problem is decomposed into smaller sub-problems, one or more of which are simpler versions of
More informationUnit 7 Number System and Bases. 7.1 Number System. 7.2 Binary Numbers. 7.3 Adding and Subtracting Binary Numbers. 7.4 Multiplying Binary Numbers
Contents STRAND B: Number Theory Unit 7 Number System and Bases Student Text Contents Section 7. Number System 7.2 Binary Numbers 7.3 Adding and Subtracting Binary Numbers 7.4 Multiplying Binary Numbers
More informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 12
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 12 p. 1/27 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science
More informationLecture 6,
Lecture 6, 4.16.2009 Today: Review: Basic Set Operation: Recall the basic set operator,!. From this operator come other set quantifiers and operations:!,!,!,! \ Set difference (sometimes denoted, a minus
More informationBasic operators, Arithmetic, Relational, Bitwise, Logical, Assignment, Conditional operators. JAVA Standard Edition
Basic operators, Arithmetic, Relational, Bitwise, Logical, Assignment, Conditional operators JAVA Standard Edition Java - Basic Operators Java provides a rich set of operators to manipulate variables.
More informationAdvanced Induction. Drawing hands by Escher. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1
Advanced Induction Drawing hands by Escher Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1 Logistics Midterm is done grading but entering into Moodle/tallying/analysis will
More informationPrelim One Solution. CS211 Fall Name. NetID
Name NetID Prelim One Solution CS211 Fall 2005 Closed book; closed notes; no calculators. Write your name and netid above. Write your name clearly on each page of this exam. For partial credit, you must
More information1. Suppose you are given a magic black box that somehow answers the following decision problem in polynomial time:
1. Suppose you are given a magic black box that somehow answers the following decision problem in polynomial time: Input: A CNF formula ϕ with n variables x 1, x 2,..., x n. Output: True if there is an
More informationUNC Charlotte 2010 Comprehensive
00 Comprehensive March 8, 00. A cubic equation x 4x x + a = 0 has three roots, x, x, x. If x = x + x, what is a? (A) 4 (B) 8 (C) 0 (D) (E) 6. For which value of a is the polynomial P (x) = x 000 +ax+9
More informationECE G205 Fundamentals of Computer Engineering Fall Exercises in Preparation to the Midterm
ECE G205 Fundamentals of Computer Engineering Fall 2003 Exercises in Preparation to the Midterm The following problems can be solved by either providing the pseudo-codes of the required algorithms or the
More informationCS 151 Complexity Theory Spring Final Solutions. L i NL i NC 2i P.
CS 151 Complexity Theory Spring 2017 Posted: June 9 Final Solutions Chris Umans 1. (a) The procedure that traverses a fan-in 2 depth O(log i n) circuit and outputs a formula runs in L i this can be done
More informationDO NOT RE-DISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 104, Graph Theory Homework 2 Solutions February 7, 2013 Introduction to Graph Theory, West Section 1.2: 26, 38, 42 Section 1.3: 14, 18 Section 2.1: 26, 29, 30 DO NOT RE-DISTRIBUTE
More information1 / 43. Today. Finish Euclid. Bijection/CRT/Isomorphism. Fermat s Little Theorem. Review for Midterm.
1 / 43 Today Finish Euclid. Bijection/CRT/Isomorphism. Fermat s Little Theorem. Review for Midterm. 2 / 43 Finding an inverse? We showed how to efficiently tell if there is an inverse. Extend euclid to
More informationMath 485, Graph Theory: Homework #3
Math 485, Graph Theory: Homework #3 Stephen G Simpson Due Monday, October 26, 2009 The assignment consists of Exercises 2129, 2135, 2137, 2218, 238, 2310, 2313, 2314, 2315 in the West textbook, plus the
More informationCSCI 270: Introduction to Algorithms and Theory of Computing Fall 2017 Prof: Leonard Adleman Scribe: Joseph Bebel
CSCI 270: Introduction to Algorithms and Theory of Computing Fall 2017 Prof: Leonard Adleman Scribe: Joseph Bebel We will now discuss computer programs, a concrete manifestation of what we ve been calling
More informationMidterm 2 Solutions. CS70 Discrete Mathematics and Probability Theory, Spring 2009
CS70 Discrete Mathematics and Probability Theory, Spring 2009 Midterm 2 Solutions Note: These solutions are not necessarily model answers. Rather, they are designed to be tutorial in nature, and sometimes
More informationRecursion. Chapter 7. Copyright 2012 by Pearson Education, Inc. All rights reserved
Recursion Chapter 7 Contents What Is Recursion? Tracing a Recursive Method Recursive Methods That Return a Value Recursively Processing an Array Recursively Processing a Linked Chain The Time Efficiency
More informationU.C. Berkeley CS170 : Algorithms, Fall 2013 Midterm 1 Professor: Satish Rao October 10, Midterm 1 Solutions
U.C. Berkeley CS170 : Algorithms, Fall 2013 Midterm 1 Professor: Satish Rao October 10, 2013 Midterm 1 Solutions 1 True/False 1. The Mayan base 20 system produces representations of size that is asymptotically
