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- 1. CSCD 300 Data Structures Donald Shell’s Sorting Algorithm Originally developed by Bill Clark, modified by Tom Capaul and Tim Rolfe 1
- 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. 2
- 3. Shell Sort -General Description Essentially a segmented insertion sort Divides an array into several smaller noncontiguous 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. 3
- 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. 4
- 5. Shell Sort - example Initial Segmenting Gap = 4 80 93 60 12 42 30 68 85 10 10 30 60 12 42 93 68 85 80 5
- 6. Shell Sort - example (2) Resegmenting Gap = 2 10 30 60 12 42 93 68 85 80 10 12 42 30 60 85 68 93 80 6
- 7. Shell Sort - example (3) Resegmenting Gap = 1 10 12 42 30 60 85 68 93 80 10 12 30 42 60 68 80 85 93 7
- 8. Gap Sequences for Shell Sort The sequence h1, h2, h3,. . . , ht 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 h1 = 1. For any gap value hk we have A[i] <= A[i + hk] An array A for which this is true is hk sorted. An array which is hk sorted and is then hk-1 sorted remains hk sorted. 8
- 9. Shell Sort - Ideal Gap Sequence Although any increasing sequence will work ( if h1 = 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. 9
- 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. 10
- 11. Shell Sort - Added Gap Sequence Donald Knuth, in his discussion of Shell’s Sort, recommended another sequence of gaps. h0 = 1 hj+1 = hj * 3 + 1 Find the hj > n, then start with hj/3 11
- 12. Link to the Java program that generated the above data. 12
- 13. Shell Sort - Time Complexity Time complexity: O(nr) with 1 < r < 2 This is better than O(n2) but generally worse than O(n log2n). 13
- 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; } } 14
- 15. ShellSort -Trace (gap = 4) [0] [1] [2] theArray 80 n: 9 gap: 4 93 60 [3] [4] [5] [6] [7] [8] 12 85 10 42 30 68 i: j: 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
- 16. ShellSort -Trace (gap = 2) [0] [1] [2] theArray [3] [4] [5] [6] [7] [8] 10 12 85 80 n: 9 gap: 2 30 60 42 93 68 i: j: 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; } 16
- 17. ShellSort -Trace (gap = 1) [0] [1] [2] theArray [3] [4] [5] [6] [7] [8] 10 30 93 80 n: 9 gap: 1 12 42 60 85 68 i: j: 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; } 17

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