DOCSLIB.ORG

Asymptotic Notation in Equations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Asymptotic Notation in Equations Intro to Algorithms: BIBLIOGRAPHY Intro to Algorithms: BIBLIOGRAPHY Table of Contents BIBLIOGRAPHY...............................................................................................................................................1 CHAPTER 1: INTRODUCTION....................................................................................................................12 1.1 Algorithms...................................................................................................................................................13 Insertion sort..........................................................................................................................................13 Pseudocode conventions........................................................................................................................15 Exercises................................................................................................................................................16 1.2 Analyzing algorithms..................................................................................................................................17 Analysis of insertion sort.......................................................................................................................17 Worst−case and average−case analysis.................................................................................................19 Order of growth......................................................................................................................................20 Exercises................................................................................................................................................20 1.3 Designing algorithms..................................................................................................................................22 1.3.1 The divide−and−conquer approach..............................................................................................22 1.3.2 Analyzing divide−and−conquer algorithms..................................................................................23 Exercises................................................................................................................................................24 Analysis of merge sort........................................................................................................................................25 1.4 Summary......................................................................................................................................................26 Exercises................................................................................................................................................26 Problems............................................................................................................................................................27 Chapter notes....................................................................................................................................................28 CHAPTER 2: GROWTH OF FUNCTIONS..................................................................................................29 2.1 Asymptotic notation....................................................................................................................................30 −notation................................................................................................................................................30 O−notation.............................................................................................................................................32 −notation................................................................................................................................................32 Asymptotic notation in equations..........................................................................................................33 o−notation..............................................................................................................................................34 −notation................................................................................................................................................35 Comparison of functions........................................................................................................................35 Exercises................................................................................................................................................36 2.2 Standard notations and common functions..............................................................................................38 Monotonicity..........................................................................................................................................38 Floors and ceilings.................................................................................................................................38 Polynomials...........................................................................................................................................38 Exponentials...........................................................................................................................................39 Logarithms.............................................................................................................................................40 Factorials................................................................................................................................................41 The iterated logarithm function.............................................................................................................42 Fibonacci numbers.................................................................................................................................42 Exercises................................................................................................................................................43 i Intro to Algorithms: BIBLIOGRAPHY Table of Contents Problems............................................................................................................................................................45 Chapter notes....................................................................................................................................................48 CHAPTER 3: SUMMATIONS........................................................................................................................49 3.1 Summation formulas and properties........................................................................................................50 Linearity.................................................................................................................................................50 Arithmetic series....................................................................................................................................50 Geometric series.....................................................................................................................................51 Harmonic series.....................................................................................................................................51 Integrating and differentiating series.....................................................................................................52 Telescoping series..................................................................................................................................52 Products.................................................................................................................................................52 Exercises................................................................................................................................................53 3.2 Bounding summations................................................................................................................................54 Mathematical induction.........................................................................................................................54 Bounding the terms................................................................................................................................55 Splitting summations.............................................................................................................................56 Approximation by integrals...................................................................................................................57 Exercises................................................................................................................................................59 Technicalities.........................................................................................................................................61 Making a good guess.............................................................................................................................62 Subtleties................................................................................................................................................63 Avoiding pitfalls....................................................................................................................................63 Changing variables.................................................................................................................................65
Load more

Recommended publications
  • An Evolutionary Approach for Sorting Algorithms
    ORIENTAL JOURNAL OF ISSN: 0974-6471 COMPUTER SCIENCE & TECHNOLOGY December 2014, An International Open Free Access, Peer Reviewed Research Journal Vol. 7, No. (3): Published By: Oriental Scientific Publishing Co., India. Pgs. 369-376 www.computerscijournal.org Root to Fruit (2): An Evolutionary Approach for Sorting Algorithms PRAMOD KADAM AND Sachin KADAM BVDU, IMED, Pune, India. (Received: November 10, 2014; Accepted: December 20, 2014) ABstract This paper continues the earlier thought of evolutionary study of sorting problem and sorting algorithms (Root to Fruit (1): An Evolutionary Study of Sorting Problem) [1]and concluded with the chronological list of early pioneers of sorting problem or algorithms. Latter in the study graphical method has been used to present an evolution of sorting problem and sorting algorithm on the time line. Key words: Evolutionary study of sorting, History of sorting Early Sorting algorithms, list of inventors for sorting. IntroDUCTION name and their contribution may skipped from the study. Therefore readers have all the rights to In spite of plentiful literature and research extent this study with the valid proofs. Ultimately in sorting algorithmic domain there is mess our objective behind this research is very much found in documentation as far as credential clear, that to provide strength to the evolutionary concern2. Perhaps this problem found due to lack study of sorting algorithms and shift towards a good of coordination and unavailability of common knowledge base to preserve work of our forebear platform or knowledge base in the same domain. for upcoming generation. Otherwise coming Evolutionary study of sorting algorithm or sorting generation could receive hardly information about problem is foundation of futuristic knowledge sorting problems and syllabi may restrict with some base for sorting problem domain1.
