Advanced Induction. Drawing hands by Escher. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1
|
|
- Imogen Bishop
- 5 years ago
- Views:
Transcription
1 Advanced Induction Drawing hands by Escher Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1
2 Logistics Midterm is done grading but entering into Moodle/tallying/analysis will take longer. You should be able to see your exam tomorrow in discussion sections, but probably won t be able to take it away. Watch for Piazza posts on status/statistics/etc. 2
3 This class More induction Induction on data structures Lists, trees, graphs, etc. manipulated by algorithms Structural induction Strengthening the hypothesis 3
4 Strengthenining inductive hypothesis << Problem 3 in HW 7 already illustrates this>> Prove that for any n > 0, 1 + 1/4 + 1/ /n 2 < 2 4
5 5
6 Another example For all n > 1, prove that the sum of the first n odd numbers is a perfect square. 6
7 7
8 Correctness of merge 8
9 Proving properties of recursive algorithms Think of the program as a recursive definition. Do induction using the recursive definition to prove algorithm correct. Works for any recursive definition (not just algorithms) Structural induction 9
10 Simple example Let S be the smallest set of integers such that: 3 is in S If x is in S, then x+6 and x 3 both belong to S. Prove that all elements of S are divisible by 3. 10
11 Lists Recursive definition of lists containing elements in A: <> is a lists(a) If x is in A and L is in lists(a), then x::l is in lists(a). Now consider the following function: Size: Lists(A) N Size(<>) = 0 Size(x::L) = 2+Size(L). Prove that Size(L) is even for any list in L(A). 11
12 Key idea prove properties about lists using its recursive definition structural induction similar to proving properties by induction over length of lists, heights of trees, etc. but more general as you can do this for any recursively defined structure (trees, doubly linked lists, bst, avl, etc.) recursive algorithms recursive definitions 12
13 Trees Binary trees over alphabet A as follows: <> is in BT(A) If L, R are in BT(A) and x is in A, then <L, x, R> is in BT(A). Example trees: < < <> a <> > b <<> c <> > > We can prove properties about trees using the above inductive definition. 13
14 Inserting into a sorted list Recursive algorithm: insert (x,l) { if L=<> then return x::<>; else y:= cons(l); L = cdr(l); if (x <= y) return x::y::l ; else return y::x::l; } Recursive function: insert(x,<>) = x::<>; insert(x,y::l)= x::y::l if x<=y y::x::l otherwise Think of a recursive algorithm as a recursive definition! Apply structural induction to prove properties of the algorithm. Example: if L is sorted then insert(x,l) is sorted. 14
15 Proof of insert on lists To prove: if L is sorted then insert(x,l) is sorted. Try induction: 15
16 Proof of insert on lists Strengthen the inductive hypothesis: If L is sorted, then insert(x,l) is sorted and is a keys stored in it is keys(l) U {x}. 16
17 17
18 Recap Recursive algms Recursive definitions Prove properties of algorithm by induction using the structure of the recursive definition. However, in practice: writing down the recursive defn explicitly is hard hard for programs that have state however, your proof of the algorithm can follow the recursive style, nevertheless. 18
19 Correctness of merge Prove correctness of merge using induction! Induction is on number of times merge calls itself. What decreases with each call? Length of L1? Length of L2? 19
20 20
21 Correctness of merge-sort 21
22 22
23 Correctness of binary search Searching a sorted array A[1..n] binary_search(int A[], int key, int imin, int imax){ if (imax < imin) return 1; else { int imid = midpoint(imin, imax); if (A[imid] > key) return binsearch(a, key, imin, mid 1); elseif (A[imid] < key) binsearch(a, key, imid+1, imax); else return imid; } } 23
24 24
25 Searching for a key in a bin search tree Binary search tree: datastructure used to search for elements fast in an ordered set. works best if tree is almost balanced. Key property: For any node n: keys(n.left) key(n) keys(n.right) 25
