# Find-The-Number. 1 Find-The-Number With Comps

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Find-The-Number 1 Find-The-Number With Comps Consider the following two-person game, which we call Find-The-Number with Comps. Player A (for answerer) has a number x between 1 and Player Q (for questioner) asks questions of the form x y? and A responds with YES or NO. They keep doing this until Q determines x. Lets see what might happen. Lets say Q first ask x 600? If A say YES then Q knows x [1, 600], so its down to 600 numbers. If A say NO then x [601, 1000], so its down to 400 numbers. So Q might luck out and get down to 400 numbers BUT Q could get stuck with 600 numbers. Lets say Q s first question is x 500? If A says YES then Q knows x [1, 500], so its down to 500 numbers. If Q says NO then A knows x [501, 1000], so its down to 500 numbers. So Q never does as well as 400 numbers, but never does as badly as 600 numbers. KEY: Q can always cut the interval in half by asking a question in the middle of the interval. Lets see how the worst case of this may go: Lets use a smaller number for exposition. Say x [1, 100]. Initially x [1, 100]. Q: is x 50? A: YES so x [1, 50] (if A said NO then x [51, 100], same size) Q: is x 5? A: YES so x [1, 5] (if A said NO then x [6, 50], same size) Q: is x 1? A: NO so x [13, 5] (if A said YES then x [1, 1], one less in size) Q: is x 19? A: YES so x [13, 19] (if A said NO then x [0, 5], one less in size) Q: is x 16? A: YES so x [13, 16] (if A said NO then x [17, 19], one less in size) Q: is x 14? A: YES so x [13, 14] (if A said NO then x [15, 16], the same size) Q: is x 13? A: YES so x = 13 (if A said NO then x = 14) This took 7 comparisons. 1

3 Find-The-number With Parallel Comps What if Q can ask TWO Comparisons at a time and wants to minimize the number-of-rounds of comparisons? Look at the interval [1, 1000]. In the last section Q HALVED the interval at every turn. Can Q do better? Q ask x 333? and x 666? 1. If the answers are x 333 and x 666 then we get x [1, 333]. Length If the answers are x 333 and x > 666 then we say LIAR LIAR, PANTS ON FIRE since there is no number x such that both x 333 and x > If the answers are x > 333 and x 666 then we get x [334, 666]. Length If the answers are x > 333 and x > 666 then we get x [667, 1000]. Length 333. The interval is cut in thirds. We can keep doing this. Theorem.1 Let A and Q play the game on [1,n]. If Q can ask comparisons at a time then by making them as close to the n/3 and n/3 points as possible as possible then the game will terminate in at most log 3 n rounds. Exercise 1 Show that if you ask comparison questions at a time then log 3 n questions are REQUIRED. Exercise What happens when you ask 3 questions at a time? p questions at a time? Upper and lower bounds. 3

4 3 Find-the-Number With Boolean Queries In the last section we noted that one of the possibilities, x < 333 and x > 666, could not happen. This is unavoidable if you ask COMPARISONS. What if you ask any YES-NO questions at all? Can you do better than log 3 n? First express the number in base. Let (n) b denote n in base b. (1000) 10 = ( ). Note that it has 9 places. Q: Is the first bit 1? Is the second bit 1? Whatever A answers you KNOW the first two bits. Keep doing this and you can find the answer out in 5 questions. Theorem 3.1 Let A and Q play the game on [1,n]. If Q can ask questions at a time then by asking questions about the bits then the game will terminate in at most log n/ steps. Proof: Every number between 1 and n can be expressed in log n bits. Let k = log n. Let x = b k 1 b k b 0. We do not know the bits b 0,...,b k 1. After asking the following questions and getting the answers, Q knows the number. Q: Does b 0 = 0? Does b 1 = 0? Q: Does b = 0? Does b 3 = 0? Q: Does b 4 = 0? Does b 5 = 0?. Q: Does b k 1 = 0? Does b k = 0? (This assume that k is even. If k is odd then the last question is just one question.) The answers to these questions tell Q all of the bits, and hence the number x. Is this better than using two comparisons at a time? We need to know if < log3 n. log n It is easy to show that log x y = log b x where b is any base. Hence log b y log 3 n = log n log 3. So we want to know if 4

