Time and space behaviour. Chapter 20
|
|
- Winfred Hamilton
- 6 years ago
- Views:
Transcription
1 Time and space behaviour Chapter 20
2 Algorithmic complexity How many steps does it take to execute an algorithm given an input of size n?
3 Complexity of functions Consider: f n = 2 n n + 13 This function has three components: a constant, 13 a term 4 n a term 2 n 2 As the value of n increases: the constant component is unchanged the 4 n component grows linearly the 2 n 2 component grows quadratically
4 Complexity of functions: big-oh O For large n, the quadratic term dominates in f. So, we say that f is of order n 2 and we write O(n 2 ) A function f::integer->integer is O(g) if there are positive integers m and d such that for all n m: f n d (g n) i.e., that f is bounded above by g: for sufficiently large n the value of f is no larger than a multiple of the function g. e.g., f from the previous slide is O(n 2 ), since for n 1: 2 n n n n n 2 19 n 2
5 Complexity of functions: big-theta Θ For a tight bound, a function f is Θ(g) when: both f is O(g) and g is O(f) When f is O(g) but g is not O(f) we write f g. Also, when f is Θ(g) we write f g. n 0 n 1 n 2 n k 2 n n 0 log n n 1 n (log n) n 2
6 28 y=10 x y=10x y=x y=
7 y=x 2 y=x log x y=x y=log x y=x
8 Bisection Given a list of length n, how many times can it be bisected before all the pieces are of length one? After p cuts the length of each piece is n/(2 p ). We are looking for p such that ( n/(2 p ) ) 1 and ( n/(2 p-1 ) ) > 1, i.e. n 2 p and n > 2 p-1 : 2 p n > 2 p-1 p log 2 n > p-1 The function giving the number of steps in terms of the length n is Θ(log 2 n)
9 We are looking for p such: p log 2 n > p-1 log 2 n > p-1 => p < 1 + log 2 n p is a function of n (the length of a list) f(n) < 1 + log 2 n A function f::integer->integer is O(g) if there are positive integers m and d such that for all n m: f n d (g n) There exists m (e.g.: 10) and d (e.g.: 3) such that f(n) d * log 2 n for all n m For example f(10) < 1 + log 2 (10) <= 3 * log 2 (10) f(11) < 1 + log 2 (11) <= 3 * log 2 (11) Therefore, F(n) is O(log 2 n) (the upper bound). Similarly we can prove the lower bound, the other direction usung the other inequality. Hence, Θ(log 2 n).
10 Size of a balanced binary tree A tree is balanced if all of its branches are the same length. Given a balanced binary tree whose branches are length b, how many nodes are there in the tree? k b = 2 b+1 1 Thus, the size of the tree is Θ(2 b ) in the length of the branches. Conversely, a balanced tree will have branches of length Θ(log 2 n) in the size of the tree.
11 Apples Given an apple a day for n days we will end up with n apples. Given n apples a day for n days we will end up with n 2 apples. Given 1 apple on the first day, 2 apples on the second day, etc., how many apples do we end up with? That is, what is the sum of the list [1..n]? (n-1) + n = n + (n-1) + (n-2) = ((n+1) + (n+1) + (n+1) + + (n+1) + (n+1)) / 2 = n (n+1) / 2 which is quadratic, Θ(n 2 ).
12 Measuring complexity Time taken to compute a result is given by the number of steps in a calculation. Space necessary for the computation. During calculation the expression being calculate grows and shrinks we need space to hold the largest expression. This is called the residency of the calculation, or its space complexity. Total space used by a computation, reflecting not just the size of the expression but the sizes of the values as well.
13 Factorial fac :: Integer -> Integer fac 0 = 1 fac n = n * fac (n-1) fac n n * fac (n-1) n * ((n-1) *... * (2 * (1 * 1))... ) n * ((n-1) *... * (2 * 1)... ) n * ((n-1) *... * 2... ) n! This takes 2n+1 steps and the largest expression contains n multiplication symbols. So, time and space is linear, Θ(n)
14 Insertion sort isort :: Ord a => [a] -> [a] isort [] = [] isort (x:xs) = ins x (isort xs) ins :: Ord a => a -> [a] -> [a] ins x [] = [x] ins x (y:ys) (x<=y) = x:y:ys otherwise = y:ins x ys
15 isort [] = [] isort (x:xs) = ins x (isort xs) isort [a 1, a 2,..., a n-1, a n ] ins a 1 (isort [a 2,..., a n-1, a n ])... ins a 1 (ins a 2 (... (ins a n-1 (ins a n []))... )) So, n steps followed by n invocations of ins.
