Implementing Merge Sort

Implementing Merge Sort

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  IMPLEMENTING MERGE-SORT
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  OUTLINE  INTRODUCTION  Divide& Conquer approach  Illustration  LITERATURE REVIEW  PROBLEM STATEMENT  CONCLUSION  REFERENCES 2
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  INTRODUCTION 3  Merge Sort: "Break the data to sort in half, sort each half separately using merge sort, and merge the halves back together into a single sorted list."  It employs a common algorithmic paradigm based on recursion, i.e. divide-and-conquer paradigm.  Divide-and-conquer algorithm works in three steps:  Divide  Conquer  Combine
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  DIVIDE & CONQUERAPPROACH 4 Fig. 1: Divide & Conquer view of one step[3] Here's how to view one step, assuming that each divide step creates two subproblems:
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  ILLUSTRATION 5 Here is adiagrammatic example for how merge sort works: Fig. 2: Merge Sort Working[5]
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  LITERATURE-REVIEW 6
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  LITERATURE-REVIEW(1/2) 7  Title: ”Formal Model Merging Applied to Class Diagram Integration”[1] Authors: Artur Boronat, Jose A. C., Isidro Ramos, Patrico Letelier  Objective: The definition of Merge operator is applied on class diagrams integration to present an automated approach for generic model merging providing support for conflict resolution and traceability between software artifacts. Keywords: Model-Driven Engineering, Model Management, model merging, conflict resolution.
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  LITERATURE-REVIEW(1/2 Cont..) 8  Introduction: Models are the main assets in the software development process.  Models collect the information that describes the information system at a high abstraction level, which permits the development of the application in an automated way.  In this process, models constitute software artifacts that experience refinements from the problem space to the solution space.
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  CASE STUDY 9  UseCase Analysis using Partial Class Diagrams:  Case Study - To illustrate how Merge operator can be used effectively to deal with the required needs like inconsistencies or conflicts among partial models which often arises, etc.  Here, part of a system for managing submissions that are received in a conference is presented.  In this example, the focus is on the fragment of the Use-Case Model (Fig. 3).
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  USE CASE MODEL 10 Softwaredevelopment methodologies based on UML propose an approach where the process is Use Case Driven, which means that all artifacts have traceability links from Use Cases (The artifacts are refined through several transformation steps). The Use Case Model must sacrifice precision to facilitate readability and validation. So, the analysis of use cases is mainly a manual activity. Fig. 3: Use Case Model[1]
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  GENERIC SEMANTICS 11  MergeOperator:  The Merge operator takes two models as input and produces a third one.  If A and B are models in a specific meta-model algebraic specification, the application of the Merge operator on them produces a model C, which consists the union of A and B.  Taking into account that duplicates are not allowed in a model, the union is disjoint.
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  LITERATURE-REVIEW(1/2 Cont..) 12  Semanticsof the Merge operator needs to introduce three concepts:  The equivalence relation;  The conflict resolution strategy;  The refreshment of a construct. UML CASE tools permit the arrangement of Use Cases and their corresponding partial Class Diagram into the same package, i.e. no option is provided to obtain the global Class Diagram from the partial ones. The Rational Rose Model Integration tool provides an ad-hoc solution to merge UML models. Once the merged model is generated, there is no way to relate the obtained model to the partial source models in order to keep some degree of traceability.
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  LITERATURE-REVIEW(2/2) 13  Title: “Heuristicand pattern based Merge Sort”[2] Author: Manouchehr Zadahmaf jafarlou, Parisa Yousefzadeh fard  Objective: The aim of this study is to present a stable and adaptable Merge Sort algorithm that uses the design patterns to reduce computational complexity of swaps and memory usage.  Keywords: Merge sort, Design patterns, computational complexity.
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  LITERATURE-REVIEW(2/2 Cont..) 14  Introduction: Although many consider that sorting algorithm is 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, best, worst and average case analysis, etc.
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  15  Sorting algorithmsused in computer science are often classified as:  Computational complexity.  Memory usage.  Recursion.  Stability: stable sorting algorithms maintain the relative order of records with equal values.  Whether or not they are a comparison sort. A comparison sort examines the data only by comparing two elements with a comparison operator.  General method: insertion, exchange, selection, merging, etc.  Adaptability: Whether or not the pre sorted-ness of the input affects the running time. CLASSIFICATION
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  LITERATURE-REVIEW(2/2 Cont..) 16  Heuristicand pattern based Merge Sort implementation: Fig. 4. Heuristic and pattern based Merge Sort implementation[2]
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  LITERATURE-REVIEW(2/2 Cont..) 17  Firstalgorithm:  First element of status is the index of first element of next partial array and the second one represent "the number of elements of partial array“ (+1) for ascending partial array or (-1) for descending partial array.  Used when the numbers of arrays are so many that cannot move to volatile memory.  Ex.: Given array -4, -3, 0, 1, 3, 8, 9, 14, 5, 6, 8, 14, 2, 1, -3, 1 ..  if the (-4, -3, 0, 1, 3, 8, 9, 14) be the part of array that sorted before with stMS, (5, 6, 8, 14) will be the next partial ordered array and so the status array is (8, +4).
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  LITERATURE-REVIEW(2/2 Cont..) 18  Secondalgorithm:  “The first index of first entry of partial arrays" (+1) for ascending arrays and "the first index of first entry of partial arrays" (-1) for descending arrays.  Causes a high performance by eliminating of first steps of merge sort algorithm.  For example for array -8, -4, 0, 4, 3, 1, 0, -2, 5, 7, 9, 10, 5, 4, 2, 1, 3  For status: +1, -4, +8, -12, +15; Partial arrays are indexed as 0->3 (ascending), 4->7 (descending), 8->11 (ascending), 12->14 (descending), 15->15 (ascending). Then status send as argument to stgMS and partial arrays will be the building blocks for stgM algorithm.
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  LITERATURE-REVIEW(2/2 Cont..) 19  Thirdalgorithm:  “The number of partial arrays " (+1) for ascending arrays and "the number of partial arrays" (-1) for descending arrays.  Here, stgMS is a Huffman coding algorithm used to Optimize the merging process by choosing two partial arrays, that sum of elements of them is less than others before each merging.  Then these two partial arrays send as argument to stgM algorithm and merged.
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  PROBLEM STATEMENT 20  Whenthe Use Case Model has many use cases, managing traceability between each use case and the corresponding elements in the resulting class diagram can be a difficult task.  Regarding traceability, this strategy is a sensible solution, but when several team members work in parallel with different use cases, inconsistencies or conflicts among partial models often arise, which must be solved when obtaining the integrated model.
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  CONCLUSION 21  Merge sortis an appropriate algorithm with O(nlgn) Computational complexity, but petitioning of array to one element partial arrays and then merging them cause increasing complexity in time order, system software and hardware work.  The presented algorithm eliminates these extra work using patterns.
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  REFERENCES 22 1. Artur Boronat,Jose A. C., Isidro Ramos, Patrico Letelier,” Formal Model Merging Applied to Class Diagram Integration”, © 2006-Elsevier. 2. Manouchehr Zadahmaf jafarlou, Parisa Yousefzadeh fard,” Heuristic and pattern based Merge Sort”, © 2010-Elsevier. 3. https://www.khanacademy.org/computing/computer-science/algorithms/merge- sort/a/divide-and-conquer-algorithms [09/12/2015, 22:33] 4. http://computationaltales.blogspot.in/2011/10/merge-sort-and-lines-of- kindergarteners.html [09/12/2015, 22:13] 5. http://coding-geek.com/how-databases-work/ [09/12/2015, 22:28]