5 log n 1 log 3 < log n log 3. is roughly For n large enough (bigger than 10 certainly works) this inequality holds. Exercise 3 What if Q asks p Boolean questions at a time. How many rounds does Q need to find the number? Can you do better than log n. We explore this in the next section. 4 Lower Bounds with Boolean Queries Earlier we showed that if you use COMPARISONS then the problem requires log n comparisons. What if you asked ANY Boolean queries. Can you do better than log n? But to even state this requires more care. We need to use Decision trees. Note the following: 1. There are at least n leaves since there are n answers 1,,...,n.. Since the questions are Boolean every internal node has two children. If a Decision tree has depth 1 then it has 1 = leaves. If a Decision tree has depth then it has = 4 leaves. etc. Lemma 4.1 If a Decision tree has depth d then it has d leaves. Exercise 4 Proof Lemma 4.1. Theorem 4. Let A and Q play the game on [1,n] where Q can ask any Boolean queries whatsoever. No matter how clever Q is, A can answer the questions so that A is forced to take log n steps. 5

7 Theorem 5.1 Let A and Q play the game on [1,n]. If Q always guesses a number as close to the midpoint as possible, and asks it twice, then the game will terminate in at most log n + 1 steps. Exercise 6 Prove Theorem 5.1. Can we do better than this? Since A will be able to lie once, we ll allow Q to make Boolean Queries. Theorem 5. Let A and Q play the game on [1,n]. Assume that Q can ask Boolean queries and that A can lie once. Then there is a strategy for Q that finds the number in log n + log log n + O(1) steps. Proof: The questions will be of the form x C where C is some set. For example, x {1,, 3,...,17} {n 10,n 7,n}? could be a question. Assume that the answer to this question was NO. It could still be the case that (say) x = 17 since this could have been the question that was lied about. If the next question was x {1, 3, 5, 17, 18, 19, 0,n 7} and the answer was NO then we know that x / {1, 3, 17,n 7}. That is because these elements were TWICE declared to not be x. We will keep track of two sets: (1) the set of all y that have NEVER been declared to NOT be x, and () the set of all y that have ONCE been declared to NOT be x (these could still be x since A can lie once). If an element has been declared NOT x twice or more, then it is no longer considered since it is not x. Let S 0 = {1,...,n} and T 0 =. Throughout the algorithm (1) S i will be elements that, after i questions, may still be x and have NEVER been declared to NOT be x, () T i will be elements that, after i questions, may still be x but have ONCE been declared to NOT be x. Since A can lie once, its possible that x T i. We pursue the first few steps of the algorithm, and one set of possible answers, as a thought experiment. We will return to doing it formally later. Lets say S 0 = {1,...,100}. I first ask Is x {1,...,50}? The answer is YES. Now S 1 = {1,...,50} (these have never been declared NOT x) T 1 = {51,...,100} (these have been declared NOT x only once) 7

9 If the answer is NO then everything in S 1 1 has been declared NOT x once, and everything in T 1 1 has been declared NOT x twice. Hence S = S 1 T = (T 1 S 1 1) T 1 1. More generally, we will ask a question about a set formed from part of the S and part of the T and then shift down as appropriate. How big is S 0,S 1,S,... Note that S 0 = n, S 1 = n/, S = n/, etc. Hence in log n the S pile will have only one element in it. We present the first phase of the algorithm. ALGORITHM: Phase 1. S 0 = {1,...,n} T 0 =. k = log n For i = 1 to k S i 1 = Si 1 1 Si 1 (near even divide) T i 1 = T 1 i 1 T Q: x Si 1 1 Ti 1 1 if answer is YES then S i = Si 1 1 i 1 (near even divide) T i = (Ti 1 1 Si 1) Ti 1. if answer is NO then S i = Si 1 T i = (T i 1 S 1 i 1) T 1 i 1. END OF PHASE 1. When this part of the algorithm is completed S i has at most 1 element y in it. Ask if x = y three times. If the majority say YES then x = y and we only used log n +3 queries. If the first answer is NO then put the element into T i. Now we have to look at the T pile. How big is it? Let s i = S i and t i = T i. Note that for any i k. s i = 1 s i 1 t i = 1 t i s i 1 We first solve for s i : s i = 1 s i 1 = 1 s i = 1 3 s i 3 = 1 i s i i = 1 i s 0 = 1 i n. 9

