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Algorithms, Searching and Sorting
Python for Data Scientists L8: Algorithms, searching and sorting 1 Iterative and recursion algorithms 2 Iterative algorithms • looping constructs (while and for loops) lead to iterative algorithms • can capture computation in a set of state variables that update on each iteration through loop 3 Iterative algorithms • “multiply x * y” is equivalent to “add x to itself y times” • capture state by • result 0 • an iteration number starts at y y y-1 and stop when y = 0 • a current value of computation (result) result result + x 4 Iterative algorithms def multiplication_iter(x, y): result = 0 while y > 0: result += x y -= 1 return result 5 Recursion The process of repeating items in a self-similar way 6 Recursion • recursive step : think how to reduce problem to a simpler/smaller version of same problem • base case : • keep reducing problem until reach a simple case that can be solved directly • when y = 1, x*y = x 7 Recursion : example x * y = x + x + x + … + x y items = x + x + x + … + x y - 1 items = x + x * (y-1) Recursion reduction def multiplication_rec(x, y): if y == 1: return x else: return x + multiplication_rec(x, y-1) 8 Recursion : example 9 Recursion : example 10 Recursion : example 11 Recursion • each recursive call to a function creates its own scope/environment • flow of control passes back to previous scope once function call returns value 12 Recursion vs Iterative algorithms • recursion may be simpler, more intuitive • recursion may be efficient for programmer but not for computers def fib_iter(n): if n == 0: return 0 O(n) elif n == 1: return 1 def fib_recur(n): else: if n == 0: O(2^n) a = 0 return 0 b = 1 elif n == 1: for i in range(n-1): return 1 tmp = a else: a = b return fib_recur(n-1) + fib_recur(n-2) b = tmp + b return b 13 Recursion : Proof by induction How do we know that our recursive code will work ? → Mathematical Induction To prove a statement indexed on integers is true for all values of n: • Prove it is true when n is smallest value (e.g. -
Decomposition, Abstraction, Functions
SEARCHING AND SORTING ALGORITHMS 6.00.01X LECTURE 1 SEARCH ALGORITHMS . search algorithm – method for finding an item or group of items with specific properties within a collection of items . collection could be implicit ◦ example – find square root as a search problem ◦ exhaustive enumeration ◦ bisection search ◦ Newton-Raphson . collection could be explicit ◦ example – is a student record in a stored collection of data? SEARCHING ALGORITHMS . linear search • brute force search • list does not have to be sorted . bisection search • list MUST be sorted to give correct answer • will see two different implementations of the algorithm 6.00.01X LECTURE 3 LINEAR SEARCH ON UNSORTED LIST def linear_search(L, e): found = False for i in range(len(L)): if e == L[i]: found = True return found . must look through all elements to decide it’s not there . O(len(L)) for the loop * O(1) to test if e == L[i] . overall complexity is O(n) – where n is len(L) 6.00.01X LECTURE 4 CONSTANT TIME LIST ACCESS . if list is all ints … ◦ ith element at ◦ base + 4*I .if list is heterogeneous ◦ indirection ◦ references to other objects … 6.00.01X LECTURE 5 6.00.01X LECTURE 6 LINEAR SEARCH ON SORTED LIST def search(L, e): for i in range(len(L)): if L[i] == e: return True if L[i] > e: return False return False . must only look until reach a number greater than e . O(len(L)) for the loop * O(1) to test if e == L[i] . overall complexity is O(n) – where n is len(L) 6.00.01X LECTURE 7 USE BISECTION SEARCH 1. -
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. -
3 Algorithms and Data Structures
3 Algorithms and Data Structures Program ®le for this chapter: algs What's wrong with this procedure? to process.sentence :sent output fput (process.word first :sent) (process.sentence bf :sent) end If you said ªIt's a recursive procedure without a stop rule,º you've solved a simple problem in analysis of algorithms. This branch of computer science is concerned with the correctness and the ef®ciency of programs. The ®eld is called analysis of algorithms rather than analysis of programs because the emphasis is on the meaning of a program (how you might express the program in English) and not on the details of its expression in a particular language. For