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Sorting, Sets, and Selecting - Ed. 2. and 3.: Chapter 10 - Ed. 4.: Chapter 11

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Sorting, Sets, and Selection Merge-sort -Divide-and-conquer strategy -Merge process and merge-sort Sets -Set ADT -Set operations Quick-sort -Simple quick-sort -In-place quick-sort

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Merge Sort

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Therefore, the merge sort involves three steps in sorting a sequence S with n elements: 1. Divide: If S has zero or one element, return S immediately; it is already sorted. Otherwise, split S into two sequences S 1 and S 2, each containing about half of the elements in S; that is, S 1 contains the first n/2 elements of S, and S 2 contains the rest. 2. Recur: Recursively sort sequence S 1 and S 2. 3. Conquer: Put back the elements into S by merging the sorted sequences S 1 and S 2 into a sorted sequence.

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The figure here shows how the sequence is divided in the previous example. 6385244517315096 6385244517315096 8524634517315096 8524634517319650

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The figure here shows how the sequences are merged in the previous example. 6385244517 31 5096 6385 24 4517315096 8524 63 451731 5096 8524634517319650

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take the first element comparison Put remaining elements in S1 into S Put remaining elements in S2 into S

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The Running Time of Merge Sort The running time of merge sort is proportional to nlogn where n is the size of the sequence. running time = O(nlogn). Recall that the running time for bubble sort is proportional to the square of n. Therefore merge sort is much faster than bubble sort.

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public class Comparator { public int compare(Object v1, Object v2) { if (((Integer) v1).intValue() < (Integer) v2).intValue()) return -1; else if (((Integer) v1).intValue() == ((Integer) v2).intValue()) return 0; return 1; } public boolean isLessThan (Object v1, Object v2) { Integer u1 = (Integer) v1; Integer u2 = (Integer) v2; if (u1.intValue() < u2.intValue()) return true; return false; }

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public boolean isEqualTo (Object v1, Object v2) { Integer u1 = (Integer) v1; Integer u2 = (Integer) v2; if (u1.intValue() == u2.intValue()) return true; return false; } public boolean isLargerThan (Object v1, Object v2) { Integer u1 = (Integer) v1; Integer u2 = (Integer) v2; if (u1.intValue() > u2.intValue()) return true; return false; } public boolean isLessThanOrEqualTo (Object v1, Object v2) { Integer u1 = (Integer) v1; Integer u2 = (Integer) v2; if (u1.intValue() == u2.intValue() or (u1.intValue() < u2.intValue()) ) return true; return false; }

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ALGORITHM ANALYSIS AND DESIGN INTRODUCTION TO ALGORITHMS CS 413 Divide and Conquer Algortihms: Binary search, merge sort.

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