26 Searching for a key in a bin search tree find (k, T) { if T=nil return false; else if key(root(t))=k return true; else if k<key(root(t)) return find(k, T.left); else return find(k, T.right); } Prove inductively that find(k,t) return true iff k is stored in T. 26
27 27
28 Take away Inductive arguments often require strengthening Properties of recursively defined sets/functions can be proved using induction that follows the structure of the definition. Recursive algorithms can be seen as recursive definitions/functions. Learn how to see that in a recursive algorithm. Prove properties about recursive algorithms using structural induction that implicitly looks at the structure of the recursion in the algorithm. 28
Strategies for Proofs
Strategies for Proofs Landscape with House and Ploughman Van Gogh Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1 Goals of this lecture A bit more logic Reviewing Implication
More informationCS 220: Discrete Structures and their Applications. Recursive objects and structural induction in zybooks
CS 220: Discrete Structures and their Applications Recursive objects and structural induction 6.9 6.10 in zybooks Using recursion to define objects We can use recursion to define functions: The factorial
More information10/9/17. Using recursion to define objects. CS 220: Discrete Structures and their Applications
Using recursion to define objects CS 220: Discrete Structures and their Applications Recursive objects and 6.9 6.10 in zybooks We can use recursion to define functions: The factorial function can be defined
More informationCOT 3100 Spring 2010 Midterm 2
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
More informationCS 112 Introduction to Computing II. Wayne Snyder Computer Science Department Boston University
CS 112 Introduction to Computing II Wayne Snyder Department Boston University Today: Efficiency of binary trees; Balanced Trees 2-3 Trees Next Time: 2-3 Trees continued B-Trees and External Search Efficiency
More informationMIDTERM EXAM (HONORS SECTION)
Data Structures Course (V22.0102.00X) Professor Yap Fall 2010 MIDTERM EXAM (HONORS SECTION) October 19, 2010 SOLUTIONS Problem 1 TRUE OR FALSE QUESTIONS (4 Points each) Brief justification is required
More informationAVL Trees. See Section 19.4of the text, p
AVL Trees See Section 19.4of the text, p. 706-714. AVL trees are self-balancing Binary Search Trees. When you either insert or remove a node the tree adjusts its structure so that the remains a logarithm
More informationUniversity of Illinois at Urbana-Champaign Department of Computer Science. Second Examination
University of Illinois at Urbana-Champaign Department of Computer Science Second Examination CS 225 Data Structures and Software Principles Spring 2014 7-10p, Tuesday, April 8 Name: NetID: Lab Section
More informationQ1 Q2 Q3 Q4 Q5 Q6 Total
Name: SSN: Computer Science Foundation Exam May 5, 006 Computer Science Section 1A Q1 Q Q3 Q4 Q5 Q6 Total KNW KNW KNW ANL,DSN KNW DSN You have to do all the 6 problems in this section of the exam. Partial
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 informationData Structures in Java
Data Structures in Java Lecture 9: Binary Search Trees. 10/7/015 Daniel Bauer 1 Contents 1. Binary Search Trees. Implementing Maps with BSTs Map ADT A map is collection of (key, value) pairs. Keys are
More information15 150: Principles of Functional Programming Sorting Integer Lists
15 150: Principles of Functional Programming Sorting Integer Lists Michael Erdmann Spring 2018 1 Background Let s define what we mean for a list of integers to be sorted, by reference to datatypes and
More informationCS70 - Lecture 6. Graphs: Coloring; Special Graphs. 1. Review of L5 2. Planar Five Color Theorem 3. Special Graphs:
CS70 - Lecture 6 Graphs: Coloring; Special Graphs 1. Review of L5 2. Planar Five Color Theorem 3. Special Graphs: Trees: Three characterizations Hypercubes: Strongly connected! Administration You need
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 informationCSC 1052 Algorithms & Data Structures II: Recursion
CSC 1052 Algorithms & Data Structures II: Recursion Professor Henry Carter Spring 2018 Recap Stacks provide a LIFO ordered data structure Implementation tradeoffs between arrays and linked lists typically