example, the error in the procedure to process.sentence :sent if emptyp :sent [output []] output fput (process.word first :sent) (process.snetence bf :sent) end is just as fatal as the error in the ®rst example, but there isn't much of theoretical interest to say about a misspelled procedure name. On the other hand, another branch of computer science, program veri®cation, is concerned with developing techniques to ensure that a computer program really does correctly express the algorithm it is meant to express. The examples I've given so far are atypical, it's worth noting, in that you were able to see the errors in the procedures without having any real idea of what they do! Without seeing process.word you can tell that this program is doing something or other to each word of a sentence, but you don't know whether the words are being translated 107 to another language, converted from upper case to lower case letters, translated into a string of phoneme codes to be sent to a speech synthesizer, or what. -
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 . -
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. -
4Th, the Friendly Forth Compiler
4tH, the friendly Forth compiler J.L. Bezemer July 14, 2021 Contents 1 What’s new 19 I Getting Started 24 2 Overview 25 2.1 Introduction . 25 2.2 History . 25 2.3 Applications . 26 2.4 Architecture . 26 2.4.1 The 4tH language . 28 2.4.2 H-code . 28 2.4.3 H-code compiler . 29 2.4.4 Error handling . 30 2.4.5 Interfacing with C . 30 3 Installation Guide 32 3.1 About this package . 32 3.1.1 Example code . 32 3.1.2 Main program . 32 3.1.3 Unix package . 33 3.1.4 Linux package . 34 3.1.5 Android package . 38 3.1.6 MS-DOS package . 40 3.1.7 MS-Windows package . 41 3.2 Setting up your working directory . 43 3.3 Now what? . 43 3.4 Pedigree . 43 3.5 Contributors . 44 1 CONTENTS 2 3.6 Questions . 44 3.6.1 4tH website . 44 3.6.2 4tH Google group . 45 3.6.3 Newsgroup . 45 4 A guided tour 46 4.1 4tH interactive . 46 4.2 Starting up 4tH . 46 4.3 Running a program . 47 4.4 Starting an editing session . 47 4.5 Writing your first 4tH program . 48 4.6 A more complex program . 52 4.7 Advanced features . 56 4.8 Suspending a program . 63 4.9 Calculator mode . 65 4.10 Epilogue . 65 5 Frequently asked questions 66 II Primer 69 6 Introduction 70 7 4tH fundamentals 71 7.1 Making calculations without parentheses . 71 7.2 Manipulating the stack . -
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). -
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. -
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. -
Advanced Programming Language Design (Raphael A. Finkel)
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/220692467 Advanced programming language design. Book · January 1996 Source: DBLP CITATIONS READS 41 3,040 1 author: Raphael Finkel University of Kentucky 122 PUBLICATIONS 6,757 CITATIONS SEE PROFILE All content following this page was uploaded by Raphael Finkel on 16 December 2013. The user has requested enhancement of the downloaded file. ADVANCED PROGRAMMING LANGUAGE DESIGN Raphael A. Finkel ❖ UNIVERSITY OF KENTUCKY ▲ ▼▼ Addison-Wesley Publishing Company Menlo Park, California · Reading, Massachusetts New York · Don Mills, Ontario · Harlow, U.K. · Amsterdam Bonn · Paris · Milan · Madrid · Sydney · Singapore · Tokyo Seoul · Taipei · Mexico City · San Juan, Puerto Rico hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh On-line edition copyright 1996 by Addison-Wesley Publishing Company. Permission is granted to print or photo- copy this document for a fee of $0.02 per page, per copy, payable to Addison-Wesley Publishing Company. All other rights reserved. Acquisitions Editor: J. Carter Shanklin Proofreader: Holly McLean-Aldis Editorial Assistant: Christine Kulke Text Designer: Peter Vacek, Eigentype Senior Production Editor: Teri Holden Film Preparation: Lazer Touch, Inc. Copy Editor: Nick Murray Cover Designer: Yvo Riezebos Manufacturing Coordinator: Janet Weaver Printer: The Maple-Vail Book Manufacturing Group Composition and Film Coordinator: Vivian McDougal Copyright 1996 by Addison-Wesley Publishing Company All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or any other media embodiments now known, or hereafter to become known, without the prior written permission of the publisher. -
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