10 We can use this in our equation for t i. t i = 1 t i s i 1 = 1 t i 1 + n/ i = 1 (1 t i + n/ i 1 ) = 1 t i + n/ i + n/ i = 1 3 t i 3 + n/ i + n/ i + n/ i = 1 4 t i 4 + 4(n/ i ) =. = 1 i t i i + i(n/ i ) = 1 i t 0 + i(n/ i ) = 1 i 0 + i(n/ i ) = in i Our major concern is what happens when i = k = log n. t k = kn = n log k n = log log n n + O(1). So after the first k = log n questions there are log n elements in the T-pile. Since there are no elements in the S-pile that game is now that of finding a number in a set of log n numbers by using Boolean Queries. PHASE TWO: By Theorem 3.1 this can be done with log log n queries. END OF PHASE TWO Hence the entire algorithm takes log n + log log n + O(1) queries. Exercise 7 Explore what happens when Q asks 1 Boolean queries at a time and A is allowed to lie about two question once in the entire game. Exercise 8 Explore what happens when Q asks Boolean queries at a time and A is allowed to lie about one question once in the entire game. 10

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### Lecture 5 - CPA security, Pseudorandom functions

Lecture 5 - CPA security, Pseudorandom functions Boaz Barak October 2, 2007 Reading Pages 82 93 and 221 225 of KL (sections 3.5, 3.6.1, 3.6.2 and 6.5). See also Goldreich (Vol I) for proof of PRF construction.

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### 8.1 Makespan Scheduling

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: Min-Makespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Introduction to Algorithms. Part 3: P, NP Hard Problems

Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter

### Examples of decision problems:

Examples of decision problems: The Investment Problem: a. Problem Statement: i. Give n possible shares to invest in, provide a distribution of wealth on stocks. ii. The game is repetitive each time we

### 6.852: Distributed Algorithms Fall, 2009. Class 2

.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

### Algorithms complexity of recursive algorithms. Jiří Vyskočil, Marko Genyk-Berezovskyj

complexity of recursive algorithms Jiří Vyskočil, Marko Genyk-Berezovskyj 2010-2014 Recurrences A recurrence is an equation or inequality that describes a function in terms of its value on smaller inputs.

### TAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION

TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### Continued fractions and good approximations.

Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our

### COS 511: Foundations of Machine Learning. Rob Schapire Lecture #14 Scribe: Qian Xi March 30, 2006

COS 511: Foundations of Machine Learning Rob Schapire Lecture #14 Scribe: Qian Xi March 30, 006 In the previous lecture, we introduced a new learning model, the Online Learning Model. After seeing a concrete

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Balanced search trees

Lecture 8 Balanced search trees 8.1 Overview In this lecture we discuss search trees as a method for storing data in a way that supports fast insert, lookup, and delete operations. (Data structures handling

### , each of which contains a unique key value, say k i , R 2. such that k i equals K (or to determine that no such record exists in the collection).

The Search Problem 1 Suppose we have a collection of records, say R 1, R 2,, R N, each of which contains a unique key value, say k i. Given a particular key value, K, the search problem is to locate the

### Computer Algorithms. Merge Sort. Analysis of Merge Sort. Recurrences. Recurrence Relations. Iteration Method. Lecture 6

Computer Algorithms Lecture 6 Merge Sort Sorting Problem: Sort a sequence of n elements into non-decreasing order. Divide: Divide the n-element sequence to be sorted into two subsequences of n/ elements

### Notes from Week 1: Algorithms for sequential prediction

CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking

### Using Generalized Forecasts for Online Currency Conversion

Using Generalized Forecasts for Online Currency Conversion Kazuo Iwama and Kouki Yonezawa School of Informatics Kyoto University Kyoto 606-8501, Japan {iwama,yonezawa}@kuis.kyoto-u.ac.jp Abstract. El-Yaniv

### Pseudocode Analysis. COMP3600/6466 Algorithms Lecture 5. Recursive Algorithms. Asymptotic Bounds for Recursions

COMP600/6466 Algorithms Lecture 5 Pseudocode Analysis Iterative Recursive S 0 Dr. Hassan Hijazi Prof. Weifa Liang Recursive Algorithms Asymptotic Bounds for Recursions Total running time = Sum of times

