# Introduction. An Introduction to Algorithms and Data Structures

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Introduction An Introduction to Algorithms and Data Structures

2 Overview Aims This course is an introduction to the design, analysis and wide variety of algorithms (a topic often called Algorithmics ). We shall look at the specification of computational tasks, varieties of algorithms for tasks, demonstrating (through informal proof) that algorithms perform given tasks, the structure of algorithms and, finally, measures for comparing the performance of algorithms. We also consider the implementation of algorithms and relevant data and program structures, and principles of program design.

3 Contents 1 Idea of Algorithms: algorithms and programs. Some history. Formalising algorithms, programming languages and machine models. 2 Specification and correctness. Informal correctness proofs. 3 Performance of algorithms. The issues. Performance measures. Complexity: Counting, worst- best- and average-cases. Recurrence relations for computing complexity. Asymptotic approximation. 4 Searching and sorting algorithms. 5 Trees and graphs and associated algorithms. 6 Hashing

4 Books Good books are: SEDGEWICK, R. Algorithms in C (ISBN ) Addison-Wesley A general, good quality, book on algorithms, with very good detail. CORMEN, T. H. et al, Introduction to Algorithms, (ISBN ) MIT Press This is an encyclopaedic reference, well written and extremely thorough. HAREL, D. Algorithmics; the Spirit of Computing 2nd Edition, Addison Wesley A good informal (and readable) introduction to the subject. You are also urged to consult some of the classic texts in the area, chief of which is: Knuth, D. E. Fundamental Algorithms, Addison Wesley 1968.

5 Algorithms What are they? Not really a defined idea. Informally, a recipe, or procedure, or construction; algorithms themselves consist of precise descriptions of procedures. The word: Its history lies in Arabicized Persian, from al-kuwarizmi ( the man from Kuwarizm - modern-day Khiva, in Uzbekistan, south of the Aral Sea), the mathematician Abu Ja far Muhammad ibn Musa (c780-c850 living in Baghdad) who wrote Hisab al-jabr wa l-muqabala ( Rules for restoring and equating ). The European translation of his works introduced the Hindu-Arabic notation for numbers. This is our standard notation for numbers: 124 means 1 hundred, 2 tens and 4 units! It is positional (Hindu) and has a zero (Arabic). The standard algorithms for arithmetic are expressed in this notation.

6 Complexity of Algorithms Idea: We want to compare the performance of different algorithms for a given task in terms of some measure of efficiency. For example: We may want to find how long an algorithm (coded as a program) takes to compute a result. But: This varies with (1) program code, (2) programming language, (3) compiler, (4) processor etc (operating system, weather...). Time itself is not a useful measure. We seek: Measures that are intrinsic to an algorithm (ie independent of all the conditions above), but nevertheless give a real assessment of the performance of algorithms.

7 Complexity measures Such measures are called complexity measures and consist simply of counting, as follows: Two common measures The TIME COMPLEXITY of an algorithm consists of a count of the number of time-critical operations as the size of the input varies. The SPACE COMPLEXITY of an algorithm consists of a count of the number of space cells as the size of the input varies. The choice of which operations are time critical and what constitutes a space cell, varies according to application, as we shall see. NB Complexity is about performance measures, not about the intricacy of programs.

8 Complexity Measures Worst-case, best-case and average-case measures: Suppose we are given an algorithm and we wish to assess its performance on inputs of size N. Usually, even with inputs of a fixed size, the performance can vary considerably according to the contents of the input and the arrangement of data. However, we can identify: The best-case performance for each size N - the smallest time complexity (or space complexity) over all inputs of size N. The worst-case performance for each size N - the largest time complexity (or space complexity) over all inputs of size N. These are extremal measures, and we would like to be able to assess the performance on average, but this is more difficult to compute as we shall see.

9 An Example: Searching TASK: Given an item, and a sequence/list of items find whether or not the item occurs in the sequence. [Part of a family of problems: find all occurrences, find number of occurrences, find first occurrence etc.] Algorithm 1: Sequential search If sequence empty then stop with failure. Otherwise, compare given item with head of sequence, using equality test. If found then stop with success, else continue down tail of sequence, successively testing for equality, until success or sequence becomes empty.

10 Performance The performance of this algorithm clearly varies with the length of the sequence. Let N be the length. For time complexity, which operation is time critical? Usually, it is the equality test of the item against members of the sequence. An alternative is the operation of access to an item in the sequence, but for this algorithm each access is followed by one equality test so they are equivalent.

11 Performance Given a sequence of length N, how many equality tests are performed? If the item sought is found at the head of the sequence then 1 equality test only is required: Best-case time complexity = 1. This is constant time i.e. independent of N. If the item sought is found at the end of the sequence or is not in the sequence then N equality tests are required: Worst-case time complexity = N. This is linear. The best-case is the lower bound and the worst case is the upper bound of any actual performance.

12 Average case Average-case time complexity Is either of these - the best-case or the worst-case - a realistic measure of what happens on average? Let us try to define average-case time complexity. This is a difficult concept and often difficult to compute. It involves probability and infinite distributions. What is an average sequence of integers? If all integers are equally probable in the sequence then the probability that any given integer is in the sequence is zero! (Probability zero does not mean impossible when there are infinite distributions around.)

13 For sequential search on sequences of random integers this means Average-case time complexity = N. This is linear and the same as worst-case. Note: The average case is NOT the average of the best-case and worst-case, but is an average over all possible input sequences and their probabilities of occurrence. If the sought item is known to occur exactly once in the sequence and we seek its position, then the average case is N/2.