16 ins x [] = [x] ins x (y:ys) (x<=y) = x:y:ys otherwise = y:ins x ys Assume[a 1,,..., a n ] is sorted. ins a [a 1, a 2,..., a n-1, a n ] Best case: a<=a 1, takes one step Worst case: a>a n, takes n steps Average case: takes n/2 steps
17 So for isort: ins a 1 (ins a 2 (... (ins a n-1 (ins a n []))... )) Best case: each ins takes one step, so n more steps, so 2n steps overall which is O(n) Worst case: first ins 1 step, second 2,, so overall O(n 2 ) Average case: ins s will take 1/2 + 2/2 + + (n-1)/2 + n/2 steps, so overall O(n 2 ) isort takes quadratic time in most cases, linear for (almost) sorted lists. Space usage is linear in all cases.
18 ++ [a 1, a 2,..., a n-1, a n ] ++ x a 1 : ([a 2,..., a n-1, a n ] ++ x) a 1 : (a 2 : ([a 3,..., a n-1, a n ] ++ x)) n-3 steps a 1 : (a 2 :... : (a n :x)) So, the time taken is linear in the length of the first list.
19 Quick sort qsort :: Ord a => [a] -> [a] qsort [] = [] qsort (x:xs) = qsort [z z<-xs,z<=x] ++ [x] ++ qsort [z z<-xs,z>x] When the list is sorted and without duplicates the calculation goes: qsort [a 1, a 2,..., a n-1, a n ] n steps [] ++ [a 1 ] ++ qsort [a 2,..., a n-1, a n ] n-1 steps a 1 : ([] ++ [a 2 ] ++ qsort [a 3,..., a n-1, a n ]) n-2 steps a 1 : (a 2 : (a 3 :... a n : [])) [a 1, a 2,..., a n-1, a n ] So, overall n steps, so qsort is quadratic for sorted lists: O(n 2 )
20 qsort :: Ord a => [a] -> [a] qsort [] = [] qsort (x:xs) = qsort [z z<-xs,z<=x] ++ [x] ++ qsort [z z<-xs,z>x] In the average case: qsort [a 1, a 2,..., a n-1, a n ] qsort [b 1,..., b n/2 ] ++ [a 1 ] ++ qsort [c 1,..., c n/2 ] Forming the two sublists takes O(n 1 ) steps, and O(n 1 ) steps to join the results. It takes O(log 2 n) bisections to obtain singleton lists. So qsort is on average O(n(log 2 n)).
21 Logarithmic behaviour is characteristic of divide-and-conquer algorithms: we split the problem into two smaller problems, solve them and recombine the results. They reach their base case in O(log 2 n) rather than O(n) steps.
22 Lazy evaluation arguments to functions are evaluated only when this is necessary for evaluation to continue an argument is not necessarily evaluated fully: only the parts that are needed are examined an argument is evaluated at most once (expressions are replaced by graphs and calculation is done over the graphs) evaluation order is from the outside in (for nested functions, e.g.: f 1 e 1 (f 2 e 2 5)) and from left to right (e.g.: f 1 e 1 + f 2 e 2 ).
23 Space behaviour: lazy evaluation Rule of thumb for space is the size of the largest expression produced during evaluation. This is accurate for computing numbers or Booleans, but not for data structures. Lazy evaluation means partial results are outputted, and discarded, once computed.
24 Space behaviour: lazy evaluation Consider: [m.. n] n >= m = m:[m+1.. n] otherwise = [] [1..n]?? n >= 1 1:[1+1.. n]?? n >= 1+1?? n >= 2 1:[2.. n] 1:2:[2+1.. n] 1:2:3: :n:[] The underlined pieces are printed and discarded as soon as possible. To measure space complexity we look at the non-underlined part (the residual evaluation), which is of constant size. So, space complexity is O(n 0 ).
25 Space behaviour: where clauses exam1 = [1..n] ++ [1..n] Takes time O(n 1 ) and space O(n 0 ) but calculates [1..n] twice! exam2 = list ++ list where list = [1..n] After evaluating list, the whole of the list is stored, giving space complexity O(n 1 )
26 Space behaviour: where clauses exam3 = [1..n] ++ [last [1..n]] exam4 = list ++ [last list] where list = [1..n] Space is O(n 0 ) Space is O(n 1 ) This is a space leak, since we only need one element of list.
27 Space behaviour: where clauses Avoiding redundant computation is (usually) always sensible, but it comes at the cost of space.
