An Improved Shellsort Algorithm
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Secure Multi-Party Sorting and Applications
Secure Multi-Party Sorting and Applications Kristján Valur Jónsson1, Gunnar Kreitz2, and Misbah Uddin2 1 Reykjavik University 2 KTH—Royal Institute of Technology Abstract. Sorting is among the most fundamental and well-studied problems within computer science and a core step of many algorithms. In this article, we consider the problem of constructing a secure multi-party computing (MPC) protocol for sorting, building on previous results in the field of sorting networks. Apart from the immediate uses for sorting, our protocol can be used as a building-block in more complex algorithms. We present a weighted set intersection algorithm, where each party inputs a set of weighted ele- ments and the output consists of the input elements with their weights summed. As a practical example, we apply our protocols in a network security setting for aggregation of security incident reports from multi- ple reporters, specifically to detect stealthy port scans in a distributed but privacy preserving manner. Both sorting and weighted set inter- section use O`n log2 n´ comparisons in O`log2 n´ rounds with practical constants. Our protocols can be built upon any secret sharing scheme supporting multiplication and addition. We have implemented and evaluated the performance of sorting on the Sharemind secure multi-party computa- tion platform, demonstrating the real-world performance of our proposed protocols. Keywords. Secure multi-party computation; Sorting; Aggregation; Co- operative anomaly detection 1 Introduction Intrusion Detection Systems (IDS) [16] are commonly used to detect anoma- lous and possibly malicious network traffic. Incidence reports and alerts from such systems are generally kept private, although much could be gained by co- operative sharing [30]. -
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. -
Selected Sorting Algorithms
Selected Sorting Algorithms CS 165: Project in Algorithms and Data Structures Michael T. Goodrich Some slides are from J. Miller, CSE 373, U. Washington Why Sorting? • Practical application – People by last name – Countries by population – Search engine results by relevance • Fundamental to other algorithms • Different algorithms have different asymptotic and constant-factor trade-offs – No single ‘best’ sort for all scenarios – Knowing one way to sort just isn’t enough • Many to approaches to sorting which can be used for other problems 2 Problem statement There are n comparable elements in an array and we want to rearrange them to be in increasing order Pre: – An array A of data records – A value in each data record – A comparison function • <, =, >, compareTo Post: – For each distinct position i and j of A, if i < j then A[i] ≤ A[j] – A has all the same data it started with 3 Insertion sort • insertion sort: orders a list of values by repetitively inserting a particular value into a sorted subset of the list • more specifically: – consider the first item to be a sorted sublist of length 1 – insert the second item into the sorted sublist, shifting the first item if needed – insert the third item into the sorted sublist, shifting the other items as needed – repeat until all values have been inserted into their proper positions 4 Insertion sort • Simple sorting algorithm. – n-1 passes over the array – At the end of pass i, the elements that occupied A[0]…A[i] originally are still in those spots and in sorted order. -
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 . -
Faster Shellsort Sequences: a Genetic Algorithm Application
Faster Shellsort Sequences: A Genetic Algorithm Application Richard Simpson Shashidhar Yachavaram [email protected] [email protected] Department of Computer Science Midwestern State University 3410 Taft, Wichita Falls, TX 76308 Phone: (940) 397-4191 Fax: (940) 397-4442 Abstract complex problems that resist normal research approaches. This paper concerns itself with the application of Genetic The recent popularity of genetic algorithms (GA's) and Algorithms to the problem of finding new, more efficient their application to a wide variety of problems is a result of Shellsort sequences. Using comparison counts as the main their ease of implementation and flexibility. Evolutionary criteria for quality, several new sequences have been techniques are often applied to optimization and related discovered that sort one megabyte files more efficiently than search problems where the fitness of a potential result is those presently known. easily established. Problems of this type are