14 Example: Binary Search Algorithm 2: Binary Search Suppose the items in the sequence are integers - or any type with a given (total) order relation on it - and the sequence is already in ascending (i.e. non-descending) order. Binary search is: 1 If the sequence is empty, return false. 2 If the sequence contains just one item, then return the result of the equality test. 3 Otherwise: compare given item with a middle item: If equal, then return success. If given item is less than middle item, then recursively call binary search on the sequence of items to the left of the chosen middle item. If given item is greater than middle item, then recursively call binary search on the sequence of items to the right of the chosen middle item.

15 Binary search: Program Program: Could write this recursively, as suggested, or iteratively as below. public static int binarysearch( Comparable [] seq, Comparable item ) { int left = 0; int right = seq.length -1; int middle, comparison; while (left <= right) { middle = (left+right)/2; comparison = seq[middle].compareto(item); } if (comparison == 0) return middle; else if (comparison < 0) right = middle-1; else left = middle+1; } return -1;

16 Note The use of Comparable class and compareto method to capture the genericity of the algorithm. The comparison here is a three way comparison, including equality, but other forms are possible. The algorithm returns either 1 for false, or the position of the first found occurrence. The need for arrays, rather than, say linked lists, because we need random access i.e. access to all parts (esp middle item) in constant-time. Compare sequential search where either arrays or linked lists would be appropriate.

17 Correctness The correctness of binary search The argument for the correctness of the algorithm goes as follows: Suppose we seek an item in a sequence of length N. We proceed by induction on N. If N = 0 then the algorithm returns 1 (false) as required.

18 Correctness (continued) Otherwise, suppose the algorithm is correct on all ascending sequences of length less than N. If we can show it is correct on all ascending sequences of length N, then by induction, it is correct for all finite ascending sequences. On an ascending sequence of length N, it first selects an item (the middle item) and compares it with the given item, if equal it returns the position, as required. If not equal, then because the sequence is in ascending order, if the item sought is less than the middle item, then it can only occur in the left subsequence, which itself is an ascending subsequence, but of length at least 1 less than N, so by induction it will correctly find the item, or return false. Similarly, for the case when the item is greater than the middle item.

19 Correctness (continued) Induction is a characteristic method for iterative and recursive algorithms as it allows us to establish that something holds assuming that the remaining part of the computation - the recursive call, or remaining loop computation - is correct. Notice that correctness depends upon the ascending order, but NOT on which element is chosen as the middle element (in fact any element would do for correctness).

20 Performance Performance of binary search: time complexity We count the number of comparisons. The algorithm is recursive. How do we count the number of operations in a recursive algorithm? The worst case is when each attempt to match to a middle element fails and we then approximately half the task. This is the key observation. If we continue halving at each test until one or no elements is left, how many tests are required? Suppose the list is of length 2 M. The first test failing means that we search in a sequence of length approximately 2 M /2 = 2 M 1. So approximately M + 1 comparison tests are required in the worst case.

21 Logarithms Logarithmic performance In general, for sequence length N, how many tests are required in the worst case? That is, how often do we need to halve the sequence to end up with the final test? The answer is approximately log 2 (N). Reminder: The logarithm to base 2 of a number N is the power of 2 which equals N. Thus 2 log 2 (N) = N. In general, the logarithm is defined by: A log A (B) = B.

22 Logarithms (continued) Logarithms grow much, much slower that linear - so algorithms that performs logarithmically are very fast: Linear performance: If the input size doubles, the complexity doubles. Logarithmic performance: If the input size doubles, the complexity increases by 1! Input size Logarithmic Linear Quadratic Exponential N log 2 (N) N N 2 2 N , , ,000, , , ,000,

23 Summary Thus binary search is very efficient even in the worst case (we can picture this by saying that half of the material is disposed at each test without looking at it). It is also very space efficient. It can be implemented in-place - without the need for space in addition to that occupied by the input. However, it does need the items to be in ascending order.

24 Arithmetic with logarithms The following rules hold for logarithms (all direct from the definition): log A (A) = 1, log A (B C) = log A (B) + log A (C) log A (1/B) = log A (B), log B (C) = log B (A) log A (C). The second rule above is that for which logarithms were developed - doing multiplication by addition. The final rule allows us to change base of logarithms by multiplying by a constant.

25 Recurrence relations Recurrence relations... are a general method of calculating the complexity of recursive and iterative algorithms. As an example, consider binary search. We introduce the notation that C N is the number of comparison operations on input of size N in the worst case. How do we calculate C N?

26 Looking at the algorithm, we see that in the worst case, the number of operations on input size N is 1 (the equality test with the middle element) and then a worst case search on input size approximately N/2. We can this write for N > 0, and for C 0 = 0. C N = C N/2 + 1 This is called a recurrence relation. It is a recursive expression for a numerical quantity, computing complexity measures by following the recursive structure of an algorithm. (Notice that there is some approximation going on here which we will discuss later.)

27 Recurrence relations have solutions. Sometimes more than one, sometimes one, sometimes none. The cases we deal with derived for computing the complexity of algorithms have unique solutions (or a unique family of solutions). The solution of the above recurrence relation is C N = log 2 (N) + 1. We can see this simply by substituting: Left-hand side is log 2 (N) + 1. Right-hand side is C N/2 + 1 = (log 2 (N/2) + 1) + 1 = log 2 (N 1/2) = log 2 (N) + log 2 (1/2) = log 2 (N) log 2 (2) = log 2 (N) = log 2 (N) + 1. Hence left-hand and right-hand sides are equal, as required.