### Depth-Independent Lower Bounds on the Communication Complexity of Read-Once Boolean Formulas

Depth-Independent Lower Bounds on the Communication Complexity of Read-Once Boolean Formulas Rahul Jain 1, Hartmut Klauck 2, and Shengyu Zhang 3 1 Centre for Quantum Technologies and Department of Computer

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Cost Model: Work, Span and Parallelism. 1 The RAM model for sequential computation:

CSE341T 08/31/2015 Lecture 3 Cost Model: Work, Span and Parallelism In this lecture, we will look at how one analyze a parallel program written using Cilk Plus. When we analyze the cost of an algorithm

### Math 421, Homework #5 Solutions

Math 421, Homework #5 Solutions (1) (8.3.6) Suppose that E R n and C is a subset of E. (a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as defined in Definition

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### K-Cover of Binary sequence

K-Cover of Binary sequence Prof Amit Kumar Prof Smruti Sarangi Sahil Aggarwal Swapnil Jain Given a binary sequence, represent all the 1 s in it using at most k- covers, minimizing the total length of all

### Data Structures Fibonacci Heaps, Amortized Analysis

Chapter 4 Data Structures Fibonacci Heaps, Amortized Analysis Algorithm Theory WS 2012/13 Fabian Kuhn Fibonacci Heaps Lacy merge variant of binomial heaps: Do not merge trees as long as possible Structure:

### Recurrence Relations

Recurrence Relations Introduction Determining the running time of a recursive algorithm often requires one to determine the big-o growth of a function T (n) that is defined in terms of a recurrence relation.

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### Strong Induction. Catalan Numbers.

.. The game starts with a stack of n coins. In each move, you divide one stack into two nonempty stacks. + + + + + + + + + + If the new stacks have height a and b, then you score ab points for the move.

### Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Intro. to the Divide-and-Conquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4

Intro. to the Divide-and-Conquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4 I. Algorithm Design and Divide-and-Conquer There are various strategies we

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### The van Hoeij Algorithm for Factoring Polynomials

The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial

### Solutions of Equations in One Variable. Fixed-Point Iteration II

Solutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

### The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge,

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints

### NIM Games: Handout 1

NIM Games: Handout 1 Based on notes by William Gasarch 1 One-Pile NIM Games Consider the following two-person game in which players alternate making moves. There are initially n stones on the board. During

### 4.4 The recursion-tree method

4.4 The recursion-tree method Let us see how a recursion tree would provide a good guess for the recurrence = 3 4 ) Start by nding an upper bound Floors and ceilings usually do not matter when solving

### 2 When is a 2-Digit Number the Sum of the Squares of its Digits?

When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property

CmSc 250 Intro to Algorithms Chapter 6. Transform and Conquer Binary Heaps 1. Definition A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called

### Lecture 6 Online and streaming algorithms for clustering

CSE 291: Unsupervised learning Spring 2008 Lecture 6 Online and streaming algorithms for clustering 6.1 On-line k-clustering To the extent that clustering takes place in the brain, it happens in an on-line

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### Game Theory: Supermodular Games 1

Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution

### Primality - Factorization

Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

### Question 1 Formatted: Formatted: Formatted: Formatted:

In many situations in life, we are presented with opportunities to evaluate probabilities of events occurring and make judgments and decisions from this information. In this paper, we will explore four

### Single machine models: Maximum Lateness -12- Approximation ratio for EDD for problem 1 r j,d j < 0 L max. structure of a schedule Q...

Lecture 4 Scheduling 1 Single machine models: Maximum Lateness -12- Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule 0 Q 1100 11 00 11 000 111 0 0 1 1 00 11 00 11 00

### 2 Primality and Compositeness Tests

Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 1635-1642 On Factoring R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 http://www.math.ucalgary.ca/

### CSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph

Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1)

COMP 50 Fall 016 9 - Mathematical induction Sept 6, 016 You will see many examples in this course and upcoming courses of algorithms for solving various problems It many cases, it will be obvious that

### CS420/520 Algorithm Analysis Spring 2010 Lecture 04

CS420/520 Algorithm Analysis Spring 2010 Lecture 04 I. Chapter 4 Recurrences A. Introduction 1. Three steps applied at each level of a recursion a) Divide into a number of subproblems that are smaller

### 15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012

15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012 1 A Take-Away Game Consider the following game: there are 21 chips on the table.

### NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics

NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer

### Lecture 4: BK inequality 27th August and 6th September, 2007

CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Cpt S 223. School of EECS, WSU

Priority Queues (Heaps) 1 Motivation Queues are a standard mechanism for ordering tasks on a first-come, first-served basis However, some tasks may be more important or timely than others (higher priority)

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### CSC2420 Fall 2012: Algorithm Design, Analysis and Theory

CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms

### T(n)=1 for sufficiently small n

Chapter 4. Recurrences Outline Offers three methods for solving recurrences, that is for obtaining asymptotic bounds on the solution In the substitution method,, we guess a bound and then use mathematical

### Union-Find Algorithms. network connectivity quick find quick union improvements applications

Union-Find Algorithms network connectivity quick find quick union improvements applications 1 Subtext of today s lecture (and this course) Steps to developing a usable algorithm. Define the problem. Find

### Randomized algorithms

Randomized algorithms March 10, 2005 1 What are randomized algorithms? Algorithms which use random numbers to make decisions during the executions of the algorithm. Why would we want to do this?? Deterministic

### Degrees that are not degrees of categoricity

Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

### 4.1 Introduction - Online Learning Model

Computational Learning Foundations Fall semester, 2010 Lecture 4: November 7, 2010 Lecturer: Yishay Mansour Scribes: Elad Liebman, Yuval Rochman & Allon Wagner 1 4.1 Introduction - Online Learning Model

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

### CS 561, Lecture 3 - Recurrences. Jared Saia University of New Mexico

CS 561, Lecture 3 - Recurrences Jared Saia University of New Mexico Recurrence Relations Oh how should I not lust after eternity and after the nuptial ring of rings, the ring of recurrence - Friedrich

### Take-Away Games MICHAEL ZIEVE

Games of No Chance MSRI Publications Volume 29, 1996 Take-Away Games MICHAEL ZIEVE Abstract. Several authors have considered take-away games where the players alternately remove a positive number of counters

### Lesson 13: Proofs in Geometry

211 Lesson 13: Proofs in Geometry Beginning with this lesson and continuing for the next few lessons, we will explore the role of proofs and counterexamples in geometry. To begin, recall the Pythagorean

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### Lecture 4 Online and streaming algorithms for clustering

CSE 291: Geometric algorithms Spring 2013 Lecture 4 Online and streaming algorithms for clustering 4.1 On-line k-clustering To the extent that clustering takes place in the brain, it happens in an on-line

### ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### CS2223 Algorithms B Term 2013 Exam 2 Solutions

CS2223 Algorithms B Term 2013 Exam 2 Solutions Dec. 3, 2013 By Prof. Carolina Ruiz Dept. of Computer Science WPI PROBLEM 1: Solving Recurrences (15 points) Solve the recurrence T (n) = 2T (n/3) + n using

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### Guessing Game: NP-Complete?

Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

### GREATEST COMMON DIVISOR

DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their

### THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES

THE SEARCH FOR NATURAL DEFINABILITY IN THE TURING DEGREES ANDREW E.M. LEWIS 1. Introduction This will be a course on the Turing degrees. We shall assume very little background knowledge: familiarity with

### 1 Construction of CCA-secure encryption

CSCI 5440: Cryptography Lecture 5 The Chinese University of Hong Kong 10 October 2012 1 Construction of -secure encryption We now show how the MAC can be applied to obtain a -secure encryption scheme.

### MONEY MANAGEMENT. Guy Bower delves into a topic every trader should endeavour to master - money management.

MONEY MANAGEMENT Guy Bower delves into a topic every trader should endeavour to master - money management. Many of us have read Jack Schwager s Market Wizards books at least once. As you may recall it

### A Practical Scheme for Wireless Network Operation

A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving

### A single minimal complement for the c.e. degrees

A single minimal complement for the c.e. degrees Andrew Lewis Leeds University, April 2002 Abstract We show that there exists a single minimal (Turing) degree b < 0 s.t. for all c.e. degrees 0 < a < 0,

### Factoring & Primality

Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in