More informationPrelim 2 Solutions. CS 2110, November 20, 2014, 7:30 PM Extra Total Question True/False Short Answer
Prelim 2 Solutions CS 2110, November 20, 2014, 7:30 PM 1 2 3 4 5 Extra Total Question True/False Short Answer Complexity Induction Trees Graphs Extra Credit Max 20 10 15 25 30 5 100 Score Grader The exam
More informationSection 1.4 Proving Conjectures: Deductive Reasoning
Section 1.4 Proving Conjectures: Deductive Reasoning May 9 10:15 AM 1 Definition: Proof: A mathematical argument showing that a statement is valid in all cases, or that no counterexample exists. Generalization:
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 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 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 informationRecursion defining an object (or function, algorithm, etc.) in terms of itself. Recursion can be used to define sequences
Section 5.3 1 Recursion Recursion defining an object (or function, algorithm, etc.) in terms of itself. Recursion can be used to define sequences Previously sequences were defined using a specific formula,
More informationECE 250 Algorithms and Data Structures
ECE 250 Algorithms and Data Structures Sections 001 and 002 FINAL EXAMINATION Douglas Wilhelm Harder dwharder@uwaterloo.ca EIT 4018 x37023 2014-04-16T09:00P2H30M Rooms: PAC 7, 8 If you are writing a supplemental
More informationRecall our recursive multiply algorithm:
Recall our recursive multiply algorithm: PRECONDITION: x and y are both binary bit arrays of length n, n a power of 2. POSTCONDITION: Returns a binary bit array equal to the product of x and y. REC MULTIPLY
More informationRecursion defining an object (or function, algorithm, etc.) in terms of itself. Recursion can be used to define sequences
Section 5.3 1 Recursion 2 Recursion Recursion defining an object (or function, algorithm, etc.) in terms of itself. Recursion can be used to define sequences Previously sequences were defined using a specific
More informationUniversity of Illinois at Urbana-Champaign Department of Computer Science. Second Examination
University of Illinois at Urbana-Champaign Department of Computer Science Second Examination CS 225 Data Structures and Software Principles Fall 2011 9a-11a, Wednesday, November 2 Name: NetID: Lab Section
More informationby the evening of Tuesday, Feb 6
Homework 1 Due 14 February Handout 6 CSCI 334: Spring 2018 Notes This homework has three types of problems: Self Check: You are strongly encouraged to think about and work through these questions, and
More informationSolutions to Exam Data structures (X and NV)
Solutions to Exam Data structures X and NV 2005102. 1. a Insert the keys 9, 6, 2,, 97, 1 into a binary search tree BST. Draw the final tree. See Figure 1. b Add NIL nodes to the tree of 1a and color it
More informationCOMP Analysis of Algorithms & Data Structures
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Binary Search Trees CLRS 12.2, 12.3, 13.2, read problem 13-3 University of Manitoba COMP 3170 - Analysis of Algorithms & Data Structures
More informationProblem Set 5 Due: Friday, November 2
CS231 Algorithms Handout # 19 Prof. Lyn Turbak October 26, 2001 Wellesley College Problem Set 5 Due: Friday, November 2 Important: On Friday, Nov. 2, you will receive a take-home midterm exam that is due
More informationECE 250 Algorithms and Data Structures
ECE 250 Algorithms and Data Structures Sections 001 and 002 FINAL EXAMINATION Douglas Wilhelm Harder dwharder@uwaterloo.ca EIT 4018 x37023 2015-4-15T16:00/18:30 Rooms: PAC 6, 7 IF YOU ARE NOT ENROLLED
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 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 informationII (Sorting and) Order Statistics
II (Sorting and) Order Statistics Heapsort Quicksort Sorting in Linear Time Medians and Order Statistics 8 Sorting in Linear Time The sorting algorithms introduced thus far are comparison sorts Any comparison
More informationNote that this is a rep invariant! The type system doesn t enforce this but you need it to be true. Should use repok to check in debug version.