28 Solutions Finding solutions Given a solution, we can check it. But our task is to find solutions of recurrence relations. How do we do this? A simple approach is to expand the recurrence and see what happens: C N = C N/2 + 1 = (C N/4 + 1) + 1 = ((C N/8 + 1) + 1) + 1 = = C where there are log 2 (N) number of 1 s. Hence C N = log 2 (N) + 1 as required. In general, algorithms which halve the input at each step have complexity in terms of logarithms base 2 (if a third at each step then log base 3 etc).

29 Rates of growth Algorithm complexity: Approximation and rates of growth In the above analysis we have used approximations: Saying divided approximately in half and even using log 2 (N) (which in general is not an integer), rather than ceiling(log 2 (N)) (the least integer above log 2 (N). Why is this? It is because we are usually only concerned with the rate of growth of the complexity as the input size increases. Complexity (that is the number of operations undertaken) at a fixed size of input, does NOT in general convert to a running time for an algorithm. However, if the operations are chosen correctly, then the rate of growth of the complexity WILL correspond to the rate of growth of the running time.

30 Rates of growth Rate of growth of numerical expressions For complexity log 2 (N) + 1 is the 1 significant? No. Whatever the constant is, if the input size doubles the complexity increases by 1. For small N, the constant is a significant part of the complexity, so that log 2 (N) + 1 and log 2 (N) + 10 are different, but as N increases the constant looses its significance. For complexity N N as N increases so does the role of the N 2 part, and the expression grows like N 2 for large N. For N log 2 (N) only the linear term N is significant for large N.

31 Rates of growth In general, for small N the behaviour of an algorithm may be erratic, involving overheads of setting up data structures, initialization, small components of the complexity, different operations may be important, etc. We are interested in the rate of growth of expressions as N increases, i.e. what happens at large N?

32 When N doubles: log 2 (N) increases by N doubles 12 N 2 quadruples ( 4) 2 N 3 increases eightfold 2 N squares (because 2 2 N = (2 N ) 2 ) Exponential algorithms are infeasible - they cannot be run on substantial inputs to provide results. Note: Exponential includes factorial N!

33 Rates of growth When N doubles, what happens to N N ? For small N it is approximately constant (because is big!). For intermediate N it is linear (because 1000 N dominates over N 2 ) But eventually N 2 dominates the other terms and the behaviour becomes closer to quadratic as N increases. We introduce a notation for this and say that N N is O(N 2 ) (Oh, not zero!) and read this as the expression is of order N 2. This is called BIG-OH notation.

34 Rates of growth Defining order of growth Definition A function f (N) is of order g(n), written f (N) is O(g(N)), just when there is a constant c such that, for all N sufficiently large f (N) c g(n) i.e. c. N 0. N N 0.f (N) c g(n). We say g(n) eventually dominates f (N). Examples 1 3 N 2 is O(N 2 ) 2 N log 2 (N) + N + 1 is O(N log 2 (N)) 3 log 3 (N) is O(log 2 (N)) because log 3 (N) = log 3 (2) log 2 (N)

35 Recurrence relations Recurrence relations and their solutions Some recurrence relations C N = C N/2 + a where a is a constant. Do constant amount of work and then halve the remainder Solution: C N = O(log N). C N = C N 1 + a where a is a constant. Do constant amount of work and then repeat on input of size one less Typically, this is a simple loop with a constant amount of work per execution. Example: sequential search. Solution: C N = O(N). C N = C N 1 + a N where a is a constant. Do amount of work proportional to N and then repeat on input of size one less Typically, two nested loops.

36 Feasible vs Infeasible algorithms Infeasible Algorithms Algorithms with complexities O(1), O(N), O(N 2 ), O(N 3 ), O(N 4 )... and between are called polynomial, and are considered as feasible - ie can be run on large inputs (though, of course, cubic, quartic etc can be very slow). Exponential algorithms are called infeasible and take considerable computational resources even for small inputs (and on large inputs can often exceed the capacity of the known universe!).

37 Feasible vs Infeasible algorithms Examples of Infeasible Algorithms... are often based on unbounded exhaustive backtracking to perform an exhaustive search. Backtracking is trial-and-error : making decisions until it is clear that a solution is not possible, then undoing previous decisions, to try another route to reach a solution. Unbounded means that we cannot fix how far back we may need to undo decisions. The decisions generate a tree and exhaustive search means traversing this tree to visit all the nodes.

38 Examples: Infeasible algorithms Example: Travelling around graphs Consider graphs (directed or undirected), we can consider various path finding problems, such as: Hamiltonian cycle (after Sir William Hamilton) Starting at a node, seek a path which travels around the graph, finishing at the start node and visiting all nodes just once. Travelling Salesperson For graphs with distances assigned to edges, seek a minimum length Hamiltonian cycle. Only known algorithms for both problems are by unbounded exhaustive backtracking and are therefore exponential.

39 Why exponential? C N = N C N 1 + a because, in the worst case, we make one of N initial choices (the first node visited) and then have a problem of size N 1, the choice requires a computation of fixed amount a. The solution is C N = O(N!) = O(N N ).

40 Other examples: Knapsack problems, games and puzzles Given an empty box of fixed dimensions, and a number of blocks of various dimensions: The task is to choose some of the blocks and arrange them in the box so that they occupy a maximum amount of space i.e. the empty (unused) space is minimized. Algorithms are by exhaustive search.

