Sorting Algorithm 1 Sorting Algorithm
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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. -
Coursenotes 4 Non-Adaptive Sorting Batcher's Algorithm
4. Non Adaptive Sorting Batcher’s Algorithm 4.1 Introduction to Batcher’s Algorithm Sorting has many important applications in daily life and in particular, computer science. Within computer science several sorting algorithms exist such as the “bubble sort,” “shell sort,” “comb sort,” “heap sort,” “bucket sort,” “merge sort,” etc. We have actually encountered these before in section 2 of the course notes. Many different sorting algorithms exist and are implemented differently in each programming language. These sorting algorithms are relevant because they are used in every single computer program you run ranging from the unimportant computer solitaire you play to the online checking account you own. Indeed without sorting algorithms, many of the services we enjoy today would simply not be available. Batcher’s algorithm is a way to sort individual disorganized elements called “keys” into some desired order using a set number of comparisons. The keys that will be sorted for our purposes will be numbers and the order that we will want them to be in will be from greatest to least or vice a versa (whichever way you want it). The Batcher algorithm is non-adaptive in that it takes a fixed set of comparisons in order to sort the unsorted keys. In other words, there is no change made in the process during the sorting. Unlike other methods of sorting where you need to remember comparisons (tournament sort), Batcher’s algorithm is useful because it requires less thinking. Non-adaptive sorting is easy because outcomes of comparisons made at one point of the process do not affect which comparisons will be made in the future. -
CS321 Spring 2021
CS321 Spring 2021 Lecture 2 Jan 13 2021 Admin • A1 Due next Saturday Jan 23rd – 11:59PM Course in 4 Sections • Section I: Basics and Sorting • Section II: Hash Tables and Basic Data Structs • Section III: Binary Search Trees • Section IV: Graphs Section I • Sorting methods and Data Structures • Introduction to Heaps and Heap Sort What is Big O notation? • A way to approximately count algorithm operations. • A way to describe the worst case running time of algorithms. • A tool to help improve algorithm performance. • Can be used to measure complexity and memory usage. Bounds on Operations • An algorithm takes some number of ops to complete: • a + b is a single operation, takes 1 op. • Adding up N numbers takes N-1 ops. • O(1) means ‘on order of 1’ operation. • O( c ) means ‘on order of constant’. • O( n) means ‘ on order of N steps’. • O( n2) means ‘ on order of N*N steps’. How Does O(k) = O(1) O(n) = c * n for some c where c*n is always greater than n for some c. O( k ) = c*k O( 1 ) = cc * 1 let ccc = c*k c*k = c*k* 1 therefore O( k ) = c * k * 1 = ccc *1 = O(1) O(n) times for sorting algorithms. Technique O(n) operations O(n) memory use Insertion Sort O(N2) O( 1 ) Bubble Sort O(N2) O(1) Merge Sort N * log(N) O(1) Heap Sort N * log(N) O(1) Quicksort O(N2) O(logN) Memory is in terms of EXTRA memory Primary Notation Types • O(n) = Asymptotic upper bound. -
Rate of Growth Linear Vs Logarithmic Growth O() Complexity Measure
Where were we…. Rate of Growth • Comparing worst case performance of algorithms. We don't know how long the steps actually take; we only know it is some constant time. We can • Do it in machine-independent way. just lump all constants together and forget about • Time usage of an algorithm: how many basic them. steps does an algorithm perform, as a function of the input size. What we are left with is the fact that the time in • For example: given an array of length N (=input linear search grows linearly with the input, while size), how many steps does linear search perform? in binary search it grows logarithmically - much slower. Linear vs logarithmic growth O() complexity measure Linear growth: Logarithmic growth: Input size T(N) = c log N Big O notation gives an asymptotic upper bound T(N) = N* c 2 on the actual function which describes 10 10c c log 10 = 4c time/memory usage of the algorithm: logarithmic, 2 linear, quadratic, etc. 100 100c c log2 100 = 7c The complexity of an algorithm is O(f(N)) if there exists a constant factor K and an input size N0 1000 1000c c log2 1000 = 10c such that the actual usage of time/memory by the 10000 10000c c log 10000 = 16c algorithm on inputs greater than N0 is always less 2 than K f(N). Upper bound example In other words f(N)=2N If an algorithm actually makes g(N) steps, t(N)=3+N time (for example g(N) = C1 + C2log2N) there is an input size N' and t(N) is in O(N) because for all N>3, there is a constant K, such that 2N > 3+N for all N > N' , g(N) ≤ K f(N) Here, N0 = 3 and then the algorithm is in O(f(N). -
1 Lower Bound on Runtime of Comparison Sorts