    [Show full text]
  • Sorting Algorithm 1 Sorting Algorithm
    Sorting algorithm 1 Sorting algorithm In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order. Efficient sorting is important for optimizing the use of other algorithms (such as search and merge algorithms) that require sorted lists to work correctly; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions: 1. The output is in nondecreasing order (each element is no smaller than the previous element according to the desired total order); 2. The output is a permutation, or reordering, of the input. Since the dawn of computing, the sorting problem has attracted a great deal of research, perhaps due to the complexity of solving it efficiently despite its simple, familiar statement. For example, bubble sort was analyzed as early as 1956.[1] Although many consider it a solved problem, useful new sorting algorithms are still being invented (for example, library sort was first published in 2004). Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide and conquer algorithms, data structures, randomized algorithms, best, worst and average case analysis, time-space tradeoffs, and lower bounds. Classification Sorting algorithms used in computer science are often classified by: • Computational complexity (worst, average and best behaviour) of element comparisons in terms of the size of the list . For typical sorting algorithms good behavior is and bad behavior is .
    [Show full text]
  • Evaluation of Sorting Algorithms, Mathematical and Empirical Analysis of Sorting Algorithms
    International Journal of Scientific & Engineering Research Volume 8, Issue 5, May-2017 86 ISSN 2229-5518 Evaluation of Sorting Algorithms, Mathematical and Empirical Analysis of sorting Algorithms Sapram Choudaiah P Chandu Chowdary M Kavitha ABSTRACT:Sorting is an important data structure in many real life applications. A number of sorting algorithms are in existence till date. This paper continues the earlier thought of evolutionary study of sorting problem and sorting algorithms concluded with the chronological list of early pioneers of sorting problem or algorithms. Latter in the study graphical method has been used to present an evolution of sorting problem and sorting algorithm on the time line. An extensive analysis has been done compared with the traditional mathematical methods of ―Bubble Sort, Selection Sort, Insertion Sort, Merge Sort, Quick Sort. Observations have been obtained on comparing with the existing approaches of All Sorts. An “Empirical Analysis” consists of rigorous complexity analysis by various sorting algorithms, in which comparison and real swapping of all the variables are calculatedAll algorithms were tested on random data of various ranges from small to large. It is an attempt to compare the performance of various sorting algorithm, with the aim of comparing their speed when sorting an integer inputs.The empirical data obtained by using the program reveals that Quick sort algorithm is fastest and Bubble sort is slowest. Keywords: Bubble Sort, Insertion sort, Quick Sort, Merge Sort, Selection Sort, Heap Sort,CPU Time. Introduction In spite of plentiful literature and research in more dimension to student for thinking4. Whereas, sorting algorithmic domain there is mess found in this thinking become a mark of respect to all our documentation as far as credential concern2.
    [Show full text]
  • Sorting Algorithm 1 Sorting Algorithm
    Sorting algorithm 1 Sorting algorithm A sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order. Efficient sorting is important for optimizing the use of other algorithms (such as search and merge algorithms) which require input data to be in sorted lists; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions: 1. The output is in nondecreasing order (each element is no smaller than the previous element according to the desired total order); 2. The output is a permutation (reordering) of the input. Since the dawn of computing, the sorting problem has attracted a great deal of research, perhaps due to the complexity of solving it efficiently despite its simple, familiar statement. For example, bubble sort was analyzed as early as 1956.[1] Although many consider it a solved problem, useful new sorting algorithms are still being invented (for example, library sort was first published in 2006). Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide and conquer algorithms, data structures, randomized algorithms, best, worst and average case analysis, time-space tradeoffs, and upper and lower bounds. Classification Sorting algorithms are often classified by: • Computational complexity (worst, average and best behavior) of element comparisons in terms of the size of the list (n). For typical serial sorting algorithms good behavior is O(n log n), with parallel sort in O(log2 n), and bad behavior is O(n2).
