Elementary Sorting Methods • Simple Sorting Methods
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Optimizing Sorting with Genetic Algorithms ∗
Optimizing Sorting with Genetic Algorithms ∗ Xiaoming Li, Mar´ıa Jesus´ Garzaran´ and David Padua Department of Computer Science University of Illinois at Urbana-Champaign {xli15, garzaran, padua}@cs.uiuc.edu http://polaris.cs.uiuc.edu Abstract 1 Introduction Although compiler technology has been extraordinarily The growing complexity of modern processors has made successful at automating the process of program optimiza- the generation of highly efficient code increasingly difficult. tion, much human intervention is still needed to obtain high- Manual code generation is very time consuming, but it is of- quality code. One reason is the unevenness of compiler im- ten the only choice since the code generated by today’s com- plementations. There are excellent optimizing compilers for piler technology often has much lower performance than some platforms, but the compilers available for some other the best hand-tuned codes. A promising code generation platforms leave much to be desired. A second, and perhaps strategy, implemented by systems like ATLAS, FFTW, and more important, reason is that conventional compilers lack SPIRAL, uses empirical search to find the parameter values semantic information and, therefore, have limited transfor- of the implementation, such as the tile size and instruction mation power. An emerging approach that has proven quite schedules, that deliver near-optimal performance for a par- effective in overcoming both of these limitations is to use ticular machine. However, this approach has only proven library generators. These systems make use of semantic in- successful on scientific codes whose performance does not formation to apply transformations at all levels of abstrac- depend on the input data. -
Space-Efficient Data Structures, Streams, and Algorithms
Andrej Brodnik Alejandro López-Ortiz Venkatesh Raman Alfredo Viola (Eds.) Festschrift Space-Efficient Data Structures, Streams, LNCS 8066 and Algorithms Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday 123 Lecture Notes in Computer Science 8066 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany Andrej Brodnik Alejandro López-Ortiz Venkatesh Raman Alfredo Viola (Eds.) Space-Efficient Data Structures, Streams, and Algorithms Papers in Honor of J. Ian Munro on the Occasion of His 66th Birthday 13 Volume Editors Andrej Brodnik University of Ljubljana, Faculty of Computer and Information Science Ljubljana, Slovenia and University of Primorska, -
Advanced Topics in Sorting
Advanced Topics in Sorting complexity system sorts duplicate keys comparators 1 complexity system sorts duplicate keys comparators 2 Complexity of sorting Computational complexity. Framework to study efficiency of algorithms for solving a particular problem X. Machine model. Focus on fundamental operations. Upper bound. Cost guarantee provided by some algorithm for X. Lower bound. Proven limit on cost guarantee of any algorithm for X. Optimal algorithm. Algorithm with best cost guarantee for X. lower bound ~ upper bound Example: sorting. • Machine model = # comparisons access information only through compares • Upper bound = N lg N from mergesort. • Lower bound ? 3 Decision Tree a < b yes no code between comparisons (e.g., sequence of exchanges) b < c a < c yes no yes no a b c b a c a < c b < c yes no yes no a c b c a b b c a c b a 4 Comparison-based lower bound for sorting Theorem. Any comparison based sorting algorithm must use more than N lg N - 1.44 N comparisons in the worst-case. Pf. Assume input consists of N distinct values a through a . • 1 N • Worst case dictated by tree height h. N ! different orderings. • • (At least) one leaf corresponds to each ordering. Binary tree with N ! leaves cannot have height less than lg (N!) • h lg N! lg (N / e) N Stirling's formula = N lg N - N lg e N lg N - 1.44 N 5 Complexity of sorting Upper bound. Cost guarantee provided by some algorithm for X. Lower bound. Proven limit on cost guarantee of any algorithm for X. -
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 . -
Ankara Yildirim Beyazit University Graduate