41 Other examples: Games and puzzles Many games and puzzles have only infeasible algorithms. That is what makes them interesting. Compare noughts-and-crosses, which has a simple algorithm. For example, the Polyominos puzzle: Given a range of pieces, of polyomino shape (ie consisting of squares of constant size fixed together orthogonally, choose some and fit them onto a chessboard to cover all squares exactly (or on any other fixed shape board).

42 Complexity theory Question: Given a computational task, what performance of algorithms can we expect for it? Two approaches: Invent a range of good algorithms. Look at the nature of the task and try to determine what range of algorithms can exist. Example: Matrix multiplication The standard algorithm for multiplication of two N N matrices is (obviously) O(N 3 ). For a long while, this was believed to be the best possible but... Strassen s algorithm (1968) is O(N )!! and slightly better ones

43 Example: Sorting The task of sorting is to rearrange a list (of integers, say) into ascending order. We shall see later that there are algorithms with worst-case time complexity O(N log(n)) for this task. Moreover this is a lower bound: we can show that any general sorting algorithm must be at least O(N log(n)). There is no computational gap.

44 Famous unsolved problem The examples of tasks which appear to have only exhaustive search algorithms (travelling salesperson, knapsack etc) are believed to be intrinsically exponential (ie there are no faster algorithms). This is the essence of the famous conjecture in complexity theory. P NP (P stands for polynomial - those problems which have polynomial algorithms, and NP stands for non-deterministic polynomial, and is a class of problems including travelling salesperson, knapsack problems etc.)

### Computer Science 385 Analysis of Algorithms Siena College Spring Topic Notes: Limitations of Algorithms

Computer Science 385 Analysis of Algorithms Siena College Spring 2011 Topic Notes: Limitations of Algorithms We conclude with a discussion of the limitations of the power of algorithms. That is, what kinds

### O Notation (Big Oh) We want to give an upper bound on the amount of time it takes to solve a problem.

O Notation (Big Oh) We want to give an upper bound on the amount of time it takes to solve a problem. defn: v(n) = O(f(n)) constants c and n 0 such that v(n) c f(n) whenever n > n 0 Termed complexity:

### Analysis of Algorithms

October 1, 2015 Analysis of Algorithms CS 141, Fall 2015 1 Analysis of Algorithms: Issues Correctness/Optimality Running time ( time complexity ) Memory requirements ( space complexity ) Power I/O utilization

### Lecture 29: Tractable and Intractable Problems

Lecture 29: Tractable and Intractable Problems Aims: To look at the ideas of polynomial and exponential functions and algorithms; and tractable and intractable problems. Page 1 of 10 To look at ways of

### Topic 17. Analysis of Algorithms

Topic 17 Analysis of Algorithms Analysis of Algorithms- Review Efficiency of an algorithm can be measured in terms of : Time complexity: a measure of the amount of time required to execute an algorithm

### N/4 + N/2 + N = 2N 2.

CS61B Summer 2006 Instructor: Erin Korber Lecture 24, 7 Aug. 1 Amortized Analysis For some of the data structures we ve discussed (namely hash tables and splay trees), it was claimed that the average time

### CS 350 Algorithms and Complexity

1 CS 350 Algorithms and Complexity Fall 2015 Lecture 15: Limitations of Algorithmic Power Introduction to complexity theory Andrew P. Black Department of Computer Science Portland State University Lower

### 1 Computational problems

80240233: Computational Complexity Lecture 1 ITCS, Tsinghua Univesity, Fall 2007 9 October 2007 Instructor: Andrej Bogdanov Notes by: Andrej Bogdanov The aim of computational complexity theory is to study

### Analysis of Algorithms I: Asymptotic Notation, Induction, and MergeSort

Analysis of Algorithms I: Asymptotic Notation, Induction, and MergeSort Xi Chen Columbia University We continue with two more asymptotic notation: o( ) and ω( ). Let f (n) and g(n) are functions that map

### Running Time Evaluation

Running Time Evaluation Quadratic Vs. Linear Time Lecturer: Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 19 1 Running time 2 Examples 3 Big-Oh, Big-Omega, and Big-Theta Tools 4 Time

### Algorithms and Complexity Theory. Chapter 8: Introduction to Complexity. Computer Science - Durban - September 2005

Algorithms and Complexity Theory Chapter 8: Introduction to Complexity Jules-R Tapamo Computer Science - Durban - September 2005 Contents 1 Introduction 2 1.1 Dynamic programming...................................

### Unit 1A: Computational Complexity

Unit 1A: Computational Complexity Course contents: Computational complexity NP-completeness Algorithmic Paradigms Readings Chapters 3, 4, and 5 Unit 1A 1 O: Upper Bounding Function Def: f(n)= O(g(n)) if

### 1 Reductions and Expressiveness

15-451/651: Design & Analysis of Algorithms November 3, 2015 Lecture #17 last changed: October 30, 2015 In the past few lectures we have looked at increasingly more expressive problems solvable using efficient

### P, NP, NP-Complete, and NPhard

P, NP, NP-Complete, and NPhard Problems Zhenjiang Li 21/09/2011 Outline Algorithm time complicity P and NP problems NP-Complete and NP-Hard problems Algorithm time complicity Outline What is this course

### Computational complexity and some Graph Theory

Graph Theory Lars Hellström January 22, 2014 Contents of todays lecture An important concern when choosing the algorithm to use for something (after basic requirements such as correctness and stability)