    [Show full text]
  • Design and Analysis of Optimized Stooge Sort Algorithm
    International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019 Design and Analysis of Optimized Stooge Sort Algorithm Amit Kishor, Pankaj Pratap Singh ABSTRACT: Data sorting has many advantages and applications in And some sorting algorithms are not, like Heap Sort [1], software and web development. Search engines use sorting techniques to Quick Sort [2], etc. Stable Sort algorithms helps in sorting sort the result before it is presented to the user. The words in a dictionary are the identical elements in the order in which they appear at in sorted order so that the words can be found easily .There are many sorting the time of input. algorithms that are used in many domains to perform some operation and An algorithm is a mathematical procedure which serves obtain the desired output. But there are some sorting algorithms that take large time in sorting the data. This huge time can be vulnerable the computation or construction of the solution to the given to the operation. Every sorting algorithm has the different sorting problem [5]. There are some algorithm like Hill Climbing technique to sort the given data, Stooge sort is a sorting algorithm which algorithms which cannot be predicted [16]. Hill Climbing sorts the data recursively. Stooge sort takes comparatively more time as algorithm can find the solution faster in some cases or else it compared to many other sorting algorithms. Stooge sort works recursively can also take a large time. Sorting algorithms are prominent to sort the data element but the Optimized Stooge sort does not use in introductory computer science classes, where the number recursive process.
    [Show full text]
  • Csci 231 Homework 3
    CSci 231 Homework 3 Quicksort and Linear Time Sorting CLRS Chapter 7 and 8 Write and justify your answers on this sheet in the space provided.1 1. (CLRS 7.2-2) What is the running time of Quicksort when all elements of arrary A have the same value? 2. (CLRS 7.2-3) Briefly sketch why the runnning time of Quicksort is Θ(n2) when the arrary A contains distinct elements and is sorted in decreasing order. 1Collaboration is allowed, even encouraged, provided that the names of the collaborators are listed along with the solutions. Write up the solutions on your own. 3. (CLRS 7-3) Professors Dewey, Cheatham, and Howe have proposed the following “elegant” sorting algorithm: Stooge-Sort(A, i, j) if A[i] > A[j] then exchange A[i] ↔ A[j] if i + 1 ≥ j then return k ← b(j − i + 1)/3c Stooge-Sort(A, i, j − k) Stooge-Sort(A, i + k, j) Stooge-Sort(A, i, j − k) a. Argue that Stooge-Sort(A, 1, length[A]) correctly sorts the input array A[1..n], where n = length[A]. b. Give a recurrence for the worst-case running time of Stooge-Sort and a tight asymp- totic (Θ-notation) bound on the worst-case running time. c. Compare the worst-case running time of Stooge-Sort with that of insertion sort, merge sort, heapsort, sock sort, and quicksort. Do the professors deserve tenure? 2 4. (CLRS 8.3-2) Which of the following sorting algorithms are stable: insertion sort, merge sort, quicksort? Give a simple scheme that makes any sorting algorithm stable.