28 Saving space? fac 0 = 1 fac n = n * fac (n-1) Has O(n 1 ) space complexity from: n * ((n-1) *... * (2 * (1 * 1))... ) before it is evaluated. Alternative is to perform multiplication as we go: newfac :: Integer -> Integer newfac n = afac n 1 afac :: Integer -> Integer -> Integer afac 0 p = p afac n p = afac (n-1) (p*n)
29 newfac :: Integer -> Integer newfac n = afac n 1 afac :: Integer -> Integer -> Integer afac 0 p = p afac n p = afac (n-1) (p*n) newfac n afac n 1 afac (n-1) (1*n)?? (n-1) == 0 False afac (n-2) (1*n*(n-1)) afac 0 (1*n*(n-1)*(n-2)* *2*1) (1*n*(n-1)*(n-2)* *2*1) Still forms a large unevaluated expression, since its value is not needed until the end.
30 Consider: afac 0 p = p afac n p (p==p) = afac (n-1) (p*n) The guard test forces evaluation of the intermediate multiplications. This version has constant space behaviour. afac 4 1 afac (4-1) (1*4)?? (4-1) == 0 False?? (1*4) == (1*4)?? 4 == 4 True afac (3-1) (4*3)?? (3-1) == 0 False?? (4*3) == (4*3)?? 12 == 12 True afac (2-1) (12*2) afac 0 (24*1) (24*1) 24
31 Strictness A function is strict in an argument if the result is undefined whenever the argument is undefined. Examples: (+) is strict in both arguments (&&) is strict in only its first argument: True && x = x False && x = False A function that is not strict in an argument is said to be non-strict or lazy in that argument.
32
The Limits of Sorting Divide-and-Conquer Comparison Sorts II
The Limits of Sorting Divide-and-Conquer Comparison Sorts II CS 311 Data Structures and Algorithms Lecture Slides Monday, October 12, 2009 Glenn G. Chappell Department of Computer Science University of
More informationElementary maths for GMT. Algorithm analysis Part II
Elementary maths for GMT Algorithm analysis Part II Algorithms, Big-Oh and Big-Omega An algorithm has a O( ) and Ω( ) running time By default, we mean the worst case running time A worst case O running
More informationasymptotic growth rate or order compare two functions, but ignore constant factors, small inputs
Big-Oh 1 asymptotic growth rate or order 2 compare two functions, but ignore constant factors, small inputs asymptotic growth rate or order 2 compare two functions, but ignore constant factors, small inputs
More informationCS240 Fall Mike Lam, Professor. Algorithm Analysis
CS240 Fall 2014 Mike Lam, Professor Algorithm Analysis Algorithm Analysis Motivation: what and why Mathematical functions Comparative & asymptotic analysis Big-O notation ("Big-Oh" in textbook) Analyzing
More informationFundamental mathematical techniques reviewed: Mathematical induction Recursion. Typically taught in courses such as Calculus and Discrete Mathematics.
Fundamental mathematical techniques reviewed: Mathematical induction Recursion Typically taught in courses such as Calculus and Discrete Mathematics. Techniques introduced: Divide-and-Conquer Algorithms
More informationElementary maths for GMT. Algorithm analysis Part I
Elementary maths for GMT Algorithm analysis Part I Algorithms An algorithm is a step-by-step procedure for solving a problem in a finite amount of time Most algorithms transform input objects into output
More informationCS240 Fall Mike Lam, Professor. Algorithm Analysis
CS240 Fall 2014 Mike Lam, Professor Algorithm Analysis HW1 Grades are Posted Grades were generally good Check my comments! Come talk to me if you have any questions PA1 is Due 9/17 @ noon Web-CAT submission
More informationAdam Blank Lecture 2 Winter 2017 CSE 332. Data Structures and Parallelism
Adam Blank Lecture 2 Winter 2017 CSE 332 Data Structures and Parallelism CSE 332: Data Structures and Parallelism Algorithm Analysis 1 Outline 1 Comparing Algorithms 2 Asymptotic Analysis Comparing Programs
More informationComplexity of Algorithms
CSCE 222 Discrete Structures for Computing Complexity of Algorithms Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Overview Example - Fibonacci Motivating Asymptotic Run Time Analysis Asymptotic
More informationData Structures and Algorithms Key to Homework 1
Data Structures and Algorithms Key to Homework 1 January 31, 2005 15 Define an ADT for a set of integers (remember that a set may not contain duplicates) Your ADT should consist of the functions that can
More informationCSE373: Data Structures and Algorithms Lecture 4: Asymptotic Analysis. Aaron Bauer Winter 2014
CSE373: Data Structures and Algorithms Lecture 4: Asymptotic Analysis Aaron Bauer Winter 2014 Previously, on CSE 373 We want to analyze algorithms for efficiency (in time and space) And do so generally