### (tree searching technique) (Boolean formulas) satisfying assignment: (X 1, X 2 )

Algorithms Chapter 5: The Tree Searching Strategy - Examples 1 / 11 Chapter 5: The Tree Searching Strategy 1. Ex 5.1Determine the satisfiability of the following Boolean formulas by depth-first search

### Computer Sciences Department

Computer Sciences Department 1 Reference Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER Computer Sciences Department 3 ADVANCED TOPICS IN C O M P U T A B I L I T Y

### Module 1: Analyzing the Efficiency of Algorithms

Module 1: Analyzing the Efficiency of Algorithms Dr. Natarajan Meghanathan Associate Professor of Computer Science Jackson State University Jackson, MS 39217 E-mail: natarajan.meghanathan@jsums.edu Based

### NP-Completeness. Until now we have been designing algorithms for specific problems

NP-Completeness 1 Introduction Until now we have been designing algorithms for specific problems We have seen running times O(log n), O(n), O(n log n), O(n 2 ), O(n 3 )... We have also discussed lower

### Module 1: Analyzing the Efficiency of Algorithms

Module 1: Analyzing the Efficiency of Algorithms Dr. Natarajan Meghanathan Professor of Computer Science Jackson State University Jackson, MS 39217 E-mail: natarajan.meghanathan@jsums.edu What is an Algorithm?

### MA008/MIIZ01 Design and Analysis of Algorithms Lecture Notes 2

MA008 p.1/36 MA008/MIIZ01 Design and Analysis of Algorithms Lecture Notes 2 Dr. Markus Hagenbuchner markus@uow.edu.au. MA008 p.2/36 Content of lecture 2 Examples Review data structures Data types vs. data

### A point p is said to be dominated by point q if p.x=q.x&p.y=q.y 2 true. In RAM computation model, RAM stands for Random Access Model.

In analysis the upper bound means the function grows asymptotically no faster than its largest term. 1 true A point p is said to be dominated by point q if p.x=q.x&p.y=q.y 2 true In RAM computation model,

### Q = Set of states, IE661: Scheduling Theory (Fall 2003) Primer to Complexity Theory Satyaki Ghosh Dastidar

IE661: Scheduling Theory (Fall 2003) Primer to Complexity Theory Satyaki Ghosh Dastidar Turing Machine A Turing machine is an abstract representation of a computing device. It consists of a read/write

### Analysis of Algorithms

September 29, 2017 Analysis of Algorithms CS 141, Fall 2017 1 Analysis of Algorithms: Issues Correctness/Optimality Running time ( time complexity ) Memory requirements ( space complexity ) Power I/O utilization

### Circuits. CSE 373 Data Structures

Circuits CSE 373 Data Structures Readings Reading Alas not in your book. So it won t be on the final! Circuits 2 Euler Euler (1707-1783): might be the most prolific mathematician of all times (analysis,

### Data Structures and Algorithms Chapter 2

1 Data Structures and Algorithms Chapter 2 Werner Nutt 2 Acknowledgments The course follows the book Introduction to Algorithms, by Cormen, Leiserson, Rivest and Stein, MIT Press [CLRST]. Many examples

### Space Complexity. The space complexity of a program is how much memory it uses.

Space Complexity The space complexity of a program is how much memory it uses. Measuring Space When we compute the space used by a TM, we do not count the input (think of input as readonly). We say that

### CS Analysis of Recursive Algorithms and Brute Force

CS483-05 Analysis of Recursive Algorithms and Brute Force Instructor: Fei Li Room 443 ST II Office hours: Tue. & Thur. 4:30pm - 5:30pm or by appointments lifei@cs.gmu.edu with subject: CS483 http://www.cs.gmu.edu/

### CS 4104 Data and Algorithm Analysis. Recurrence Relations. Modeling Recursive Function Cost. Solving Recurrences. Clifford A. Shaffer.

Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 2010,2017 by Clifford A. Shaffer Data and Algorithm Analysis Title page Data and Algorithm Analysis Clifford A. Shaffer Spring

### Introduction to Divide and Conquer

Introduction to Divide and Conquer Sorting with O(n log n) comparisons and integer multiplication faster than O(n 2 ) Periklis A. Papakonstantinou York University Consider a problem that admits a straightforward

### Big O 2/14/13. Administrative. Does it terminate? David Kauchak cs302 Spring 2013

/4/3 Administrative Big O David Kauchak cs3 Spring 3 l Assignment : how d it go? l Assignment : out soon l CLRS code? l Videos Insertion-sort Insertion-sort Does it terminate? /4/3 Insertion-sort Loop

### Algorithms for pattern involvement in permutations

Algorithms for pattern involvement in permutations M. H. Albert Department of Computer Science R. E. L. Aldred Department of Mathematics and Statistics M. D. Atkinson Department of Computer Science D.

### A Brief Introduction to Proofs

A Brief Introduction to Proofs William J. Turner October, 010 1 Introduction Proofs are perhaps the very heart of mathematics. Unlike the other sciences, mathematics adds a final step to the familiar scientific

### AMA1D01C Mathematics in the Islamic World

Hong Kong Polytechnic University 2017 Introduction Major mathematician: al-khwarizmi (780-850), al-uqlidisi (920-980), abul-wafa (940-998), al-karaji (953-1029), al-biruni (973-1048), Khayyam (1048-1131),

### Midterm Exam. CS 3110: Design and Analysis of Algorithms. June 20, Group 1 Group 2 Group 3

Banner ID: Name: Midterm Exam CS 3110: Design and Analysis of Algorithms June 20, 2006 Group 1 Group 2 Group 3 Question 1.1 Question 2.1 Question 3.1 Question 1.2 Question 2.2 Question 3.2 Question 3.3

### Theory of Computation

Theory of Computation Dr. Sarmad Abbasi Dr. Sarmad Abbasi () Theory of Computation 1 / 33 Lecture 20: Overview Incompressible strings Minimal Length Descriptions Descriptive Complexity Dr. Sarmad Abbasi

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

Algorithm Analysis 1 Purpose Why bother analyzing code; isn t getting it to work enough? Estimate time and memory in the average case and worst case Identify bottlenecks, i.e., where to reduce time Compare

### The next sequence of lectures in on the topic of Arithmetic Algorithms. We shall build up to an understanding of the RSA public-key cryptosystem.

CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 10 The next sequence of lectures in on the topic of Arithmetic Algorithms. We shall build up to an understanding of the RSA public-key cryptosystem.

### Fermat s Little Theorem. Fermat s little theorem is a statement about primes that nearly characterizes them.

Fermat s Little Theorem Fermat s little theorem is a statement about primes that nearly characterizes them. Theorem: Let p be prime and a be an integer that is not a multiple of p. Then a p 1 1 (mod p).

### b + O(n d ) where a 1, b > 1, then O(n d log n) if a = b d d ) if a < b d O(n log b a ) if a > b d

CS161, Lecture 4 Median, Selection, and the Substitution Method Scribe: Albert Chen and Juliana Cook (2015), Sam Kim (2016), Gregory Valiant (2017) Date: January 23, 2017 1 Introduction Last lecture, we

### Data Structures and Algorithms. Asymptotic notation

Data Structures and Algorithms Asymptotic notation Estimating Running Time Algorithm arraymax executes 7n 1 primitive operations in the worst case. Define: a = Time taken by the fastest primitive operation

### 1 Terminology and setup

15-451/651: Design & Analysis of Algorithms August 31, 2017 Lecture #2 last changed: August 29, 2017 In this lecture, we will examine some simple, concrete models of computation, each with a precise definition

### Examination paper for TDT4120 Algorithms and Data Structures

Examination paper for TDT4120 Algorithms and Data Structures Academic contact during examination Magnus Lie Hetland Phone 91851949 Examination date December 7, 2013 Examination time (from to) 0900 1300

### Advanced Implementations of Tables: Balanced Search Trees and Hashing

Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the

### Solutions. Problem 1: Suppose a polynomial in n of degree d has the form

Assignment 1 1. Problem 3-1 on p. 57 2. Problem 3-2 on p. 58 3. Problem 4-5 on p. 86 4. Problem 6-1 on p. 142 5. Problem 7-4 on p. 162 6. Prove correctness (including halting) of SelectionSort (use loop

### Growth of Functions (CLRS 2.3,3)

Growth of Functions (CLRS 2.3,3) 1 Review Last time we discussed running time of algorithms and introduced the RAM model of computation. Best-case running time: the shortest running time for any input

### CS 61B Asymptotic Analysis Fall 2017

CS 61B Asymptotic Analysis Fall 2017 1 More Running Time Give the worst case and best case running time in Θ( ) notation in terms of M and N. (a) Assume that slam() Θ(1) and returns a boolean. 1 public

### Santa Claus Schedules Jobs on Unrelated Machines

Santa Claus Schedules Jobs on Unrelated Machines Ola Svensson (osven@kth.se) Royal Institute of Technology - KTH Stockholm, Sweden March 22, 2011 arxiv:1011.1168v2 [cs.ds] 21 Mar 2011 Abstract One of the

Advanced Algorithmics (6EAP) MTAT.03.238 Order of growth maths Jaak Vilo 2017 fall Jaak Vilo 1 Program execution on input of size n How many steps/cycles a processor would need to do How to relate algorithm

### Once they had completed their conquests, the Arabs settled down to build a civilization and a culture. They became interested in the arts and

The Islamic World We know intellectual activity in the Mediterranean declined in response to chaos brought about by the rise of the Roman Empire. We ve also seen how the influence of Christianity diminished

### Mm7 Intro to distributed computing (jmp) Mm8 Backtracking, 2-player games, genetic algorithms (hps) Mm9 Complex Problems in Network Planning (JMP)

Algorithms and Architectures II H-P Schwefel, Jens M. Pedersen Mm6 Advanced Graph Algorithms (hps) Mm7 Intro to distributed computing (jmp) Mm8 Backtracking, 2-player games, genetic algorithms (hps) Mm9

### /633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programming II Date: 10/12/17

601.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programming II Date: 10/12/17 12.1 Introduction Today we re going to do a couple more examples of dynamic programming. While

### Problem. Problem Given a dictionary and a word. Which page (if any) contains the given word? 3 / 26

Binary Search Introduction Problem Problem Given a dictionary and a word. Which page (if any) contains the given word? 3 / 26 Strategy 1: Random Search Randomly select a page until the page containing

### Complexity and NP-completeness

Lecture 17 Complexity and NP-completeness Supplemental reading in CLRS: Chapter 34 As an engineer or computer scientist, it is important not only to be able to solve problems, but also to know which problems

### Divide-and-Conquer. Reading: CLRS Sections 2.3, 4.1, 4.2, 4.3, 28.2, CSE 6331 Algorithms Steve Lai

Divide-and-Conquer Reading: CLRS Sections 2.3, 4.1, 4.2, 4.3, 28.2, 33.4. CSE 6331 Algorithms Steve Lai Divide and Conquer Given an instance x of a prolem, the method works as follows: divide-and-conquer

### This is a recursive algorithm. The procedure is guaranteed to terminate, since the second argument decreases each time.