    [Show full text]
  • Bogo Sort, Stooge Sort, Bubble Sort Reading: Weiss Ch
    CSE 373 Sorting 1: Bogo Sort, Stooge Sort, Bubble Sort reading: Weiss Ch. 7 slides created by Marty Stepp http://www.cs.washington.edu/373/ © University of Washington, all rights reserved. 1 Sorting • sorting : Rearranging the values in an array or collection into a specific order (usually into their "natural ordering"). one of the fundamental problems in computer science can be solved in many ways: • there are many sorting algorithms • some are faster/slower than others • some use more/less memory than others • some work better with specific kinds of data • some can utilize multiple computers / processors, ... comparison-based sorting : determining order by comparing pairs of elements: •<, >, compareTo , … 2 Sorting methods in Java • The Arrays and Collections classes in java.util have a static method sort that sorts the elements of an array/list String[] words = {"foo", "bar", "baz", "ball"}; Arrays.sort(words); System.out.println(Arrays.toString(words)); // [ball, bar, baz, foo] List<String> words2 = new ArrayList<String>(); for (String word : words) { words2.add(word); } Collections.sort(words2); System.out.println(words2); // [ball, bar, baz, foo] 3 Collections class Method name Description binarySearch( list , value ) returns the index of the given value in a sorted list (< 0 if not found) copy( listTo , listFrom ) copies listFrom 's elements to listTo emptyList() , emptyMap() , returns a read-only collection of the emptySet() given type that has no elements fill( list , value ) sets every element in the list to have the given
    [Show full text]
  • Performance Analysis of Sorting Algorithms
    Masaryk University Faculty of Informatics Performance analysis of Sorting Algorithms Bachelor's Thesis Matej Hul´ın Brno, Fall 2017 Declaration Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Matej Hul´ın Advisor: prof. RNDr. Ivana Cern´a,CSc.ˇ i Acknowledgement I would like to thank my advisor, prof. RNDr. Ivana Cern´a,CSc.ˇ for her advice and support throughout the thesis work. iii Abstract The aim of the bachelor thesis was to provide an extensive overview of sorting algorithms ranging from the pessimal and quadratic time solutions, up to those, that are currently being used in contemporary standard libraries. All of the chosen algorithms and some of their variants are described in separate sections, that include pseudocodes, images and discussion of their asymptotic space and time complexity in the best, average and the worst case. The second and main goal of the thesis was to perform a performance analysis. For this reason, the selected algorithm variants, that perform optimally, have been implemented in C++. All of them have undergone several performance analysis tests, that involved 8 sequence types with different kind and level of presortedness. The measurements showed, that timsort is not necessarily faster than mergesort on sequences, that feature only very limited amount of presortedness. In addition, the results revealed, that timsort is com- petetive with quicksorts on sequences of larger objects, that require a lot of comparisons.
    [Show full text]
  • R2 = Iso Lef T R1
    ALGORITHMS -CSE202-SPRING 2000 Divide and Conquer and Sorting, Some Solutions Name Tree Isomorphism 0 T Problem Give an efficient algorithm to decide , given two rooted binary trees T and , 0 T whether T and are isomorphic as rooted trees. Solution 0 0 0 0 T r oot T L; R ; L ;R Let r oot be the root of and be the root of , and let be the left and 0 0 T T T f right subtrees of T and respectively. If embeds in , then the isomorphism must 0 0 r oot L L map r oot to , and must map every member of either entirely to or entirely to 0 Iso R and vice versa. Thus, we can define the isomorphism relationship on sub-trees as NIL;NIL=True Isox; N I L=IsoN I L; x=False x 6= NIL follows: Iso , if and otherwise Isor ;r = 1 2 IsoLef tr ; Lef tr AN D I soRig htr ; R ig htr 1 2 1 2 OR IsoLef tr ; R ig htr AN D I soRig htr ; R ig htr : 1 2 1 2 2 n The above immediately translates into a program, and the time, I claim, is O . 0 2 T; y 2 T x; y The reason is that for any nodes x , the above calls itself on at most y P once (in fact, precisely when x and are at the same depths of their trees.) Let be 0 0 x P R oot y the path from R oot to and be that from to ; in a tree there is exactly one’ such path.
    [Show full text]
  • Parallel Sorting (A Taste of Parallel Algorithms!) Spcl.Inf.Ethz.Ch @Spcl Eth
    spcl.inf.ethz.ch @spcl_eth TORSTEN HOEFLER Parallel Programming Parallel Sorting (A taste of parallel algorithms!) spcl.inf.ethz.ch @spcl_eth Today: Parallel Sorting (one of the most fun problems in CS) 2 spcl.inf.ethz.ch @spcl_eth Literature . D.E. Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 5.3.4: Networks for Sorting, pp. 219–247. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 1990. ISBN 0-262-03293-7. Chapter 27: Sorting Networks, pp.704–724. google "chapter 27 sorting networks" 3 spcl.inf.ethz.ch @spcl_eth How fast can we sort? Heapsort & Mergesort have O(n log n) worst-case run time Quicksort has O(n log n) average-case run time These bounds are all tight, actually (n log n) So maybe we can dream up another algorithm with a lower asymptotic complexity, such as O(n) or O(n log log n) This is unfortunately IMPOSSIBLE! But why? 4 spcl.inf.ethz.ch @spcl_eth Permutations Assume we have n elements to sort For simplicity, also assume none are equal (i.e., no duplicates) How many permutations of the elements (possible orderings)? Example, n=3 a[0]<a[1]<a[2] a[0]<a[2]<a[1] a[1]<a[0]<a[2] a[1]<a[2]<a[0] a[2]<a[0]<a[1] a[2]<a[1]<a[0] In general, n choices for first, n-1 for next, n-2 for next, etc.