ANKARA YILDIRIM BEYAZIT UNIVERSITY GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES DEVELOPMENT OF SORTING AND SEARCHING ALGORITHMS Ph. D Thesis By ADNAN SAHER MOHAMMED Department of Computer Engineering December 2017 ANKARA DEVELOPMENT OF SORTING AND SEARCHING ALGORITHMS A Thesis Submitted to The Graduate School of Natural and Applied Sciences of Ankara Yıldırım Beyazıt University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Computer Engineering, Department of Computer Engineering By ADNAN SAHER MOHAMMED December 2017 ANKARA PH.D. THESIS EXAMINATION RESULT FORM We have read the thesis entitled “Development of Sorting and Searching Algorithms” completed by Adnan Saher Mohammed under the supervision of Prof. Dr. Fatih V. ÇELEBİ and we certify that in our opinion it is fully adequate, in scope and in quality, as a thesis for the degree of Doctor of Philosophy. Prof. Dr. Fatih V. ÇELEBİ (Supervisor) Prof. Dr. Şahin EMRAH Yrd. Doç. Dr. Hacer YALIM KELEŞ (Jury member) (Jury member) Yrd. Doç. Dr. Özkan KILIÇ Yrd. Doç. Dr. Shafqat Ur REHMAN (Jury member) (Jury member) Prof. Dr. Fatih V. ÇELEBİ Director Graduate School of Natural and Applied Sciences ii ETHICAL DECLARATION I hereby declare that, in this thesis which has been prepared in accordance with the Thesis Writing Manual of Graduate School of Natural and Applied Sciences, • All data, information and documents are obtained in the framework of academic and ethical rules, • All information, documents and assessments are presented in accordance with scientific ethics and morals, • All the materials that have been utilized are fully cited and referenced, • No change has been made on the utilized materials, • All the works presented are original, and in any contrary case of above statements, I accept to renounce all my legal rights. -
Advanced Topics in Sorting List RSS News Items in Reverse Chronological Order
Sorting applications Sorting algorithms are essential in a broad variety of applications Organize an MP3 library. Display Google PageRank results. Advanced Topics in Sorting List RSS news items in reverse chronological order. Find the median. Find the closest pair. Binary search in a database. [email protected] Identify statistical outliers. Find duplicates in a mailing list. [email protected] Data compression. Computer graphics. Computational biology. Supply chain management. Load balancing on a parallel computer. http://www.4shared.com/file/79096214/fb2ed224/lect01.html . Sorting algorithms Which algorithm to use? Many sorting algorithms to choose from Applications have diverse attributes Stable? Internal sorts Multiple keys? Insertion sort, selection sort, bubblesort, shaker sort. Quicksort, mergesort, heapsort, samplesort, shellsort. Deterministic? Solitaire sort, red-black sort, splaysort, Dobosiewicz sort, psort, ... Keys all distinct? External sorts Multiple key types? Poly-phase mergesort, cascade-merge, oscillating sort. Linked list or arrays? Radix sorts Large or small records? Distribution, MSD, LSD. 3-way radix quicksort. Is your file randomly ordered? Parallel sorts Need guaranteed performance? Bitonic sort, Batcher even-odd sort. Smooth sort, cube sort, column sort. Cannot cover all combinations of attributes. GPUsort. 1 Case study 1 Case study 2 Problem Problem Sort a huge randomly-ordered file of small Sort a huge file that is already almost in records. order. Example Example Process transaction records for a phone Re-sort a huge database after a few company. changes. Which sorting method to use? Which sorting method to use? 1. Quicksort: YES, it's designed for this problem 1. Quicksort: probably no, insertion simpler and faster 2. -
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). -
Comparison Sorts Name Best Average Worst Memory Stable Method Other Notes Quicksort Is Usually Done in Place with O(Log N) Stack Space