8 Modular Arithmetic We introduce an operator mod. Let d be a positive integer. For c a nonnegative integer, the value c mod d is the remainder when c is divided by d. For example, c mod d = 0 if and only

### Ma/CS 117c Handout # 5 P vs. NP

Ma/CS 117c Handout # 5 P vs. NP We consider the possible relationships among the classes P, NP, and co-np. First we consider properties of the class of NP-complete problems, as opposed to those which are

### Determine the size of an instance of the minimum spanning tree problem.

3.1 Algorithm complexity Consider two alternative algorithms A and B for solving a given problem. Suppose A is O(n 2 ) and B is O(2 n ), where n is the size of the instance. Let n A 0 be the size of the

### A strongly polynomial algorithm for linear systems having a binary solution

A strongly polynomial algorithm for linear systems having a binary solution Sergei Chubanov Institute of Information Systems at the University of Siegen, Germany e-mail: sergei.chubanov@uni-siegen.de 7th

### Introduction to Algorithms

Lecture 1 Introduction to Algorithms 1.1 Overview The purpose of this lecture is to give a brief overview of the topic of Algorithms and the kind of thinking it involves: why we focus on the subjects that

### Notes on induction proofs and recursive definitions

Notes on induction proofs and recursive definitions James Aspnes December 13, 2010 1 Simple induction Most of the proof techniques we ve talked about so far are only really useful for proving a property

### CIS 121 Data Structures and Algorithms with Java Spring Big-Oh Notation Monday, January 22/Tuesday, January 23

CIS 11 Data Structures and Algorithms with Java Spring 018 Big-Oh Notation Monday, January /Tuesday, January 3 Learning Goals Review Big-Oh and learn big/small omega/theta notations Discuss running time

CMPSCI611: NP Completeness Lecture 17 Essential facts about NP-completeness: Any NP-complete problem can be solved by a simple, but exponentially slow algorithm. We don t have polynomial-time solutions

### Lecture 20: PSPACE. November 15, 2016 CS 1010 Theory of Computation

Lecture 20: PSPACE November 15, 2016 CS 1010 Theory of Computation Recall that PSPACE = k=1 SPACE(nk ). We will see that a relationship between time and space complexity is given by: P NP PSPACE = NPSPACE

### Asymptotic Running Time of Algorithms

Asymptotic Complexity: leading term analysis Asymptotic Running Time of Algorithms Comparing searching and sorting algorithms so far: Count worst-case of comparisons as function of array size. Drop lower-order

### NP Complete Problems. COMP 215 Lecture 20

NP Complete Problems COMP 215 Lecture 20 Complexity Theory Complexity theory is a research area unto itself. The central project is classifying problems as either tractable or intractable. Tractable Worst

### Exercise Sheet #1 Solutions, Computer Science, A M. Hallett, K. Smith a k n b k = n b + k. a k n b k c 1 n b k. a k n b.

Exercise Sheet #1 Solutions, Computer Science, 308-251A M. Hallett, K. Smith 2002 Question 1.1: Prove (n + a) b = Θ(n b ). Binomial Expansion: Show O(n b ): (n + a) b = For any c 1 > b = O(n b ). Show

### P vs NP & Computational Complexity

P vs NP & Computational Complexity Miles Turpin MATH 89S Professor Hubert Bray P vs NP is one of the seven Clay Millennium Problems. The Clay Millenniums have been identified by the Clay Mathematics Institute

### Concrete models and tight upper/lower bounds

Lecture 3 Concrete models and tight upper/lower bounds 3.1 Overview In this lecture, we will examine some simple, concrete models of computation, each with a precise definition of what counts as a step,

### The space complexity of a standard Turing machine. The space complexity of a nondeterministic Turing machine

298 8. Space Complexity The space complexity of a standard Turing machine M = (Q,,,, q 0, accept, reject) on input w is space M (w) = max{ uav : q 0 w M u q av, q Q, u, a, v * } The space complexity of

### 1 Acceptance, Rejection, and I/O for Turing Machines

1 Acceptance, Rejection, and I/O for Turing Machines Definition 1.1 (Initial Configuration) If M = (K,Σ,δ,s,H) is a Turing machine and w (Σ {, }) then the initial configuration of M on input w is (s, w).

### Reductions. Reduction. Linear Time Reduction: Examples. Linear Time Reductions

Reduction Reductions Problem X reduces to problem Y if given a subroutine for Y, can solve X. Cost of solving X = cost of solving Y + cost of reduction. May call subroutine for Y more than once. Ex: X

### when r x when x 5 9 Was the expression evaluated correctly using the order of operations? If not, find and correct the error.