CS 161 Lecture 7 { Sorting in Linear Time Jessica Su (some parts copied from CLRS) 1 Lower bound on runtime of comparison sorts So far we've looked at several sorting algorithms { insertion sort, merge sort, heapsort, and quicksort. Merge sort and heapsort run in worst-case O(n log n) time, and quicksort runs in expected O(n log n) time. One wonders if there's something special about O(n log n) that causes no sorting algorithm to surpass it. As a matter of fact, there is! We can prove that any comparison-based sorting algorithm must run in at least Ω(n log n) time. By comparison-based, I mean that the algorithm makes all its decisions based on how the elements of the input array compare to each other, and not on their actual values. One example of a comparison-based sorting algorithm is insertion sort. Recall that insertion sort builds a sorted array by looking at the next element and comparing it with each of the elements in the sorted array in order to determine its proper position. Another example is merge sort. When we merge two arrays, we compare the first elements of each array and place the smaller of the two in the sorted array, and then we make other comparisons to populate the rest of the array. In fact, all of the sorting algorithms we've seen so far are comparison sorts. But today we will cover counting sort, which relies on the values of elements in the array, and does not base all its decisions on comparisons. -
An Efficient Evolutionary Algorithm for Solving Incrementally Structured
An Efficient Evolutionary Algorithm for Solving Incrementally Structured Problems Jason Ansel Maciej Pacula Saman Amarasinghe Una-May O'Reilly MIT - CSAIL July 14, 2011 Jason Ansel (MIT) PetaBricks July 14, 2011 1 / 30 Our goal is to make programs run faster We use evolutionary algorithms to search for faster programs The PetaBricks language defines search spaces of algorithmic choices Who are we? I do research in programming languages (PL) and compilers The PetaBricks language is a collaboration between: A PL / compiler research group A evolutionary algorithms research group A applied mathematics research group Jason Ansel (MIT) PetaBricks July 14, 2011 2 / 30 The PetaBricks language defines search spaces of algorithmic choices Who are we? I do research in programming languages (PL) and compilers The PetaBricks language is a collaboration between: A PL / compiler research group A evolutionary algorithms research group A applied mathematics research group Our goal is to make programs run faster We use evolutionary algorithms to search for faster programs Jason Ansel (MIT) PetaBricks July 14, 2011 2 / 30 Who are we? I do research in programming languages (PL) and compilers The PetaBricks language is a collaboration between: A PL / compiler research group A evolutionary algorithms research group A applied mathematics research group Our goal is to make programs run faster We use evolutionary algorithms to search for faster programs The PetaBricks language defines search spaces of algorithmic choices Jason Ansel (MIT) PetaBricks July 14, 2011 -
CS 758/858: Algorithms
CS 758/858: Algorithms ■ COVID Prof. Wheeler Ruml Algorithms TA Sumanta Kashyapi This Class Complexity http://www.cs.unh.edu/~ruml/cs758 4 handouts: course info, schedule, slides, asst 1 2 online handouts: programming tips, formulas 1 physical sign-up sheet/laptop (for grades, piazza) Wheeler Ruml (UNH) Class 1, CS 758 – 1 / 25 COVID ■ COVID Algorithms This Class Complexity ■ check your Wildcat Pass before coming to campus ■ if you have concerns, let me know Wheeler Ruml (UNH) Class 1, CS 758 – 2 / 25 ■ COVID Algorithms ■ Algorithms Today ■ Definition ■ Why? ■ The Word ■ The Founder This Class Complexity Algorithms Wheeler Ruml (UNH) Class 1, CS 758 – 3 / 25 Algorithms Today ■ ■ COVID web: search, caching, crypto Algorithms ■ networking: routing, synchronization, failover ■ Algorithms Today ■ machine learning: data mining, recommendation, prediction ■ Definition ■ Why? ■ bioinformatics: alignment, matching, clustering ■ The Word ■ ■ The Founder hardware: design, simulation, verification ■ This Class business: allocation, planning, scheduling Complexity ■ AI: robotics, games Wheeler Ruml (UNH) Class 1, CS 758 – 4 / 25 Definition ■ COVID Algorithm Algorithms ■ precisely defined ■ Algorithms Today ■ Definition ■ mechanical steps ■ Why? ■ ■ The Word terminates ■ The Founder ■ input and related output This Class Complexity What might we want to know about it? Wheeler Ruml (UNH) Class 1, CS 758 – 5 / 25 Why? ■ ■ COVID Computer scientist 6= programmer Algorithms ◆ ■ Algorithms Today understand program behavior ■ Definition ◆ have confidence in results, performance ■ Why? ■ The Word ◆ know when optimality is abandoned ■ The Founder ◆ solve ‘impossible’ problems This Class ◆ sets you apart (eg, Amazon.com) Complexity ■ CPUs aren’t getting faster ■ Devices are getting smaller ■ Software is the differentiator ■ ‘Software is eating the world’ — Marc Andreessen, 2011 ■ Everything is computation Wheeler Ruml (UNH) Class 1, CS 758 – 6 / 25 The Word: Ab¯u‘Abdall¯ah Muh.ammad ibn M¯us¯aal-Khw¯arizm¯ı ■ COVID 780-850 AD Algorithms Born in Uzbekistan, ■ Algorithms Today worked in Baghdad. -
