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Sort Algorithms 15-110 - Friday 2/28 Learning Objectives
Sort Algorithms 15-110 - Friday 2/28 Learning Objectives • Recognize how different sorting algorithms implement the same process with different algorithms • Recognize the general algorithm and trace code for three algorithms: selection sort, insertion sort, and merge sort • Compute the Big-O runtimes of selection sort, insertion sort, and merge sort 2 Search Algorithms Benefit from Sorting We use search algorithms a lot in computer science. Just think of how many times a day you use Google, or search for a file on your computer. We've determined that search algorithms work better when the items they search over are sorted. Can we write an algorithm to sort items efficiently? Note: Python already has built-in sorting functions (sorted(lst) is non-destructive, lst.sort() is destructive). This lecture is about a few different algorithmic approaches for sorting. 3 Many Ways of Sorting There are a ton of algorithms that we can use to sort a list. We'll use https://visualgo.net/bn/sorting to visualize some of these algorithms. Today, we'll specifically discuss three different sorting algorithms: selection sort, insertion sort, and merge sort. All three do the same action (sorting), but use different algorithms to accomplish it. 4 Selection Sort 5 Selection Sort Sorts From Smallest to Largest The core idea of selection sort is that you sort from smallest to largest. 1. Start with none of the list sorted 2. Repeat the following steps until the whole list is sorted: a) Search the unsorted part of the list to find the smallest element b) Swap the found element with the first unsorted element c) Increment the size of the 'sorted' part of the list by one Note: for selection sort, swapping the element currently in the front position with the smallest element is faster than sliding all of the numbers down in the list. -
PROC SORT (Then And) NOW Derek Morgan, PAREXEL International
Paper 143-2019 PROC SORT (then and) NOW Derek Morgan, PAREXEL International ABSTRACT The SORT procedure has been an integral part of SAS® since its creation. The sort-in-place paradigm made the most of the limited resources at the time, and almost every SAS program had at least one PROC SORT in it. The biggest options at the time were to use something other than the IBM procedure SYNCSORT as the sorting algorithm, or whether you were sorting ASCII data versus EBCDIC data. These days, PROC SORT has fallen out of favor; after all, PROC SQL enables merging without using PROC SORT first, while the performance advantages of HASH sorting cannot be overstated. This leads to the question: Is the SORT procedure still relevant to any other than the SAS novice or the terminally stubborn who refuse to HASH? The answer is a surprisingly clear “yes". PROC SORT has been enhanced to accommodate twenty-first century needs, and this paper discusses those enhancements. INTRODUCTION The largest enhancement to the SORT procedure is the addition of collating sequence options. This is first and foremost recognition that SAS is an international software package, and SAS users no longer work exclusively with English-language data. This capability is part of National Language Support (NLS) and doesn’t require any additional modules. You may use standard collations, SAS-provided translation tables, custom translation tables, standard encodings, or rules to produce your sorted dataset. However, you may only use one collation method at a time. USING STANDARD COLLATIONS, TRANSLATION TABLES AND ENCODINGS A long time ago, SAS would allow you to sort data using ASCII rules on an EBCDIC system, and vice versa. -
Computational Geometry – Problem Session Convex Hull & Line Segment Intersection
Computational Geometry { Problem Session Convex Hull & Line Segment Intersection LEHRSTUHL FUR¨ ALGORITHMIK · INSTITUT FUR¨ THEORETISCHE INFORMATIK · FAKULTAT¨ FUR¨ INFORMATIK Guido Bruckner¨ 04.05.2018 Guido Bruckner¨ · Computational Geometry { Problem Session Modus Operandi To register for the oral exam we expect you to present an original solution for at least one problem in the exercise session. • this is about working together • don't worry if your idea doesn't work! Guido Bruckner¨ · Computational Geometry { Problem Session Outline Convex Hull Line Segment Intersection Guido Bruckner¨ · Computational Geometry { Problem Session Definition of Convex Hull Def: A region S ⊆ R2 is called convex, when for two points p; q 2 S then line pq 2 S. The convex hull CH(S) of S is the smallest convex region containing S. Guido Bruckner¨ · Computational Geometry { Problem Session Definition of Convex Hull Def: A region S ⊆ R2 is called convex, when for two points p; q 2 S then line pq 2 S. The convex hull CH(S) of S is the smallest convex region containing S. Guido Bruckner¨ · Computational Geometry { Problem Session Definition of Convex Hull Def: A region S ⊆ R2 is called convex, when for two points p; q 2 S then line pq 2 S. The convex hull CH(S) of S is the smallest convex region containing S. In physics: Guido Bruckner¨ · Computational Geometry { Problem Session Definition of Convex Hull Def: A region S ⊆ R2 is called convex, when for two points p; q 2 S then line pq 2 S. The convex hull CH(S) of S is the smallest convex region containing S. -