More informationTesting. Wouter Swierstra and Alejandro Serrano. Advanced functional programming - Lecture 2. [Faculty of Science Information and Computing Sciences]
Testing Advanced functional programming - Lecture 2 Wouter Swierstra and Alejandro Serrano 1 Program Correctness 2 Testing and correctness When is a program correct? 3 Testing and correctness When is a
More information11. Divide and Conquer
The whole is of necessity prior to the part. Aristotle 11. Divide and Conquer 11.1. Divide and Conquer Many recursive algorithms follow the divide and conquer philosophy. They divide the problem into smaller
More informationAlgorithm Analysis. (Algorithm Analysis ) Data Structures and Programming Spring / 48
Algorithm Analysis (Algorithm Analysis ) Data Structures and Programming Spring 2018 1 / 48 What is an Algorithm? An algorithm is a clearly specified set of instructions to be followed to solve a problem
More informationDESIGN AND ANALYSIS OF ALGORITHMS. Unit 1 Chapter 4 ITERATIVE ALGORITHM DESIGN ISSUES
DESIGN AND ANALYSIS OF ALGORITHMS Unit 1 Chapter 4 ITERATIVE ALGORITHM DESIGN ISSUES http://milanvachhani.blogspot.in USE OF LOOPS As we break down algorithm into sub-algorithms, sooner or later we shall
More informationRun Times. Efficiency Issues. Run Times cont d. More on O( ) notation
Comp2711 S1 2006 Correctness Oheads 1 Efficiency Issues Comp2711 S1 2006 Correctness Oheads 2 Run Times An implementation may be correct with respect to the Specification Pre- and Post-condition, but nevertheless
More informationLecture 6: Sequential Sorting
15-150 Lecture 6: Sequential Sorting Lecture by Dan Licata February 2, 2012 Today s lecture is about sorting. Along the way, we ll learn about divide and conquer algorithms, the tree method, and complete
More information[ 11.2, 11.3, 11.4] Analysis of Algorithms. Complexity of Algorithms. 400 lecture note # Overview
400 lecture note #0 [.2,.3,.4] Analysis of Algorithms Complexity of Algorithms 0. Overview The complexity of an algorithm refers to the amount of time and/or space it requires to execute. The analysis
More informationUnit 1 Chapter 4 ITERATIVE ALGORITHM DESIGN ISSUES
DESIGN AND ANALYSIS OF ALGORITHMS Unit 1 Chapter 4 ITERATIVE ALGORITHM DESIGN ISSUES http://milanvachhani.blogspot.in USE OF LOOPS As we break down algorithm into sub-algorithms, sooner or later we shall
More informationPROGRAMMING IN HASKELL. CS Chapter 6 - Recursive Functions
PROGRAMMING IN HASKELL CS-205 - Chapter 6 - Recursive Functions 0 Introduction As we have seen, many functions can naturally be defined in terms of other functions. factorial :: Int Int factorial n product
More informationPseudo code of algorithms are to be read by.
Cs502 Quiz No1 Complete Solved File Pseudo code of algorithms are to be read by. People RAM Computer Compiler Approach of solving geometric problems by sweeping a line across the plane is called sweep.
More informationThen you were asked to paint another fence that is 2m high and 400m long for the price of $80. Should you accept it?
Complexity 1 Complexity Complexity is related to the mathematical relationship between: 1) some measurement of the task size, and 2) some measurement of the effort required to complete the task. A Good
More informationLecture 2: Algorithm Analysis
ECE4050/CSC5050 Algorithms and Data Structures Lecture 2: Algorithm Analysis 1 Mathematical Background Logarithms Summations Recursion Induction Proofs Recurrence Relations 2 2 Logarithm Definition: 3
More informationAnalysis of Algorithms. CSE Data Structures April 10, 2002
Analysis of Algorithms CSE 373 - Data Structures April 10, 2002 Readings and References Reading Chapter 2, Data Structures and Algorithm Analysis in C, Weiss Other References 10-Apr-02 CSE 373 - Data Structures
More informationRecall from Last Time: Big-Oh Notation
CSE 326 Lecture 3: Analysis of Algorithms Today, we will review: Big-Oh, Little-Oh, Omega (Ω), and Theta (Θ): (Fraternities of functions ) Examples of time and space efficiency analysis Covered in Chapter
More informationData Structures Question Bank Multiple Choice
Section 1. Fundamentals: Complexity, Algorthm Analysis 1. An algorithm solves A single problem or function Multiple problems or functions Has a single programming language implementation 2. A solution
More informationIntroduction to Programming: Lecture 6
Introduction to Programming: Lecture 6 K Narayan Kumar Chennai Mathematical Institute http://www.cmi.ac.in/~kumar 28 August 2012 Example: initial segments Write a Haskell function initsegs which returns