generally very difficult and often NP-Hard, so the ability to find a reasonable This paper provides a brief introduction to genetic solution to these problems within an acceptable time algorithms, a review of research on the Shellsort algorithm, constraint is clearly desirable. and concludes with a description of the GA approach and the associated results. One such problem that has been researched thoroughly is the search for Shellsort sequences. The attributes of this 2. Genetic Algorithms problem make it a prime target for the application of genetic algorithms. Noting this, the authors have designed a GA that John Holland developed the technique of GA's in the efficiently searches for Shellsort sequences that are top 1960's. -
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). -
Algorithms III
Algorithms III 1 Agenda Sorting • Simple sorting algorithms • Shellsort • Mergesort • Quicksort • A lower bound for sorting • Counting sort and Radix sort Randomization • Random number generation • Generation of random permutations • Randomized algorithms 2 Sorting Informal definition: Sorting is any process of arranging items in some sequence. Why sort? (1) Searching a sorted array is easier than searching an unsorted array. This is especially true for people. Example: Looking up a person in a telephone book. (2) Many problems may be solved more efficiently when input is sorted. Example: If two files are sorted it is possible to find all duplicates in only one pass. 3 Detecting duplicates in an array If the array is sorted, duplicates can be detected by a linear-time scan of the array: for (int i = 1; i < a.length; i++) if (a[i - 1].equals(a[i]) return true; 4 Finding the intersection of two! unsorted arrays 0 i M a: 0 j N b: 0 k c: k = 0; for (i = 0; i < M; i++) for (j = 0; j < N; j++) if (a[i].equals(b[j])) c[k++] = a[i]; Complexity: O(MN) 5 Finding the intersection of two! sorted arrays 0 i M a: 0 j N b: 0 k c: i = j = k = 0; while (i < M && j < N) if (a[i] < b[j]) i++; else if (a[i] > b[j]) j++; else { c[k++] = a[i]; i++; j++; } Complexity: O(M + N) 6 Insertion sort Problem: Given an array a of n elements a[0], a[1], ..., a[n-1]. Sort the array in increasing order. -
Goodrich: Randomized Shellsort: a Simple Oblivious Sorting Algorithm
Randomized Shellsort: A Simple Oblivious Sorting Algorithm Michael T. Goodrich Dept. of Computer Science University of California, Irvine http://www.ics.uci.edu/∼goodrich/ Abstract of the Pratt sequence. Moreover, Plaxton and Suel [35] es- Ω(n log2 n/(log log n)2) In this paper, we describe a randomized Shellsort algorithm. tablish a lower bound of for the This algorithm is a simple, randomized, data-oblivious ver- worst-case running time of Shellsort with any input sequence et al. sion of the Shellsort algorithm that always runs in O(n log n) (see also [11]) and Jiang [23] establish a lower bound Ω(pn1+1/p) time and succeeds in sorting any given input permutation of for the average-case running time of Shell- O(n log n) with very high probability. Taken together, these properties sort. Thus, the only way to achieve an average- imply applications in the design of new efficient privacy- time bound for Shellsort is to use an offset sequence of Θ(log n) preserving computations based on the secure multi-party length , and, even then, the problem of proving O(n log n) computation (SMC) paradigm. In addition, by a trivial con- an average running-time bound for a version of version of this Monte Carlo algorithm to its Las Vegas equiv- Shellsort is a long-standing open problem [42]. alent, one gets the first version of Shellsort with a running The approach we take in this paper is to consider a time that is provably O(n log n) with very high probability. -
A Comparative Study of Sorting Algorithms with FPGA Acceleration by High Level Synthesis
A Comparative Study of Sorting Algorithms with FPGA Acceleration by High Level Synthesis Yomna Ben Jmaa1,2, Rabie Ben Atitallah2, David Duvivier2, Maher Ben Jemaa1 1 NIS University of Sfax, ReDCAD, Tunisia 2 Polytechnical University Hauts-de-France, France [email protected], fRabie.BenAtitallah, [email protected], [email protected] Abstract. Nowadays, sorting is an important operation the risk of accidents, the traffic jams and pollution. for several real-time embedded applications. It is one Also, ITS improve the transport efficiency, safety of the most commonly studied problems in computer and security of passengers. They are used in science. It can be considered as an advantage for various domains such as railways, avionics and some applications such as avionic systems and decision automotive technology. At