Comparison sorts Name Best Average Worst Memory Stable Method Other notes Quicksort is usually done in place with O(log n) stack space. Most implementations on typical in- are unstable, as stable average, worst place sort in-place partitioning is case is ; is not more complex. Naïve Quicksort Sedgewick stable; Partitioning variants use an O(n) variation is stable space array to store the worst versions partition. Quicksort case exist variant using three-way (fat) partitioning takes O(n) comparisons when sorting an array of equal keys. Highly parallelizable (up to O(log n) using the Three Hungarian's Algorithmor, more Merge sort worst case Yes Merging practically, Cole's parallel merge sort) for processing large amounts of data. Can be implemented as In-place merge sort — — Yes Merging a stable sort based on stable in-place merging. Heapsort No Selection O(n + d) where d is the Insertion sort Yes Insertion number of inversions. Introsort No Partitioning Used in several STL Comparison sorts Name Best Average Worst Memory Stable Method Other notes & Selection implementations. Stable with O(n) extra Selection sort No Selection space, for example using lists. Makes n comparisons Insertion & Timsort Yes when the data is already Merging sorted or reverse sorted. Makes n comparisons Cubesort Yes Insertion when the data is already sorted or reverse sorted. Small code size, no use Depends on gap of call stack, reasonably sequence; fast, useful where Shell sort or best known is No Insertion memory is at a premium such as embedded and older mainframe applications. Bubble sort Yes Exchanging Tiny code size. -
Sorting Algorithms
Sorting Algorithms rules of the game shellsort mergesort quicksort animations Reference: Algorithms in Java, Chapters 6-8 1 Classic sorting algorithms Critical components in the world’s computational infrastructure. • Full scientific understanding of their properties has enabled us to develop them into practical system sorts. • Quicksort honored as one of top 10 algorithms of 20th century in science and engineering. Shellsort. • Warmup: easy way to break the N2 barrier. • Embedded systems. Mergesort. • Java sort for objects. • Perl, Python stable sort. Quicksort. • Java sort for primitive types. • C qsort, Unix, g++, Visual C++, Python. 2 rules of the game shellsort mergesort quicksort animations 3 Basic terms Ex: student record in a University. Sort: rearrange sequence of objects into ascending order. 4 Sample sort client Goal: Sort any type of data Example. List the files in the current directory, sorted by file name. import java.io.File; public class Files { public static void main(String[] args) { File directory = new File(args[0]); File[] files = directory.listFiles(); Insertion.sort(files); % java Files . for (int i = 0; i < files.length; i++) Insertion.class Insertion.java System.out.println(files[i]); InsertionX.class } InsertionX.java Selection.class } Selection.java Shell.class Shell.java ShellX.class ShellX.java Next: How does sort compare file names? index.html 5 Callbacks Goal. Write robust sorting library method that can sort any type of data using the data type's natural order. Callbacks. • Client passes array of objects to sorting routine. • Sorting routine calls back object's comparison function as needed. Implementing callbacks. • Java: interfaces. • C: function pointers. • C++: functors. -
Random Shuffling to Reduce Disorder in Adaptive Sorting Scheme Md
Random Shuffling to Reduce Disorder in Adaptive Sorting Scheme Md. Enamul Karim and Abdun Naser Mahmood AI and Algorithm Research Lab Department of Computer Science, University of Dhaka [email protected] Abstract In this paper we present a random shuffling scheme to apply with adaptive sorting algorithms. Adaptive sorting algorithms utilize the presortedness present in a given sequence. We have probabilistically increased the amount of presortedness present in a sequence by using a random shuffling technique that requires little computation. Theoretical analysis suggests that the proposed scheme can improve the performance of adaptive sorting. Experimental results show that it significantly reduces the amount of disorder present in a given sequence and improves the execution time of adaptive sorting algorithm as well. Keywords: Adaptive Sorting, Random Shuffle. 1. Introduction Sorting is the process of arranging numbers in either increasing (non-decreasing) or decreasing (non- increasing) order. Due to its immense importance, sorting has never ceased to appear as research topic in computer science studies. The problem of sorting has been extensively studied and many important discoveries have been made in this respect. It has been observed [Estivill-Castro and Wood 1991] that the performance of sorting algorithms can be increased if the presortedness of the data is taken into account. This led to a new type of sorting approach, which is now known as adaptive sorting. Adaptive-sorting Algorithms take the advantage of existing order in the input. Intuitively ‘run time’ of an adaptive sorting algorithm varies smoothly from O(n) to O(n log n) as the disorder or “unsortedness” of the input varies [Mehlhorn 1984]. -