Practice A For use with pages 8 13 Name the operation that would be performed first. 1. 18 2 5 1 12 2. 5 p 8 4 4 3. 27 4 (14 2 5) 4. 36 4 4 p 3 5. 8 1 10 4 5 2 3 6. 4 p (7 2 3) 2 7. 15 2 8 1 4 8. 13 2

### 11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic

11 Division Mod n, Linear Integer Equations, Random Numbers, The Fundamental Theorem of Arithmetic Bezout s Lemma Let's look at the values of 4x + 6y when x and y are integers. If x is -6 and y is 4 we

### Dynamic Programming. Data Structures and Algorithms Andrei Bulatov

Dynamic Programming Data Structures and Algorithms Andrei Bulatov Algorithms Dynamic Programming 18-2 Weighted Interval Scheduling Weighted interval scheduling problem. Instance A set of n jobs. Job j

### Finding Prime Factors

Section 3.2 PRE-ACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to

### S1 Revision/Notes Algebra Unit 2 Solving Equations

S1 Revision/Notes Algebra Unit 2 Solving Equations Algebra History While the word algebra comes from the Arabic language (al-jabr, ال ج بر literally, restoration) and much of its methods from Arabic/Islamic

### Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 27 Flow Shop Scheduling - Heuristics - Palmer, Campbell Dudek

### Discrete Structures CRN Test 3 Version 1 CMSC 2123 Spring 2011

Discrete Structures CRN 1858 Test 3 Version 1 CMSC 13 Spring 011 1. Print your name on your scantron in the space labeled NAME.. Print CMSC 13 in the space labeled SUBJECT. 3. Print the date, 5--011, in

### Mathematical Induction

Mathematical Induction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Mathematical Induction Fall 2014 1 / 21 Outline 1 Mathematical Induction 2 Strong Mathematical

### Sorting n Objects with a k-sorter. Richard Beigel. Dept. of Computer Science. P.O. Box 2158, Yale Station. Yale University. New Haven, CT

Sorting n Objects with a -Sorter Richard Beigel Dept. of Computer Science P.O. Box 258, Yale Station Yale University New Haven, CT 06520-258 John Gill Department of Electrical Engineering Stanford University

### 1 Computational Problems

Stanford University CS254: Computational Complexity Handout 2 Luca Trevisan March 31, 2010 Last revised 4/29/2010 In this lecture we define NP, we state the P versus NP problem, we prove that its formulation

### Discrete mathematics is the study of techniques, ideas and modes of

CHAPTER 1 Discrete Systems Discrete mathematics is the study of techniques, ideas and modes of reasoning that are indispensable in applied disciplines such as computer science or information technology.

### Lecture 14: Nov. 11 & 13

CIS 2168 Data Structures Fall 2014 Lecturer: Anwar Mamat Lecture 14: Nov. 11 & 13 Disclaimer: These notes may be distributed outside this class only with the permission of the Instructor. 14.1 Sorting

### Chapter 2: The Basics. slides 2017, David Doty ECS 220: Theory of Computation based on The Nature of Computation by Moore and Mertens

Chapter 2: The Basics slides 2017, David Doty ECS 220: Theory of Computation based on The Nature of Computation by Moore and Mertens Problem instances vs. decision problems vs. search problems Decision

### Alternative Combinatorial Gray Codes

Alternative Combinatorial Gray Codes Cormier-Iijima, Samuel sciyoshi@gmail.com December 17, 2010 Abstract Gray codes have numerous applications in a variety of fields, including error correction, encryption,

### DRAFT. Algebraic computation models. Chapter 14

Chapter 14 Algebraic computation models Somewhat rough We think of numerical algorithms root-finding, gaussian elimination etc. as operating over R or C, even though the underlying representation of the

### D1 Discrete Mathematics The Travelling Salesperson problem. The Nearest Neighbour Algorithm The Lower Bound Algorithm The Tour Improvement Algorithm

1 iscrete Mathematics The Travelling Salesperson problem The Nearest Neighbour lgorithm The Lower ound lgorithm The Tour Improvement lgorithm The Travelling Salesperson: Typically a travelling salesperson

### REACHING ROW FIVE IN SOLITAIRE ARMY. Simon Tatham No current affiliation.

REACHING ROW FIVE IN SOLITAIRE ARMY Simon Tatham No current affiliation simon.tatham@pobox.com Gareth Taylor Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA,

### Recursion: Introduction and Correctness

Recursion: Introduction and Correctness CSE21 Winter 2017, Day 7 (B00), Day 4-5 (A00) January 25, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 Today s Plan From last time: intersecting sorted lists and

### THIS paper is aimed at designing efficient decoding algorithms

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 7, NOVEMBER 1999 2333 Sort-and-Match Algorithm for Soft-Decision Decoding Ilya Dumer, Member, IEEE Abstract Let a q-ary linear (n; k)-code C be used

### 1 Recommended Reading 1. 2 Public Key/Private Key Cryptography Overview RSA Algorithm... 2

Contents 1 Recommended Reading 1 2 Public Key/Private Key Cryptography 1 2.1 Overview............................................. 1 2.2 RSA Algorithm.......................................... 2 3 A Number

### Mergesort and Recurrences (CLRS 2.3, 4.4)

Mergesort and Recurrences (CLRS 2.3, 4.4) We saw a couple of O(n 2 ) algorithms for sorting. Today we ll see a different approach that runs in O(n lg n) and uses one of the most powerful techniques for

### SAT-Solvers: propositional logic in action

SAT-Solvers: propositional logic in action Russell Impagliazzo, with assistence from Cameron Held October 22, 2013 1 Personal Information A reminder that my office is 4248 CSE, my office hours for CSE

### Standard forms for writing numbers

Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,

### Fast Fraction-Integer Method for Computing Multiplicative Inverse

Fast Fraction-Integer Method for Computing Multiplicative Inverse Hani M AL-Matari 1 and Sattar J Aboud 2 and Nidal F Shilbayeh 1 1 Middle East University for Graduate Studies, Faculty of IT, Jordan-Amman

### Embedded Systems Development

Embedded Systems Development Lecture 3 Real-Time Scheduling Dr. Daniel Kästner AbsInt Angewandte Informatik GmbH kaestner@absint.com Model-based Software Development Generator Lustre programs Esterel programs