different steps, several support systems because these applications need a applications need to use sorting algorithms such sorting algorithm for their implementation. However, as decision support systems, path planning [6], sorting a big number of elements and/or real-time de- cision making need high processing speed. Therefore, scheduling and so on. accelerating sorting algorithms using FPGA can be an attractive solution. In this paper, we propose an efficient However, the complexity and the targeted hardware implementation for different sorting algorithms execution platform(s) are the main performance (BubbleSort, InsertionSort, SelectionSort, QuickSort, criteria for sorting algorithms. Different platforms HeapSort, ShellSort, MergeSort and TimSort) from such as CPU (single or multi-core), GPU (Graphics high-level descriptions in the zynq-7000 platform. In Processing Unit), FPGA (Field Programmable addition, we compare the performance of different Gate Array) [14] and heterogeneous architectures algorithms in terms of execution time, standard deviation can be used. -
Register Level Sort Algorithm on Multi-Core SIMD Processors
Register Level Sort Algorithm on Multi-Core SIMD Processors Tian Xiaochen, Kamil Rocki and Reiji Suda Graduate School of Information Science and Technology The University of Tokyo & CREST, JST {xchen, kamil.rocki, reiji}@is.s.u-tokyo.ac.jp ABSTRACT simultaneously. GPU employs a similar architecture: single- State-of-the-art hardware increasingly utilizes SIMD paral- instruction-multiple-threads(SIMT). On K20, each SMX has lelism, where multiple processing elements execute the same 192 CUDA cores which act as individual threads. Even a instruction on multiple data points simultaneously. How- desktop type multi-core x86 platform such as i7 Haswell ever, irregular and data intensive algorithms are not well CPU family supports AVX2 instruction set (256 bits). De- suited for such architectures. Due to their importance, it veloping algorithms that support multi-core SIMD architec- is crucial to obtain efficient implementations. One example ture is the precondition to unleash the performance of these of such a task is sort, a fundamental problem in computer processors. Without exploiting this kind of parallelism, only science. In this paper we analyze distinct memory accessing a small fraction of computational power can be utilized. models and propose two methods to employ highly efficient Parallel sort as well as sort in general is fundamental and bitonic merge sort using SIMD instructions as register level well studied problem in computer science. Algorithms such sort. We achieve nearly 270x speedup (525M integers/s) on as Bitonic-Merge Sort or Odd Even Merge Sort are widely a 4M integer set using Xeon Phi coprocessor, where SIMD used in practice. -
Sorting Algorithms
San José State University Department of Computer Engineering CMPE 180A Data Structures and Algorithms in C++ Fall 2020 Instructor: Ron Mak Assignment #12 120 points Assigned: Tuesday, November 10 Due: Tuesday, November 17 at 5:30 PM URL: http://codecheck.it/files/20042106449v8h269ox2cydek52zp0kv5rm Canvas: Assignment 12: Sorting algorithms Sorting algorithms This assignment will give you practice coding several important sorting algorithms, and you will be able to compare their performances while sorting data of various sizes. You will sort data elements in vectors with the selection sort, insertion short, Shellsort, and quicksort algorithms, and sort data elements in a linked list with the mergesort algorithm. There will be two versions of Shellsort, a “suboptimal” version that uses the halving technique for the diminishing increment, and an “optimal” version that uses a formula suggested by famous computer scientist Don Knuth. There will also be two versions of quicksort, a “suboptimal” version that uses a bad pivoting strategy, and an “optimal” version that uses a good pivoting strategy. Class hierarchy The UML (Unified Modeling Language) class diagram on the following page shows the class hierarchy. You are provided the complete code for class SelectionSort as an example, and you will code all versions of the other algorithms. You are also provided much of the support code. The code you will write are in these classes: • InsertionSort • ShellSortSuboptimal • ShellSortOptimal • QuickSorter • QuickSortSuboptimal • QuickSortOptimal • LinkedList • MergeSort 1 These sorting algorithms are all described in Chapter 10 of the Malik textbook. You can also find many sorting tutorials on the Web. Abstract class Sorter Class Sorter is the base class of all the sorting algorithms.