An Adaptive Sorting Algorithm for Almost Sorted List
International Journal of Computer Sciences and Engineering Open Access Research Paper Volume-5, Issue-12 E-ISSN: 2347-2693 An Adaptive Sorting Algorithm for Almost Sorted List Rina Damdoo1*, Kanak Kalyani2 1* Department of CSE, RCOEM, Nagpur, India 2 Department of CSE, RCOEM, Nagpur, India *Corresponding Author: [email protected], Tel.: +91-9324425240 Available online at: www.ijcseonline.org Received: 20/Nov/2017, Revised: 29/Nov/2017, Accepted: 17/Dec/2017, Published: 31/Dec/2017 Abstract—Sorting algorithm has a great impact on computing and also attracts a grand deal of research. Even though many sorting algorithms are evolved, there is always a scope of for a new one. As an example, Bubble sort was first analyzed in 1956, but due to the complexity issues it was not wide spread. Although many consider Bubble sort a solved problem, new sorting algorithms are still being evolved as per the problem scenarios (for example, library sort was first published in 2006 and its detailed experimental analysis was done in 2015) [11]. In this paper new adaptive sorting method Paper sort is introduced. The method is simple for sorting real life objects, where the input list is almost but not completely sorted. But, one may find it complex to implement when time-and-space trade-off is considered. Keywords—Adaptive sort, Time complexity, Paper sort, Sorted list, Un-sorted list I. INTRODUCTION Now the third position element is compared with the A sorting algorithm is known as adaptive, if it takes benefit last (say L) number in the sorted list and is added at end to of already sorted elements in the list i.e. -
Msc THESIS FPGA-Based High Throughput Merge Sorter
Computer Engineering 2018 Mekelweg 4, 2628 CD Delft The Netherlands http://ce.et.tudelft.nl/ MSc THESIS FPGA-Based High Throughput Merge Sorter Xianwei Zeng Abstract As database systems have shifted from disk-based to in-memory, and the scale of the database in big data analysis increases significantly, the workloads analyzing huge datasets are growing. Adopting FP- GAs as hardware accelerators improves the flexibility, parallelism and power consumption versus CPU-only systems. The accelerators CE-MS-2018-02 are also required to keep up with high memory bandwidth provided by advanced memory technologies and new interconnect interfaces. Sorting is the most fundamental database operation. In multiple- pass merge sorting, the final pass of the merge operation requires significant throughput performance to keep up with the high mem- ory bandwidth. We study the state-of-the-art hardware-based sorters and present an analysis of our own design. In this thesis, we present an FPGA-based odd-even merge sorter which features throughput of 27.18 GB/s when merging 4 streams. Our design also presents stable throughput performance when the number of input streams is increased due to its high degree of parallelism. Thanks to such a generic design, the odd-even merge sorter does not suffer through- put drop for skewed data distributions and presents constant perfor- mance over various kinds of input distributions. Faculty of Electrical Engineering, Mathematics and Computer Science FPGA-Based High Throughput Merge Sorter THESIS submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in ELECTRICAL ENGINEERING by Xianwei Zeng born in GUANGDONG, CHINA Computer Engineering Department of Electrical Engineering Faculty of Electrical Engineering, Mathematics and Computer Science Delft University of Technology FPGA-Based High Throughput Merge Sorter by Xianwei Zeng Abstract As database systems have shifted from disk-based to in-memory, and the scale of the database in big data analysis increases significantly, the workloads analyzing huge datasets are growing.