Overview of Sorting Algorithms
Unit 7 Sorting Algorithms Simple Sorting algorithms Quicksort Improving Quicksort Overview of Sorting Algorithms Given a collection of items we want to arrange them in an increasing or decreasing order. You probably have seen a number of sorting algorithms including ¾ selection sort ¾ insertion sort ¾ bubble sort ¾ quicksort ¾ tree sort using BST's In terms of efficiency: ¾ average complexity of the first three is O(n2) ¾ average complexity of quicksort and tree sort is O(n lg n) ¾ but its worst case is still O(n2) which is not acceptable In this section, we ¾ review insertion, selection and bubble sort ¾ discuss quicksort and its average/worst case analysis ¾ show how to eliminate tail recursion ¾ present another sorting algorithm called heapsort Unit 7- Sorting Algorithms 2 Selection Sort Assume that data ¾ are integers ¾ are stored in an array, from 0 to size-1 ¾ sorting is in ascending order Algorithm for i=0 to size-1 do x = location with smallest value in locations i to size-1 swap data[i] and data[x] end Complexity If array has n items, i-th step will perform n-i operations First step performs n operations second step does n-1 operations ... last step performs 1 operatio. Total cost : n + (n-1) +(n-2) + ... + 2 + 1 = n*(n+1)/2 . Algorithm is O(n2). Unit 7- Sorting Algorithms 3 Insertion Sort Algorithm for i = 0 to size-1 do temp = data[i] x = first location from 0 to i with a value greater or equal to temp shift all values from x to i-1 one location forwards data[x] = temp end Complexity Interesting operations: comparison and shift i-th step performs i comparison and shift operations Total cost : 1 + 2 + .. -
Gscan: Accelerating Graham Scan on The
gScan:AcceleratingGrahamScanontheGPU Gang Mei Institute of Earth and Environmental Science, University of Freiburg Albertstr.23B, D-79104, Freiburg im Breisgau, Germany E-mail: [email protected] Abstract This paper presents a fast implementation of the Graham scan on the GPU. The proposed algorithm is composed of two stages: (1) two rounds of preprocessing performed on the GPU and (2) the finalization of finding the convex hull on the CPU. We first discard the interior points that locate inside a quadrilateral formed by four extreme points, sort the remaining points according to the angles, and then divide them into the left and the right regions. For each region, we perform a second round of filtering using the proposed preprocessing approach to discard the interior points in further. We finally obtain the expected convex hull by calculating the convex hull of the remaining points on the CPU. We directly employ the parallel sorting, reduction, and partitioning provided by the library Thrust for better efficiency and simplicity. Experimental results show that our implementation achieves 6x ~ 7x speedups over the Qhull implementation for 20M points. Keywords: GPU, CUDA, Convex Hull, Graham Scan, Divide-and-Conquer, Data Dependency 1.Introduction Given a set of planar points S, the 2D convex hull problem is to find the smallest polygon that contains all the points in S. This problem is a fundamental issue in computer science and has been studied extensively. Several classic convex hull algorithms have been proposed [1-7]; most of them run in O(nlogn) time. Among them, the Graham scan algorithm [1] is the first practical convex hull algorithm, while QuickHull [7] is the most efficient and popular one in practice. -
Geometric Algorithms
Geometric Algorithms primitive operations convex hull closest pair voronoi diagram References: Algorithms in C (2nd edition), Chapters 24-25 http://www.cs.princeton.edu/introalgsds/71primitives http://www.cs.princeton.edu/introalgsds/72hull 1 Geometric Algorithms Applications. • Data mining. • VLSI design. • Computer vision. • Mathematical models. • Astronomical simulation. • Geographic information systems. airflow around an aircraft wing • Computer graphics (movies, games, virtual reality). • Models of physical world (maps, architecture, medical imaging). Reference: http://www.ics.uci.edu/~eppstein/geom.html History. • Ancient mathematical foundations. • Most geometric algorithms less than 25 years old. 2 primitive operations convex hull closest pair voronoi diagram 3 Geometric Primitives Point: two numbers (x, y). any line not through origin Line: two numbers a and b [ax + by = 1] Line segment: two points. Polygon: sequence of points. Primitive operations. • Is a point inside a polygon? • Compare slopes of two lines. • Distance between two points. • Do two line segments intersect? Given three points p , p , p , is p -p -p a counterclockwise turn? • 1 2 3 1 2 3 Other geometric shapes. • Triangle, rectangle, circle, sphere, cone, … • 3D and higher dimensions sometimes more complicated. 4 Intuition Warning: intuition may be misleading. • Humans have spatial intuition in 2D and 3D. • Computers do not. • Neither has good intuition in higher dimensions! Is a given polygon simple? no crossings 1 6 5 8 7 2 7 8 6 4 2 1 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 18 4 18 4 19 4 19 4 20 3 20 3 20 1 10 3 7 2 8 8 3 4 6 5 15 1 11 3 14 2 16 we think of this algorithm sees this 5 Polygon Inside, Outside Jordan curve theorem. -