Announcements: Prelim tonight! 7:30-9:00 in Thurston 203/205 o Handed back in section tomorrow o If you have a conflict you can take the exam at 5:45 but can t leave early. Please email me so we have a
More informationPriority Queues Heaps Heapsort
Priority Queues Heaps Heapsort After this lesson, you should be able to apply the binary heap insertion and deletion algorithms by hand implement the binary heap insertion and deletion algorithms explain
More informationCS 374 Fall 2014 Homework 2 Due Tuesday, September 16, 2014 at noon
CS 374 Fall 2014 Homework 2 Due Tuesday, September 16, 2014 at noon Groups of up to three students may submit common solutions for each problem in this homework and in all future homeworks You are responsible
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 informationMotivation Computer Information Systems Storage Retrieval Updates. Binary Search Trees. OrderedStructures. Binary Search Tree
Binary Search Trees CMPUT 115 - Lecture Department of Computing Science University of Alberta Revised 21-Mar-05 In this lecture we study an important data structure: Binary Search Tree (BST) Motivation
More informationYork University AS/AK/ITEC INTRODUCTION TO DATA STRUCTURES. Midterm Sample I. Examiner: S. Chen Duration: One Hour and 30 Minutes
York University AS/AK/ITEC 2620 3.0 INTRODUCTION TO DATA STRUCTURES Midterm Sample I Examiner: S. Chen Duration: One Hour and 30 Minutes This exam is closed textbook(s) and closed notes. Use of any electronic
More informationCOMP Analysis of Algorithms & Data Structures
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Lecture 9 - Jan. 22, 2018 CLRS 12.2, 12.3, 13.2, read problem 13-3 University of Manitoba COMP 3170 - Analysis of Algorithms & Data Structures
More informationHomework 1. Notes. What To Turn In. Unix Accounts. Reading. Handout 3 CSCI 334: Spring, 2017
Homework 1 Due 14 February Handout 3 CSCI 334: Spring, 2017 Notes This homework has three types of problems: Self Check: You are strongly encouraged to think about and work through these questions, but
More informationCS1800 Discrete Structures Fall 2017 October 25, CS1800 Discrete Structures Midterm Version B
CS1800 Discrete Structures Fall 2017 October 25, 2017 Instructions: CS1800 Discrete Structures Midterm Version B 1. The exam is closed book and closed notes. You may not use a calculator or any other electronic
More informationSecond Examination Solution
University of Illinois at Urbana-Champaign Department of Computer Science Second Examination Solution CS 225 Data Structures and Software Principles Fall 2007 7p-9p, Thursday, November 8 Name: NetID: Lab
More informationCSC236 Week 5. Larry Zhang
CSC236 Week 5 Larry Zhang 1 Logistics Test 1 after lecture Location : IB110 (Last names A-S), IB 150 (Last names T-Z) Length of test: 50 minutes If you do really well... 2 Recap We learned two types of
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 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 informationFinal: CSS 342 SAMPLE. Data Structures, Algorithms, and Discrete Mathematics I
Final: CSS 342 SAMPLE Data Structures, Algorithms, and Discrete Mathematics I This is a closed-book, closed-notes exam, except for a single sheet of letter-sized, double-sided hand-written notes. Total
More informationCSE Data Structures and Introduction to Algorithms... In Java! Instructor: Fei Wang. Mid-Term Exam. CSE2100 DS & Algorithms 1
CSE 2100 Data Structures and Introduction to Algorithms...! In Java!! Instructor: Fei Wang! Mid-Term Exam CSE2100 DS & Algorithms 1 1. True or False (20%=2%x10)! (1) O(n) is O(n^2) (2) The height h of
More informationAnalysis of Algorithms
Analysis of Algorithms Concept Exam Code: 16 All questions are weighted equally. Assume worst case behavior and sufficiently large input sizes unless otherwise specified. Strong induction Consider this
More informationCS61B Lecture #21: Tree Searching. Last modified: Wed Oct 12 19:23: CS61B: Lecture #21 1
CS61B Lecture #21: Tree Searching Last modified: Wed Oct 12 19:23:14 2016 CS61B: Lecture #21 1 Divide and Conquer Much (most?) computation is devoted to finding things in response to various forms of query.