More informationPROGRAMMING IN HASKELL. Chapter 5 - List Comprehensions
PROGRAMMING IN HASKELL Chapter 5 - List Comprehensions 0 Set Comprehensions In mathematics, the comprehension notation can be used to construct new sets from old sets. {x 2 x {1...5}} The set {1,4,9,16,25}
More informationAlgorithms A Look At Efficiency
Algorithms A Look At Efficiency 1B Big O Notation 15-121 Introduction to Data Structures, Carnegie Mellon University - CORTINA 1 Big O Instead of using the exact number of operations to express the complexity
More informationHow do we compare algorithms meaningfully? (i.e.) the same algorithm will 1) run at different speeds 2) require different amounts of space
How do we compare algorithms meaningfully? (i.e.) the same algorithm will 1) run at different speeds 2) require different amounts of space when run on different computers! for (i = n-1; i > 0; i--) { maxposition
More informationMeasuring algorithm efficiency
CMPT 225 Measuring algorithm efficiency Timing Counting Cost functions Cases Best case Average case Worst case Searching Sorting O Notation O notation's mathematical basis O notation classes and notations
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 informationAlgorithm Analysis. College of Computing & Information Technology King Abdulaziz University. CPCS-204 Data Structures I
Algorithm Analysis College of Computing & Information Technology King Abdulaziz University CPCS-204 Data Structures I Order Analysis Judging the Efficiency/Speed of an Algorithm Thus far, we ve looked
More informationChapter 2: Complexity Analysis
Chapter 2: Complexity Analysis Objectives Looking ahead in this chapter, we ll consider: Computational and Asymptotic Complexity Big-O Notation Properties of the Big-O Notation Ω and Θ Notations Possible
More informationPlotting run-time graphically. Plotting run-time graphically. CS241 Algorithmics - week 1 review. Prefix Averages - Algorithm #1
CS241 - week 1 review Special classes of algorithms: logarithmic: O(log n) linear: O(n) quadratic: O(n 2 ) polynomial: O(n k ), k 1 exponential: O(a n ), a > 1 Classifying algorithms is generally done
More informationThe University of Nottingham
The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 1 MODULE, SPRING SEMESTER 2016-2017 PROGRAMMING PARADIGMS Time allowed: TWO hours THIRTY minutes Candidates may complete the front cover
More informationAlgorithms and Data Structures
Algorithms and Data Structures Spring 2019 Alexis Maciel Department of Computer Science Clarkson University Copyright c 2019 Alexis Maciel ii Contents 1 Analysis of Algorithms 1 1.1 Introduction.................................
More informationComputer Science 210 Data Structures Siena College Fall Topic Notes: Complexity and Asymptotic Analysis
Computer Science 210 Data Structures Siena College Fall 2017 Topic Notes: Complexity and Asymptotic Analysis Consider the abstract data type, the Vector or ArrayList. This structure affords us the opportunity
More informationIntroduction to Computers & Programming
16.070 Introduction to Computers & Programming Asymptotic analysis: upper/lower bounds, Θ notation Binary, Insertion, and Merge sort Prof. Kristina Lundqvist Dept. of Aero/Astro, MIT Complexity Analysis
More informationAnalysis of Algorithm. Chapter 2
Analysis of Algorithm Chapter 2 Outline Efficiency of algorithm Apriori of analysis Asymptotic notation The complexity of algorithm using Big-O notation Polynomial vs Exponential algorithm Average, best
More informationCMPSCI 187: Programming With Data Structures. Lecture 5: Analysis of Algorithms Overview 16 September 2011
CMPSCI 187: Programming With Data Structures Lecture 5: Analysis of Algorithms Overview 16 September 2011 Analysis of Algorithms Overview What is Analysis of Algorithms? L&C s Dishwashing Example Being
More information10/5/2016. Comparing Algorithms. Analyzing Code ( worst case ) Example. Analyzing Code. Binary Search. Linear Search
10/5/2016 CSE373: Data Structures and Algorithms Asymptotic Analysis (Big O,, and ) Steve Tanimoto Autumn 2016 This lecture material represents the work of multiple instructors at the University of Washington.
More informationAlgorithms and Theory of Computation. Lecture 2: Big-O Notation Graph Algorithms
Algorithms and Theory of Computation Lecture 2: Big-O Notation Graph Algorithms Xiaohui Bei MAS 714 August 15, 2017 Nanyang Technological University MAS 714 August 15, 2017 1 / 22 O, Ω, and Θ Let T, f
More informationFor searching and sorting algorithms, this is particularly dependent on the number of data elements.