Optimal Output-Sensitive Convex Hull Algorithms in Two and Three Dimensions*
Discrete Comput Geom 16:361-368 (1996) GeometryDiscrete & Computational ~) 1996Springer-Verlag New YorkInc. Optimal Output-Sensitive Convex Hull Algorithms in Two and Three Dimensions* T. M. Chan Department of Computer Science, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4 Abstract. We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O (n log h) time and O(n) space, where h denotes the number of vertices of the convex hull. 1. Introduction Given a set P ofn points in the Euclidean plane E 2 or Euclidean space E 3, we consider the problem of computing the convex hull of P, cony(P), which is defined as the smallest convex set containing P. The convex hull problem has received considerable attention in computational geometry [11], [21], [23], [25]. In E 2 an algorithm known as Graham's scan [15] achieves O(n logn) rtmning time, and in E 3 an algorithm by Preparata and Hong [24] has the same complexity. These algorithms are optimal in the worst case, but if h, the number of hull vertices, is small, then it is possible to obtain better time bounds. For example, in E 2, a simple algorithm called Jarvis's march [ 19] can construct the convex hull in O(nh) time. This bound was later improved to O(nlogh) by an algorithm due to Kirkpatrick and Seidel [20], who also provided a matching lower bound; a simplification of their algorithm has been recently reported by Chan et al. -
A Framework for Multi-Core Implementations of Divide and Conquer Algorithms and Its Application to the Convex Hull Problem ∗
CCCG 2008, Montr´eal, Qu´ebec, August 13–15, 2008 A Framework for Multi-Core Implementations of Divide and Conquer Algorithms and its Application to the Convex Hull Problem ∗ Stefan N¨aher Daniel Schmitt † Abstract 2 Divide and Conquer Algorithms We present a framework for multi-core implementations Divide and conquer algorithms solve problems by di- of divide and conquer algorithms and show its efficiency viding them into subproblems, solving each subprob- and ease of use by applying it to the fundamental geo- lem recursively and merging the corresponding results metric problem of computing the convex hull of a point to a complete solution. All subproblems have exactly set. We concentrate on the Quickhull algorithm intro- the same structure as the original problem and can be duced in [2]. In general the framework can easily be solved independently from each other, and so can eas- used for any D&C-algorithm. It is only required that ily be distributed over a number of parallel processes the algorithm is implemented by a C++ class imple- or threads. This is probably the most straightforward menting the job-interface introduced in section 3 of this parallelization strategy. However, in general it can not paper. be guaranteed that always enough subproblems exist, which leads to non-optimal speedups. This is in partic- 1 Introduction ular true for the first divide step and the final merging step but is also a problem in cases where the recur- Performance gain in computing is no longer achieved sion tree is unbalanced such that the number of open by increasing cpu clock rates but by multiple cpu cores sub-problems is smaller than the number of available working on shared memory and a common cache. -
On the Ekeland Variational Principle with Applications and Detours
Lectures on The Ekeland Variational Principle with Applications and Detours By D. G. De Figueiredo Tata Institute of Fundamental Research, Bombay 1989 Author D. G. De Figueiredo Departmento de Mathematica Universidade de Brasilia 70.910 – Brasilia-DF BRAZIL c Tata Institute of Fundamental Research, 1989 ISBN 3-540- 51179-2-Springer-Verlag, Berlin, Heidelberg. New York. Tokyo ISBN 0-387- 51179-2-Springer-Verlag, New York. Heidelberg. Berlin. Tokyo No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by INSDOC Regional Centre, Indian Institute of Science Campus, Bangalore 560012 and published by H. Goetze, Springer-Verlag, Heidelberg, West Germany PRINTED IN INDIA Preface Since its appearance in 1972 the variational principle of Ekeland has found many applications in different fields in Analysis. The best refer- ences for those are by Ekeland himself: his survey article [23] and his book with J.-P. Aubin [2]. Not all material presented here appears in those places. Some are scattered around and there lies my motivation in writing these notes. Since they are intended to students I included a lot of related material. Those are the detours. A chapter on Nemyt- skii mappings may sound strange. However I believe it is useful, since their properties so often used are seldom proved. We always say to the students: go and look in Krasnoselskii or Vainberg! I think some of the proofs presented here are more straightforward. There are two chapters on applications to PDE. -