    [Show full text]
  • CSE373: Data Structure & Algorithms Comparison Sorting
    Sorting Riley Porter CSE373: Data Structures & Algorithms 1 Introduction to Sorting • Why study sorting? – Good algorithm practice! • Different sorting algorithms have different trade-offs – No single “best” sort for all scenarios – Knowing one way to sort just isn’t enough • Not usually asked about on tech interviews... – but if it comes up, you look bad if you can’t talk about it CSE373: Data Structures2 & Algorithms More Reasons to Sort General technique in computing: Preprocess data to make subsequent operations faster Example: Sort the data so that you can – Find the kth largest in constant time for any k – Perform binary search to find elements in logarithmic time Whether the performance of the preprocessing matters depends on – How often the data will change (and how much it will change) – How much data there is CSE373: Data Structures & 3 Algorithms Definition: Comparison Sort A computational problem with the following input and output Input: An array A of length n comparable elements Output: The same array A, containing the same elements where: for any i and j where 0 i < j < n then A[i] A[j] CSE373: Data Structures & 4 Algorithms More Definitions In-Place Sort: A sorting algorithm is in-place if it requires only O(1) extra space to sort the array. – Usually modifies input array – Can be useful: lets us minimize memory Stable Sort: A sorting algorithm is stable if any equal items remain in the same relative order before and after the sort. – Items that ’compare’ the same might not be exact duplicates – Might want to sort on some, but not all attributes of an item – Can be useful to sort on one attribute first, then another one CSE373: Data Structures & 5 Algorithms Stable Sort Example Input: [(8, "fox"), (9, "dog"), (4, "wolf"), (8, "cow")] Compare function: compare pairs by number only Output (stable sort): [(4, "wolf"), (8, "fox"), (8, "cow"), (9, "dog")] Output (unstable sort): [(4, "wolf"), (8, "cow"), (8, "fox"), (9, "dog")] CSE373: Data Structures & 6 Algorithms Lots of algorithms for sorting..
    [Show full text]
  • Sorting Algorithms Cheat Sheet by Pryl Via Cheatography.Com/66402/Cs/16808
    Sorting algorithms Cheat Sheet by pryl via cheatography.com/66402/cs/16808/ Sorting algorithms and Methods Merge sort Sorting algorit​ hms Metho​ ds function merge_sort(list m) Bubble sort Exchanging ​ ​ ​ // Base case. A list of zero or one elements is sorted, by definit​ ion. Heapsort Selection ​ ​ ​ if length of m ≤ 1 then Insertion sort Insertion ​ ​ ​ ​ ​ ​ ​ r​ eturn m Introsort Partiti​ oning & Selection ​ ​ ​ // Recursive case. First, divide the list into Merge sort Merging equal-s​ ized sublists Patience sorting Insertion & Selection ​ ​ ​ // consisting of the first half and second half of Quicksort Partiti​ oning the list. Selection sort Selection ​ ​ ​ // This assumes lists start at index 0. ​ ​ ​ Timsort Insertion & Merging var left := empty list ​ ​ ​ var right := empty list Unshuffle sort Distrib​ ution and Merge ​ ​ ​ for each x with index i in m do ​ ​ ​ ​ ​ ​ ​ if i < (length of m)/2 then Best and Worst Case ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ add x to left Algor​ ithms Best Case Worst Case ​ ​ ​ ​ ​ ​ ​ else Bogosort n ∞ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ add x to right Bubble sort n n2 ​ ​ ​ // Recursi​ vely sort both sublists. Bucket sort (uniform keys) - n2k ​ ​ ​ left := merge_s​ ort​ (left) ​ ​ ​ r​ ight := merge_s​ ort​ (ri​ ght) Heap sort n log n n log n ​ ​ ​ // Then merge the now-sorted sublists. Insertion sort n n2 ​ ​ ​ r​ eturn merge(l​ eft, right) Merge sort n log n n log n 2 Quick sort n log n n Bogosort Selection sort n2 n2 while not isInOrder(deck): Shell sort n log n n4/3 ​ ​ ​ s​ huf​ fle​ (deck) Spreadsort n n(k/s+d) Timsort
    [Show full text]