Looking up a phone number, accessing a website and checking the definition of a word in a dictionary all involve searching large amounts of data. Searching algorithms all accomplish the same goal finding
More informationComputer Algorithms. Introduction to Algorithm
Computer Algorithms Introduction to Algorithm CISC 4080 Yanjun Li 1 What is Algorithm? An Algorithm is a sequence of well-defined computational steps that transform the input into the output. These steps
More informationUNIT 1 ANALYSIS OF ALGORITHMS
UNIT 1 ANALYSIS OF ALGORITHMS Analysis of Algorithms Structure Page Nos. 1.0 Introduction 7 1.1 Objectives 7 1.2 Mathematical Background 8 1.3 Process of Analysis 12 1.4 Calculation of Storage Complexity
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 informationCS 137 Part 7. Big-Oh Notation, Linear Searching and Basic Sorting Algorithms. November 10th, 2017
CS 137 Part 7 Big-Oh Notation, Linear Searching and Basic Sorting Algorithms November 10th, 2017 Big-Oh Notation Up to this point, we ve been writing code without any consideration for optimization. There
More informationCSE373: Data Structure & Algorithms Lecture 21: More Comparison Sorting. Aaron Bauer Winter 2014
CSE373: Data Structure & Algorithms Lecture 21: More Comparison Sorting Aaron Bauer Winter 2014 The main problem, stated carefully For now, assume we have n comparable elements in an array and we want
More informationThe Running Time of Programs
The Running Time of Programs The 90 10 Rule Many programs exhibit the property that most of their running time is spent in a small fraction of the source code. There is an informal rule that states 90%
More informationUniversity of Toronto Department of Electrical and Computer Engineering. Midterm Examination. ECE 345 Algorithms and Data Structures Fall 2010
University of Toronto Department of Electrical and Computer Engineering Midterm Examination ECE 345 Algorithms and Data Structures Fall 2010 Print your name and ID number neatly in the space provided below;
More informationWhy study algorithms? CS 561, Lecture 1. Today s Outline. Why study algorithms? (II)
Why study algorithms? CS 561, Lecture 1 Jared Saia University of New Mexico Seven years of College down the toilet - John Belushi in Animal House Q: Can I get a programming job without knowing something
More informationcsci 210: Data Structures Program Analysis
csci 210: Data Structures Program Analysis Summary Summary analysis of algorithms asymptotic analysis and notation big-o big-omega big-theta commonly used functions discrete math refresher Analysis of
More information(Refer Slide Time: 1:27)
Data Structures and Algorithms Dr. Naveen Garg Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture 1 Introduction to Data Structures and Algorithms Welcome to data
More information9/10/2018 Algorithms & Data Structures Analysis of Algorithms. Siyuan Jiang, Sept
9/10/2018 Algorithms & Data Structures Analysis of Algorithms Siyuan Jiang, Sept. 2018 1 Email me if the office door is closed Siyuan Jiang, Sept. 2018 2 Grades have been emailed github.com/cosc311/assignment01-userid
More informationAlgorithm Performance. (the Big-O)
Algorithm Performance (the Big-O) Lecture 6 Today: Worst-case Behaviour Counting Operations Performance Considerations Time measurements Order Notation (the Big-O) Pessimistic Performance Measure Often
More informationJava How to Program, 9/e. Copyright by Pearson Education, Inc. All Rights Reserved.
Java How to Program, 9/e Copyright 1992-2012 by Pearson Education, Inc. All Rights Reserved. Searching data involves determining whether a value (referred to as the search key) is present in the data
More informationComputer Science Approach to problem solving
Computer Science Approach to problem solving If my boss / supervisor / teacher formulates a problem to be solved urgently, can I write a program to efficiently solve this problem??? Polynomial-Time Brute
More informationTREES AND ORDERS OF GROWTH 7
TREES AND ORDERS OF GROWTH 7 COMPUTER SCIENCE 61A October 17, 2013 1 Trees In computer science, trees are recursive data structures that are widely used in various settings. This is a diagram of a simple
More informationShell CSCE 314 TAMU. Functions continued
1 CSCE 314: Programming Languages Dr. Dylan Shell Functions continued 2 Outline Defining Functions List Comprehensions Recursion 3 A Function without Recursion Many functions can naturally be defined in
More informationCS/ENGRD 2110 Object-Oriented Programming and Data Structures Spring 2012 Thorsten Joachims. Lecture 10: Asymptotic Complexity and
CS/ENGRD 2110 Object-Oriented Programming and Data Structures Spring 2012 Thorsten Joachims Lecture 10: Asymptotic Complexity and What Makes a Good Algorithm? Suppose you have two possible algorithms or