Quick Sort Algorithm Song Qin Dept
Quick Sort Algorithm Song Qin Dept. of Computer Sciences Florida Institute of Technology Melbourne, FL 32901 ABSTRACT each iteration. Repeat this on the rest of the unsorted region Given an array with n elements, we want to rearrange them in without the first element. ascending order. In this paper, we introduce Quick Sort, a Bubble sort works as follows: keep passing through the list, divide-and-conquer algorithm to sort an N element array. We exchanging adjacent element, if the list is out of order; when no evaluate the O(NlogN) time complexity in best case and O(N2) exchanges are required on some pass, the list is sorted. in worst case theoretically. We also introduce a way to approach the best case. Merge sort [4] has a O(NlogN) time complexity. It divides the 1. INTRODUCTION array into two subarrays each with N/2 items. Conquer each Search engine relies on sorting algorithm very much. When you subarray by sorting it. Unless the array is sufficiently small(one search some key word online, the feedback information is element left), use recursion to do this. Combine the solutions to brought to you sorted by the importance of the web page. the subarrays by merging them into single sorted array. 2 Bubble, Selection and Insertion Sort, they all have an O(N2) time In Bubble sort, Selection sort and Insertion sort, the O(N ) time complexity that limits its usefulness to small number of element complexity limits the performance when N gets very big. no more than a few thousand data points. -
Convex Hull in 3 Dimensions
CONVEX HULL IN 3 DIMENSIONS PETR FELKEL FEL CTU PRAGUE [email protected] https://cw.felk.cvut.cz/doku.php/courses/a4m39vg/start Based on [Berg], [Preparata], [Rourke] and [Boissonnat] Version from 16.11.2017 Talk overview Upper bounds for convex hull in 2D and 3D Other criteria for CH algorithm classification Recapitulation of CH algorithms Terminology refresh Convex hull in 3D – Terminology – Algorithms • Gift wrapping • D&C Merge • Randomized Incremental www.cguu.com www.cguu.com Computational geometry (2 / 41) Upper bounds for Convex hull algorithms O(n) for sorted points and for simple polygon 2 3 O(n log n) in E ,E with sorting – insensitive about output O(n h), O(n logh), h is number of CH facets – output sensitive –O(n2) or O(n logn) for n ~ h O(log n) for new point insertion in realtime algs. => O(n log n) for n-points O(log n) search where to insert Computational geometry (3 / 41) Other criteria for CH algorithm classification Optimality – depends on data order (or distribution) In the worst case x In the expected case Output sensitivity – depends on the result ~ O(f(h)) Extendable to higher dimensions? Off-line versus on-line – Off-line – all points available, preprocessing for search speedup – On-line – stream of points, new point pi on demand, just one new point at a time, CH valid for {p1,p2 ,…, pi } – Real-time – points come as they “want” (come not faster than optimal constant O(log n) inter-arrival delay) Parallelizable x serial Dynamic – points can be deleted Deterministic x approximate (lecture 13) Computational -
An Overview of Computational Geometry
An Overview of Computational Geometry Alex Scheffelin Contents 1 Abstract 2 2 Introduction 3 2.1 Basic Definitions . .3 3 Floating Point Error 4 3.1 Imprecision in Floating Point Operations . .4 3.2 Precision in the Calculation of the Area of a Polygon . .5 4 Speed 7 4.1 Big O Notation . .7 4.2 A More Efficient Algorithm to Calculate Intersections of Lines . .8 4.3 Recap on Speed . 15 5 How Can Pure Mathematics Influence Computational Geometry? 16 5.1 Point in Polygon Calculation . 16 5.2 Why Do We Care? . 19 6 Convex Hull Algorithms 19 6.1 What is a Convex Hull . 19 6.2 The Jarvis March Algorithm . 19 6.3 The Graham Scan Algorithm . 21 6.4 Other Efficient Algorithms . 22 6.5 Generalizations to the Convex Hull . 23 7 Conclusion 23 1 1 Abstract Computational geometry is an applied math field which can be described as the intersection of geometry and computer science. It has many real world applications, including in 3D mod- elling programs, graphics processing, etc. We will provide an overview of the methodology of computational geometry in the limited case of 2D objects, culminating in the discussion of the construction of convex hulls, a type of polygon in the plane which is determined by a set of points. Along the way we will try to explain some unique difficulties one faces when using a computer to compute things, and attempt to highlight the methods of proof used. 2 2 Introduction 2.1 Basic Definitions I will define a few geometric objects, and give example about how one might represent them in a computer.