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. -
Sorting Algorithms Correcness, Complexity and Other Properties
Sorting Algorithms Correcness, Complexity and other Properties Joshua Knowles School of Computer Science The University of Manchester COMP26912 - Week 9 LF17, April 1 2011 The Importance of Sorting Important because • Fundamental to organizing data • Principles of good algorithm design (correctness and efficiency) can be appreciated in the methods developed for this simple (to state) task. Sorting Algorithms 2 LF17, April 1 2011 Every algorithms book has a large section on Sorting... Sorting Algorithms 3 LF17, April 1 2011 ...On the Other Hand • Progress in computer speed and memory has reduced the practical importance of (further developments in) sorting • quicksort() is often an adequate answer in many applications However, you still need to know your way (a little) around the the key sorting algorithms Sorting Algorithms 4 LF17, April 1 2011 Overview What you should learn about sorting (what is examinable) • Definition of sorting. Correctness of sorting algorithms • How the following work: Bubble sort, Insertion sort, Selection sort, Quicksort, Merge sort, Heap sort, Bucket sort, Radix sort • Main properties of those algorithms • How to reason about complexity — worst case and special cases Covered in: the course book; labs; this lecture; wikipedia; wider reading Sorting Algorithms 5 LF17, April 1 2011 Relevant Pages of the Course Book Selection sort: 97 (very short description only) Insertion sort: 98 (very short) Merge sort: 219–224 (pages on multi-way merge not needed) Heap sort: 100–106 and 107–111 Quicksort: 234–238 Bucket sort: 241–242 Radix sort: 242–243 Lower bound on sorting 239–240 Practical issues, 244 Some of the exercise on pp. -
Quicksort Z Merge Sort Z T(N) = Θ(N Lg(N)) Z Not In-Place CSE 680 Z Selection Sort (From Homework) Prof
Sorting Review z Insertion Sort Introduction to Algorithms z T(n) = Θ(n2) z In-place Quicksort z Merge Sort z T(n) = Θ(n lg(n)) z Not in-place CSE 680 z Selection Sort (from homework) Prof. Roger Crawfis z T(n) = Θ(n2) z In-place Seems pretty good. z Heap Sort Can we do better? z T(n) = Θ(n lg(n)) z In-place Sorting Comppgarison Sorting z Assumptions z Given a set of n values, there can be n! 1. No knowledge of the keys or numbers we permutations of these values. are sorting on. z So if we look at the behavior of the 2. Each key supports a comparison interface sorting algorithm over all possible n! or operator. inputs we can determine the worst-case 3. Sorting entire records, as opposed to complexity of the algorithm. numbers,,p is an implementation detail. 4. Each key is unique (just for convenience). Comparison Sorting Decision Tree Decision Tree Model z Decision tree model ≤ 1:2 > z Full binary tree 2:3 1:3 z A full binary tree (sometimes proper binary tree or 2- ≤≤> > tree) is a tree in which every node other than the leaves has two c hildren <1,2,3> ≤ 1:3 >><2,1,3> ≤ 2:3 z Internal node represents a comparison. z Ignore control, movement, and all other operations, just <1,3,2> <3,1,2> <2,3,1> <3,2,1> see comparison z Each leaf represents one possible result (a permutation of the elements in sorted order). -
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. -
Improving the Performance of Bubble Sort Using a Modified Diminishing Increment Sorting
Scientific Research and Essay Vol. 4 (8), pp. 740-744, August, 2009 Available online at http://www.academicjournals.org/SRE ISSN 1992-2248 © 2009 Academic Journals Full Length Research Paper Improving the performance of bubble sort using a modified diminishing increment sorting Oyelami Olufemi Moses Department of Computer and Information Sciences, Covenant University, P. M. B. 1023, Ota, Ogun State, Nigeria. E- mail: [email protected] or [email protected]. Tel.: +234-8055344658. Accepted 17 February, 2009 Sorting involves rearranging information into either ascending or descending order. There are many sorting algorithms, among which is Bubble Sort. Bubble Sort is not known to be a very good sorting algorithm because it is beset with redundant comparisons. However, efforts have been made to improve the performance of the algorithm. With Bidirectional Bubble Sort, the average number of comparisons is slightly reduced and Batcher’s Sort similar to Shellsort also performs significantly better than Bidirectional Bubble Sort by carrying out comparisons in a novel way so that no propagation of exchanges is necessary. Bitonic Sort was also presented by Batcher and the strong point of this sorting procedure is that it is very suitable for a hard-wired implementation using a sorting network. This paper presents a meta algorithm called Oyelami’s Sort that combines the technique of Bidirectional Bubble Sort with a modified diminishing increment sorting. The results from the implementation of the algorithm compared with Batcher’s Odd-Even Sort and Batcher’s Bitonic Sort showed that the algorithm performed better than the two in the worst case scenario. The implication is that the algorithm is faster.