day11extra - COP 3503 Computer Science II CLASS NOTES - DAY...

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COP 3503 – Computer Science II CLASS NOTES - DAY #11 - Supplement “Sorting” – What does it mean? Many times the question arises as to how the theory covered in discrete structures courses (COT 3100 at UCF) applies to various areas in Computer Science. We can use sorting as an easy example to illustrate the connection. We have dealt with several different sorting techniques in the last couple of days of class and frequently made mention of the fact that we want to sort the numbers in our arrays in ascending order. While it may be clear to us what we mean by such a statement, it is far cry from a precise statement. If we are to analyze sorting algorithms with any mathematical rigor a much more mathematically precise definition must be formulated. The following paragraph or two will give you a flavor of the mathematical precision that is required to do the job correctly. For those of you already familiar with discrete structures this should look familiar, for those of you who haven’t had that pleasure yet, hang on you’re in for quite an experience at some point. Consider a arbitrary sequence S = {s 1 , s 2 , s 3 , …, s n } composed of n 0 elements drawn from a universal set U . The goal of sorting is to rearrange the elements of S to produce a new sequence, say S’ , in which the elements of S appear in order. What does it mean to appear in order . We will assume that there exists a relation < , defined over the universe U . The relation < must be a total order. A total order is defined as: A total order is a relation, say <, defined on the elements of some universal set U with the following properties: 1. For all pairs of elements (i, j) U × U , exactly one of the following is true: i < j, i = j, or j < i . (All elements are commensurate.)
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day11extra - COP 3503 Computer Science II CLASS NOTES - DAY...

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