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Chapter 7: Sorting A lgorithms

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java. Chapter 7: Sorting A lgorithms. Shell Sort. Lydia Sinapova, Simpson College. Shell Sort. Improves on insertion sort Compares elements far apart , then less far apart , finally compares adjacent elements

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Chapter 7: Sorting A lgorithms

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  1. Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 7: Sorting Algorithms Shell Sort Lydia Sinapova, Simpson College

  2. Shell Sort • Improves on insertion sort • Compares elementsfar apart, thenless far apart, finally comparesadjacentelements (as an insertion sort).

  3. Idea • arrange the data sequence in atwo-dimensional array • sort the columnsof the array • repeatthe process each time withsmaller number of columns

  4. Example 3 7 9 0 5 1 6 8 4 2 0 6 1 5 7 3 4 9 8 2 it is arranged in an array with 7 columns (left), then the columns are sorted (right): 3 7 9 0 5 1 6 3 3 2 0 5 1 5 8 4 2 0 6 1 5  7 4 4 0 6 1 6 7 3 4 9 8 2 8 7 9 9 8 2

  5. Example (cont) 3 3 2 0 0 1 0 5 1 1 2 2 5 7 4 3 3 4 4 0 6 4 5 6 1 6 8 5 6 8 7 9 9 7 7 9 8 2 8 9 one column in the last step – it is only a 6, an 8 and a 9 that have to move a little bit to their correct position

  6. Implementation • one-dimensional array that is • indexed appropriately. • an increment sequence • to determine how far apart elements to be sorted are

  7. Increment sequence Determines how far apart elements to be sorted are: h1, h2, ..., ht with h1 = 1 hk-sorted array - all elements spaced a distance hkapart are sorted relative to each other.

  8. Correctness of the algorithm Shellsort only works because an array that is hk-sorted remains hk-sorted when hk- 1-sorted. Subsequent sorts do not undo the work done by previous phases. The last step (with h = 1) - Insertion Sort on the whole array

  9. Java code for Shell sort int j, p, gap; comparable tmp; for (gap = N/2; gap > 0; gap = gap/2) for ( p = gap; p < N ; p++) { tmp = a[p]; for ( j = p ; j >= gap && tmp < a[ j- gap ]; j = j - gap) a[ j ] = a[ j - gap ];   a[j] = tmp; }

  10. Increment sequences 1. Shell's original sequence: N/2 , N/4 , ..., 1 (repeatedly divide by 2). 2. Hibbard's increments: 1, 3, 7, ..., 2k - 1 ; 3. Knuth's increments: 1, 4, 13, ..., ( 3k - 1) / 2 ; 4. Sedgewick's increments: 1, 5, 19, 41, 109, .... 9 ·4k - 9 ·2k + 1 or 4k - 3 ·2k + 1.

  11. Analysis Shellsort's worst-case performance using Hibbard's increments is Θ(n3/2).  The averageperformance is thought to be about O(n 5/4) The exact complexity of this algorithm is still being debated . for mid-sized data : nearly as well if not better than the faster (n log n) sorts.

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