More informationMATH20902: Discrete Maths, Solutions to Problem Set 1. These solutions, as well as the corresponding problems, are available at
MATH20902: Discrete Maths, Solutions to Problem Set 1 These solutions, as well as the corresponding problems, are available at https://bit.ly/mancmathsdiscrete.. (1). The upper panel in the figure below
More informationCSCE 110 Dr. Amr Goneid Exercise Sheet (7): Exercises on Recursion (Solutions)
CSCE 110 Dr. Amr Goneid Exercise Sheet (7): Exercises on Recursion (Solutions) Consider the following recursive function: int what ( int x, int y) if (x > y) return what (x-y, y); else if (y > x) return
More informationLecture 1 Contracts. 1 A Mysterious Program : Principles of Imperative Computation (Spring 2018) Frank Pfenning
Lecture 1 Contracts 15-122: Principles of Imperative Computation (Spring 2018) Frank Pfenning In these notes we review contracts, which we use to collectively denote function contracts, loop invariants,
More informationComplexity, Induction, and Recurrence Relations. CSE 373 Help Session 4/7/2016
Complexity, Induction, and Recurrence Relations CSE 373 Help Session 4/7/2016 Big-O Definition Definition: g(n) is in O( f(n) ) if there exist positive constants c and n0 such that g(n) c f(n) for all
More informationMIDTERM EXAMINATION Douglas Wilhelm Harder EIT 4018 x T09:30:00P1H20M Rooms: RCH-103 and RCH-302
ECE 250 Algorithms and Data Structures MIDTERM EXAMINATION Douglas Wilhelm Harder dwharder@uwaterloo.ca EIT 4018 x37023 2013-10-23T09:30:00P1H20M Rooms: RCH-103 and RCH-302 Instructions: Read and initial
More informationOverview. A mathema5cal proof technique Proves statements about natural numbers 0,1,2,... (or more generally, induc+vely defined objects) Merge Sort
Goals for Today Induc+on Lecture 22 Spring 2011 Be able to state the principle of induc+on Iden+fy its rela+onship to recursion State how it is different from recursion Be able to understand induc+ve proofs
More informationCPS 102: Discrete Mathematics. Quiz 3 Date: Wednesday November 30, Instructor: Bruce Maggs NAME: Prob # Score. Total 60
CPS 102: Discrete Mathematics Instructor: Bruce Maggs Quiz 3 Date: Wednesday November 30, 2011 NAME: Prob # Score Max Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1 Problem 1 [10 points] Find a minimum-cost
More informationMath236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 19 Degree Sequences Let G be a graph with vertex set V (G) = {v 1, v 2, v
More informationNotes for Recitation 8
6.04/8.06J Mathematics for Computer Science October 5, 00 Tom Leighton and Marten van Dijk Notes for Recitation 8 Build-up error Recall a graph is connected iff there is a path between every pair of its
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Midterm 1
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Midterm 1 PRINT Your Name:, (last) SIGN Your Name: (first) PRINT Your Student ID: CIRCLE your exam room: 1 Pimentel 141 Mccone
More informationInfinity and Uncountability. Countable Countably infinite. Enumeration
Infinity and Uncountability. Countable Countably infinite. Enumeration How big is the set of reals or the set of integers? Infinite! Is one bigger or smaller? Same size? Same number? Make a function f
More informationMidterm solutions. n f 3 (n) = 3
Introduction to Computer Science 1, SE361 DGIST April 20, 2016 Professors Min-Soo Kim and Taesup Moon Midterm solutions Midterm solutions The midterm is a 1.5 hour exam (4:30pm 6:00pm). This is a closed
More informationCS2223: Algorithms Sorting Algorithms, Heap Sort, Linear-time sort, Median and Order Statistics
CS2223: Algorithms Sorting Algorithms, Heap Sort, Linear-time sort, Median and Order Statistics 1 Sorting 1.1 Problem Statement You are given a sequence of n numbers < a 1, a 2,..., a n >. You need to
More informationComparing sizes of sets
Comparing sizes of sets Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A
More informationLecture 7. SchemeList, finish up; Universal Hashing introduction
Lecture 7. SchemeList, finish up; Universal Hashing introduction CS 16 February 24, 2010 1 / 15 foldleft #!/usr/bin/python def foldleft(func, slist, init): foldleft: ( * -> ) * ( SchemeList)
More informationCMPSCI 187: Programming With Data Structures. Lecture #15: Thinking Recursively David Mix Barrington 10 October 2012
CMPSCI 187: Programming With Data Structures Lecture #15: Thinking Recursively David Mix Barrington 10 October 2012 Thinking Recursively Recursive Definitions, Algorithms, and Programs Computing Factorials
More informationadjacent angles Two angles in a plane which share a common vertex and a common side, but do not overlap. Angles 1 and 2 are adjacent angles.
Angle 1 Angle 2 Angles 1 and 2 are adjacent angles. Two angles in a plane which share a common vertex and a common side, but do not overlap. adjacent angles 2 5 8 11 This arithmetic sequence has a constant
More information2009 HMMT Team Round. Writing proofs. Misha Lavrov. ARML Practice 3/2/2014
Writing proofs Misha Lavrov ARML Practice 3/2/2014 Warm-up / Review 1 (From my research) If x n = 2 1 x n 1 for n 2, solve for x n in terms of x 1. (For a more concrete problem, set x 1 = 2.) 2 (From this
More information