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 informationCS 315 Data Structures mid-term 2
CS 315 Data Structures mid-term 2 1) Shown below is an AVL tree T. Nov 14, 2012 Solutions to OPEN BOOK section. (a) Suggest a key whose insertion does not require any rotation. 18 (b) Suggest a key, if
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 informationINSTITUTE OF AERONAUTICAL ENGINEERING
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 COMPUTER SCIENCE AND ENGINEERING TUTORIAL QUESTION BANK Course Name Course Code Class Branch DATA STRUCTURES ACS002 B. Tech
More informationChapter 6. Problems All programming problems should include design pseudo code either as a separate design document on embedded comments in the code.
Chapter 6. Problems All programming problems should include design pseudo code either as a separate design document on embedded comments in the code. 1S. Write an assembly code equivalent for the following
More informationCS 315 Data Structures Spring 2012 Final examination Total Points: 80
CS 315 Data Structures Spring 2012 Final examination Total Points: 80 Name This is an open-book/open-notes exam. Write the answers in the space provided. Answer for a total of 80 points, including at least
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 informationSection 1: True / False (2 points each, 30 pts total)
Section 1: True / False (2 points each, 30 pts total) Circle the word TRUE or the word FALSE. If neither is circled, both are circled, or it impossible to tell which is circled, your answer will be considered
More informationCSE 373 APRIL 17 TH TREE BALANCE AND AVL
CSE 373 APRIL 17 TH TREE BALANCE AND AVL ASSORTED MINUTIAE HW3 due Wednesday Double check submissions Use binary search for SADict Midterm text Friday Review in Class on Wednesday Testing Advice Empty
More informationCSE 332 Autumn 2013: Midterm Exam (closed book, closed notes, no calculators)
Name: Email address: Quiz Section: CSE 332 Autumn 2013: Midterm Exam (closed book, closed notes, no calculators) Instructions: Read the directions for each question carefully before answering. We will
More informationCS 310: Tree Rotations and AVL Trees
CS 310: Tree Rotations and AVL Trees Chris Kauffman Week 12-2 Practice/Demo Sites jgrasp is so-so for seeing tree operations Play with Blanced Binary Search Trees online using the following applets (titles
More informationCSE 214 Computer Science II Searching
CSE 214 Computer Science II Searching Fall 2017 Stony Brook University Instructor: Shebuti Rayana shebuti.rayana@stonybrook.edu http://www3.cs.stonybrook.edu/~cse214/sec02/ Introduction Searching in a
More informationWeek 5 Tutorial Structural Induction
Department of Computer Science, Australian National University COMP2600 / COMP6260 Formal Methods in Software Engineering Semester 2, 2016 Week 5 Tutorial Structural Induction You should hand in attempts
More informationYork University. AP/ITEC Section M INTRODUCTION TO DATA STRUCTURES Winter Midterm Test
York University AP/ITEC 2620 3.0 Section M INTRODUCTION TO DATA STRUCTURES Winter 2016 Midterm Test Examiner: S. Chen Duration: One Hour and 30 Minutes This exam is closed textbook(s) and closed notes.