More informationData Structures and Algorithms
Data Structures and Algorithms About the course (objectives, outline, recommended reading) Problem solving Notions of Algorithmics (growth of functions, efficiency, programming model, example analysis)
More informationModule 5: Hashing. CS Data Structures and Data Management. Reza Dorrigiv, Daniel Roche. School of Computer Science, University of Waterloo
Module 5: Hashing CS 240 - Data Structures and Data Management Reza Dorrigiv, Daniel Roche School of Computer Science, University of Waterloo Winter 2010 Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module
More informationCSE 146. Asymptotic Analysis Interview Question of the Day Homework 1 & Project 1 Work Session
CSE 146 Asymptotic Analysis Interview Question of the Day Homework 1 & Project 1 Work Session Comparing Algorithms Rough Estimate Ignores Details Or really: independent of details What are some details
More informationHOMEWORK FILE SOLUTIONS
Data Structures Course (CSCI-UA 102) Professor Yap Spring 2012 HOMEWORK FILE February 13, 2012 SOLUTIONS 1 Homework 2: Due on Thu Feb 16 Q1. Consider the following function called crossproduct: int crossproduct(int[]
More informationData Structures and Algorithms. Part 2
1 Data Structures and Algorithms Part 2 Werner Nutt 2 Acknowledgments The course follows the book Introduction to Algorithms, by Cormen, Leiserson, Rivest and Stein, MIT Press [CLRST]. Many examples displayed
More information1 The smallest free number
1 The smallest free number Introduction Consider the problem of computing the smallest natural number not in a given finite set X of natural numbers. The problem is a simplification of a common programming
More informationClassic Data Structures Introduction UNIT I
ALGORITHM SPECIFICATION An algorithm is a finite set of instructions that, if followed, accomplishes a particular task. All algorithms must satisfy the following criteria: Input. An algorithm has zero
More informationBasic Data Structures (Version 7) Name:
Prerequisite Concepts for Analysis of Algorithms Basic Data Structures (Version 7) Name: Email: Concept: mathematics notation 1. log 2 n is: Code: 21481 (A) o(log 10 n) (B) ω(log 10 n) (C) Θ(log 10 n)
More informationFinal Examination: Topics and Sample Problems
Computer Science 52 Final Examination: Topics and Sample Problems Spring Semester, 2015 In examinations the foolish ask questions that the wise cannot answer. Oscar Wilde, 1894 Time and Place Wednesday,
More informationCS S-02 Algorithm Analysis 1
CS245-2008S-02 Algorithm Analysis 1 02-0: Algorithm Analysis When is algorithm A better than algorithm B? 02-1: Algorithm Analysis When is algorithm A better than algorithm B? Algorithm A runs faster 02-2:
More informationCSI33 Data Structures
Outline Department of Mathematics and Computer Science Bronx Community College August 31, 2015 Outline Outline 1 Chapter 1 Outline Textbook Data Structures and Algorithms Using Python and C++ David M.
More informationAlgorithm Analysis and Design
Algorithm Analysis and Design Dr. Truong Tuan Anh Faculty of Computer Science and Engineering Ho Chi Minh City University of Technology VNU- Ho Chi Minh City 1 References [1] Cormen, T. H., Leiserson,
More informationCS 61B Asymptotic Analysis Fall 2018
CS 6B Asymptotic Analysis Fall 08 Disclaimer: This discussion worksheet is fairly long and is not designed to be finished in a single section Some of these questions of the level that you might see on
More informationW4231: Analysis of Algorithms
W4231: Analysis of Algorithms From Binomial Heaps to Fibonacci Heaps Fibonacci Heaps 10/7/1999 We first prove that in Binomial Heaps insert and find-min take amortized O(1) time, while still having insert,
More informationLecture 5: Running Time Evaluation
Lecture 5: Running Time Evaluation Worst-case and average-case performance Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 13 1 Time complexity 2 Time growth 3 Worst-case 4 Average-case
More informationCS126 Final Exam Review
CS126 Final Exam Review Fall 2007 1 Asymptotic Analysis (Big-O) Definition. f(n) is O(g(n)) if there exists constants c, n 0 > 0 such that f(n) c g(n) n n 0 We have not formed any theorems dealing with
More informationPROGRAM EFFICIENCY & COMPLEXITY ANALYSIS
Lecture 03-04 PROGRAM EFFICIENCY & COMPLEXITY ANALYSIS By: Dr. Zahoor Jan 1 ALGORITHM DEFINITION A finite set of statements that guarantees an optimal solution in finite interval of time 2 GOOD ALGORITHMS?
More informationCS:3330 (22c:31) Algorithms
What s an Algorithm? CS:3330 (22c:31) Algorithms Introduction Computer Science is about problem solving using computers. Software is a solution to some problems. Algorithm is a design inside a software.
