1. 1 CSCD 300 Data Structures Donald Shell’s Sorting Algorithm Originally developed by Bill Clark, modified by Tom Capaul and Tim Rolfe

2. 2 Shell Sort - Introduction More properly, Shell’s Sort Created in 1959 by Donald Shell Link to a local copy of the article: Donald Shell, “A High-Speed Sorting Procedure”, Communications of the ACM Vol 2, No. 7 (July 1959), 30-32 Originally Shell built his idea on top of Bubble Sort (link to article flowchart), but it has since been transported over to Insertion Sort.link to article flowchart

3. 3 Shell Sort -General Description Essentially a segmented insertion sort Divides an array into several smaller non- contiguous segments The distance between successive elements in one segment is called a gap. Each segment is sorted within itself using insertion sort. Then resegment into larger segments (smaller gaps) and repeat sort. Continue until only one segment (gap = 1) - final sort finishes array sorting.

4. 4 Shell Sort -Background General Theory: Makes use of the intrinsic strengths of Insertion sort. Insertion sort is fastest when: The array is nearly sorted. The array contains only a small number of data items. Shell sort works well because: It always deals with a small number of elements. Elements are moved a long way through array with each swap and this leaves it more nearly sorted.

5. 5 Shell Sort - example 80 93 60 68 1285 42 30 10 Initial Segmenting Gap = 4 10 30 60 68 1285 42 93 80

6. 6 Shell Sort - example (2) 103060681285429380 Resegmenting Gap = 2 101242683093608580

7. 7 Shell Sort - example (3) 101230804285606893 101242683093608580 Resegmenting Gap = 1

8. 8 Gap Sequences for Shell Sort The sequence h 1, h 2, h 3,..., h t is a sequence of increasing integer values which will be used as a sequence (from right to left) of gap values. Any sequence will work as long as it is increasing and h 1 = 1. For any gap value h k we have A[i] <= A[i + h k ] An array A for which this is true is h k sorted. An array which is h k sorted and is then h k-1 sorted remains h k sorted.

9. 9 Shell Sort - Ideal Gap Sequence Although any increasing sequence will work ( if h 1 = 1): Best results are obtained when all values in the gap sequence are relatively prime (sequence does not share any divisors). Obtaining a relatively prime sequence is often not practical in a program so practical solutions try to approximate relatively prime sequences.

10. 10 Shell Sort - Practical Gap Sequences Three possibilities presented: 1) Shell's suggestion - first gap is N/2 - successive gaps are previous value divided by 2. Odd gaps only - like Shell method except if division produces an even number add 1. better performance than 1) since all odd values eliminates the factor 2. 2.2 method - like Odd gaps method (add 1 to even division result) but use a divisor of 2.2 and truncate. best performance of all - most nearly a relatively prime sequence.

11. 11 Shell Sort - Added Gap Sequence Donald Knuth, in his discussion of Shell’s Sort, recommended another sequence of gaps. h 0 = 1 h j+1 = h j * 3 + 1 Find the h j > n, then start with h j /3

12. 12 Link to the Java program that generated the above data.

13. 13 Shell Sort - Time Complexity Time complexity: O(n r ) with 1 < r < 2 This is better than O(n 2 ) but generally worse than O(n log 2 n).

14. 14 Shellsort - Code public static void shellSort( Comparable[ ] theArray, int n ) { // shellSort: sort first n items in array theArray for( int gap = n / 2; gap > 0; gap = gap / 2 ) for( int i = gap; i < n; i++ ) { Comparable tmp = theArray[ i ]; int j = i; for( ; j >= gap && tmp.compareTo(theArray[ j - gap ]) < 0 ; j -= gap ) theArray[ j ] = theArray[ j - gap ]; theArray[ j ] = tmp; }

15. 15 ShellSort -Trace (gap = 4) for( int gap = n / 2; gap > 0; gap = gap / 2 ) for( int i = gap; i < n; i++ ) { Comparable tmp = theArray[ i ]; int j = i; for( ; j >= gap && tmp.compareTo(theArray[ j - gap ]) < 0 ; j -= gap ) theArray[ j ] = theArray[ j - gap ]; theArray[ j ] = tmp; } 8093 60 681285423010 [0] [1] [2] [3] [4] [5] [6] [7] [8] n: 9 gap: 4 i: j: theArray

16. 16 ShellSort -Trace (gap = 2) for( int gap = n / 2; gap > 0; gap = gap / 2 ) for( int i = gap; i < n; i++ ) { Comparable tmp = theArray[ i ]; int j = i; for( ; j >= gap && tmp.compareTo(theArray[ j - gap ]) < 0 ; j -= gap ) theArray[ j ] = theArray[ j - gap ]; theArray[ j ] = tmp; } [0] [1] [2] [3] [4] [5] [6] [7] [8] n: 9 gap: 2 i: j: theArray 103060681285429380

17. 17 ShellSort -Trace (gap = 1) for( int gap = n / 2; gap > 0; gap = gap / 2 ) for( int i = gap; i < n; i++ ) { Comparable tmp = theArray[ i ]; int j = i; for( ; j >= gap && tmp.compareTo(theArray[ j - gap ]) < 0 ; j -= gap ) theArray[ j ] = theArray[ j - gap ]; theArray[ j ] = tmp; } [0] [1] [2] [3] [4] [5] [6] [7] [8] n: 9 gap: 1 i: j: theArray 101242683093608580