More informationCOMP Analysis of Algorithms & Data Structures
COMP 3170 - Analysis of Algorithms & Data Structures Shahin Kamali Lecture 9 - Jan. 22, 2018 CLRS 12.2, 12.3, 13.2, read problem 13-3 University of Manitoba 1 / 12 Binary Search Trees (review) Structure
More informationPrelim 2. CS 2110, November 20, 2014, 7:30 PM Extra Total Question True/False Short Answer
Prelim 2 CS 2110, November 20, 2014, 7:30 PM 1 2 3 4 5 Extra Total Question True/False Short Answer Complexity Induction Trees Graphs Extra Credit Max 20 10 15 25 30 5 100 Score Grader The exam is closed
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 informationUniversity of Illinois at Urbana-Champaign Department of Computer Science. Second Examination
University of Illinois at Urbana-Champaign Department of Computer Science Second Examination CS 225 Data Structures and Software Principles Spring 2012 7p-9p, Tuesday, April 3 Name: NetID: Lab Section
More informationCSE 332 Spring 2013: Midterm Exam (closed book, closed notes, no calculators)
Name: Email address: Quiz Section: CSE 332 Spring 2013: Midterm Exam (closed book, closed notes, no calculators) Instructions: Read the directions for each question carefully before answering. We will
More informationBT, BST, AVL. Web Resources page on textbook website:
Excercises BT, BST, AVL Web Resources page on textbook website: http://ww3.java3.datastructures.net/resources.html Follow the execution. Consider the following Java code involving a Sequence: for (int
More informationCSE332 Summer 2010: Midterm Exam Sample Solutions
CSE332 Summer 2010: Midterm Exam Sample Solutions Closed notes, closed book; calculator ok. Read the instructions for each problem carefully before answering. Problems vary in point-values, difficulty
More informationReading 8 : Recursion
CS/Math 40: Introduction to Discrete Mathematics Fall 015 Instructors: Beck Hasti, Gautam Prakriya Reading 8 : Recursion 8.1 Recursion Recursion in computer science and mathematics refers to the idea of
More informationECE 250 Data Structures and Algorithms MID-TERM EXAMINATION B /13:30-14:50 MC-4021/RCH-211
ECE 250 Data Structures and Algorithms MID-TERM EXAMINATION B 2011-02-15/13:30-14:50 MC-4021/RCH-211 Instructions: There are 63 marks. It will be marked out of 55. No aides. Turn off all electronic media
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 informationCSC148 Week 7. Larry Zhang
CSC148 Week 7 Larry Zhang 1 Announcements Test 1 can be picked up in DH-3008 A1 due this Saturday Next week is reading week no lecture, no labs no office hours 2 Recap Last week, learned about binary trees
More information8/5/10 TODAY'S OUTLINE. Recursion COMP 10 EXPLORING COMPUTER SCIENCE. Revisit search and sorting using recursion. Recursion WHAT DOES THIS CODE DO?
8/5/10 TODAY'S OUTLINE Recursion COMP 10 EXPLORING COMPUTER SCIENCE Revisit search and sorting using recursion Binary search Merge sort Lecture 8 Recursion WHAT DOES THIS CODE DO? A function is recursive
More informationRecursion I and II. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1
Recursion I and II Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1 Recursive definitions of functions 2 One simple form In other words, f(n) is defined in terms of f(n-1)
More informationInstructions. Definitions. Name: CMSC 341 Fall Question Points I. /12 II. /30 III. /10 IV. /12 V. /12 VI. /12 VII.
CMSC 341 Fall 2013 Data Structures Final Exam B Name: Question Points I. /12 II. /30 III. /10 IV. /12 V. /12 VI. /12 VII. /12 TOTAL: /100 Instructions 1. This is a closed-book, closed-notes exam. 2. You
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 informationCS 380 ALGORITHM DESIGN AND ANALYSIS
CS 380 ALGORITHM DESIGN AND ANALYSIS Lecture 12: Red-Black Trees Text Reference: Chapters 12, 13 Binary Search Trees (BST): Review Each node in tree T is a object x Contains attributes: Data Pointers to
More informationCPSC 311: Analysis of Algorithms (Honors) Exam 1 October 11, 2002
CPSC 311: Analysis of Algorithms (Honors) Exam 1 October 11, 2002 Name: Instructions: 1. This is a closed book exam. Do not use any notes or books, other than your 8.5-by-11 inch review sheet. Do not confer
More informationCSE 373 OCTOBER 11 TH TRAVERSALS AND AVL
CSE 373 OCTOBER 11 TH TRAVERSALS AND AVL MINUTIAE Feedback for P1p1 should have gone out before class Grades on canvas tonight Emails went to the student who submitted the assignment If you did not receive
More informationDIT960 Datastrukturer
DIT90 Datastrukturer suggested solutions for exam 07-0-0. The following code takes as input two arrays with elements of the same type. The arrays are called a and b. The code returns a dynamic array which
More informationProject 1. due date Sunday July 8, 2018, 12:00 noon
Queens College, CUNY, Department of Computer Science Object-oriented programming in C++ CSCI 211 / 611 Summer 2018 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2018 Project 1 due date Sunday July 8,
More informationAssume you are given a Simple Linked List (i.e. not a doubly linked list) containing an even number of elements. For example L = [A B C D E F].