More informationHaske k ll An introduction to Functional functional programming using Haskell Purely Lazy Example: QuickSort in Java Example: QuickSort in Haskell
Haskell An introduction to functional programming using Haskell Anders Møller amoeller@cs.au.dk The most popular purely functional, lazy programming language Functional programming language : a program
More informationSorting. Bubble Sort. Selection Sort
Sorting In this class we will consider three sorting algorithms, that is, algorithms that will take as input an array of items, and then rearrange (sort) those items in increasing order within the array.
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 informationComplexity of Algorithms. Andreas Klappenecker
Complexity of Algorithms Andreas Klappenecker Example Fibonacci The sequence of Fibonacci numbers is defined as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... F n 1 + F n 2 if n>1 F n = 1 if n =1 0 if n =0 Fibonacci
More informationThe Complexity of Algorithms (3A) Young Won Lim 4/3/18
Copyright (c) 2015-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationAlgorithms in Systems Engineering IE172. Midterm Review. Dr. Ted Ralphs
Algorithms in Systems Engineering IE172 Midterm Review Dr. Ted Ralphs IE172 Midterm Review 1 Textbook Sections Covered on Midterm Chapters 1-5 IE172 Review: Algorithms and Programming 2 Introduction to
More informationWeek - 03 Lecture - 18 Recursion. For the last lecture of this week, we will look at recursive functions. (Refer Slide Time: 00:05)
Programming, Data Structures and Algorithms in Python Prof. Madhavan Mukund Department of Computer Science and Engineering Indian Institute of Technology, Madras Week - 03 Lecture - 18 Recursion For the
More informationAn introduction introduction to functional functional programming programming using usin Haskell
An introduction to functional programming using Haskell Anders Møller amoeller@cs.au.dkau Haskell The most popular p purely functional, lazy programming g language Functional programming language : a program
More informationUnit 6 Chapter 15 EXAMPLES OF COMPLEXITY CALCULATION
DESIGN AND ANALYSIS OF ALGORITHMS Unit 6 Chapter 15 EXAMPLES OF COMPLEXITY CALCULATION http://milanvachhani.blogspot.in EXAMPLES FROM THE SORTING WORLD Sorting provides a good set of examples for analyzing
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 informationComputational Complexity: Measuring the Efficiency of Algorithms
Computational Complexity: Measuring the Efficiency of Algorithms Rosen Ch. 3.2: Growth of Functions Rosen Ch. 3.3: Complexity of Algorithms Walls Ch. 10.1: Efficiency of Algorithms Measuring the efficiency
More informationChoice of C++ as Language
EECS 281: Data Structures and Algorithms Principles of Algorithm Analysis Choice of C++ as Language All algorithms implemented in this book are in C++, but principles are language independent That is,
More informationChapter 6 INTRODUCTION TO DATA STRUCTURES AND ALGORITHMS
Chapter 6 INTRODUCTION TO DATA STRUCTURES AND ALGORITHMS 1 Reference books: The C Programming Language by Brian W. Kernighan and Dennis M. Ritchie Programming in C (3rd Edition) by Stephen G. Kochan. Data
More informationYou should know the first sum above. The rest will be given if you ever need them. However, you should remember that,, and.
Big-Oh Notation Formal Definitions A function is in (upper bound) iff there exist positive constants k and n 0 such that for all. A function is in (lower bound) iff there exist positive constants k and
More informationCS 580: Algorithm Design and Analysis. Jeremiah Blocki Purdue University Spring 2018
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Recap: Stable Matching Problem Definition of a Stable Matching Stable Roomate Matching Problem Stable matching does not
More informationChapter 3, Algorithms Algorithms
CSI 2350, Discrete Structures Chapter 3, Algorithms Young-Rae Cho Associate Professor Department of Computer Science Baylor University 3.1. Algorithms Definition A finite sequence of precise instructions
More informationShell CSCE 314 TAMU. Haskell Functions
1 CSCE 314: Programming Languages Dr. Dylan Shell Haskell Functions 2 Outline Defining Functions List Comprehensions Recursion 3 Conditional Expressions As in most programming languages, functions can
More informationA general introduction to Functional Programming using Haskell
A general introduction to Functional Programming using Haskell Matteo Rossi Dipartimento di Elettronica e Informazione Politecnico di Milano rossi@elet.polimi.it 1 Functional programming in a nutshell
More informationIntro. Speed V Growth
Intro Good code is two things. It's elegant, and it's fast. In other words, we got a need for speed. We want to find out what's fast, what's slow, and what we can optimize. First, we'll take a tour of
More informationCSCE 314 TAMU Fall CSCE 314: Programming Languages Dr. Flemming Andersen. Haskell Functions
1 CSCE 314: Programming Languages Dr. Flemming Andersen Haskell Functions 2 Outline Defining Functions List Comprehensions Recursion 3 Conditional Expressions As in most programming languages, functions
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 information