Question Assume you are given a Simple Linked List (i.e. not a doubly linked list) containing an even number of elements. For example L = [A B C D E F]. a) Draw the linked node structure of L, including
More informationCS1800 Discrete Structures Fall 2017 October 25, CS1800 Discrete Structures Midterm Version B
CS1800 Discrete Structures Fall 2017 October 25, 2017 Instructions: CS1800 Discrete Structures Midterm Version B 1. The exam is closed book and closed notes. You may not use a calculator or any other electronic
More informationFigure 4.1: The evolution of a rooted tree.
106 CHAPTER 4. INDUCTION, RECURSION AND RECURRENCES 4.6 Rooted Trees 4.6.1 The idea of a rooted tree We talked about how a tree diagram helps us visualize merge sort or other divide and conquer algorithms.
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 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 informationExam Data structures DAT036/DAT037/DIT960
Exam Data structures DAT036/DAT037/DIT960 Time Thursday 20 th August 2015, 14:00 18:00 Place Maskinhuset Course responsible Nick Smallbone, tel. 0707 183062 The exam consists of six questions. For a 3
More informationSorting Algorithms. CSE21 Winter 2017, Day 2 (B00), Day 1-2 (A00) January 11, 2017
Sorting Algorithms CSE21 Winter 2017, Day 2 (B00), Day 1-2 (A00) January 11, 2017 Sorting (or Ordering) Section 3.1 in Rosen vs. * Assume elements of the set to be sorted have some underlying order Why
More informationChapter 12 Supplement: Recursion with Java 1.5. Mr. Dave Clausen La Cañada High School
Chapter 12 Supplement: Recursion with Java 1.5 La Cañada High School Recursion: Definitions Recursion The process of a subprogram (method) calling itself. A clearly defined stopping state must exist. The
More informationBinary Search Trees, etc.
Chapter 12 Binary Search Trees, etc. Binary Search trees are data structures that support a variety of dynamic set operations, e.g., Search, Minimum, Maximum, Predecessors, Successors, Insert, and Delete.
More informationRecursively Defined Functions
Section 5.3 Recursively Defined Functions Definition: A recursive or inductive definition of a function consists of two steps. BASIS STEP: Specify the value of the function at zero. RECURSIVE STEP: Give
More informationComputer Science Foundation Exam
Computer Science Foundation Exam December 13, 2013 Section I A COMPUTER SCIENCE NO books, notes, or calculators may be used, and you must work entirely on your own. SOLUTION Question # Max Pts Category
More informationWeek 2. TA Lab Consulting - See schedule (cs400 home pages) Peer Mentoring available - Friday 8am-12pm, 12:15-1:30pm in 1289CS
ASSIGNMENTS h0 available and due before 10pm on Monday 1/28 h1 available and due before 10pm on Monday 2/4 p1 available and due before 10pm on Thursday 2/7 Week 2 TA Lab Consulting - See schedule (cs400
More informationGreedy Algorithms Part Three
Greedy Algorithms Part Three Announcements Problem Set Four due right now. Due on Wednesday with a late day. Problem Set Five out, due Monday, August 5. Explore greedy algorithms, exchange arguments, greedy
More informationCISC 320 Midterm Exam
Name: CISC 320 Midterm Exam Wednesday, Mar 25, 2015 There are 19 questions. The first 15 questions count 4 points each. For the others, points are individually shown. The total is 100 points. Multiple
More informationCS 163 Practice Final Exam Winter 2012
CS 163 Practice Final Exam Winter 2012 The final exam is Saturday, 21 April. Any problem from either midterm or either practice midterm may (and often does) appear again on the final. In addition, make
More information