November 25th, 2014

/*************************************************************************
 *  Compilation:  javac Shell.java
 *  Execution:    java Shell < input.txt
 *  Dependencies: StdOut.java StdIn.java
 *  Data files:   http://algs4.cs.princeton.edu/21sort/tiny.txt
 *                http://algs4.cs.princeton.edu/21sort/words3.txt
 *   
 *  Sorts a sequence of strings from standard input using shellsort.
 *
 *  Uses increment sequence proposed by Sedgewick and Incerpi.
 *  The nth element of the sequence is the smallest integer >= 2.5^n
 *  that is relatively prime to all previous terms in the sequence.
 *  For example, incs[4] is 41 because 2.5^4 = 39.0625 and 41 is
 *  the next integer that is relatively prime to 3, 7, and 16.
 *   
 *  % more tiny.txt
 *  S O R T E X A M P L E
 *
 *  % java Shell < tiny.txt
 *  A E E L M O P R S T X                 [ one string per line ]
 *    
 *  % more words3.txt
 *  bed bug dad yes zoo ... all bad yet
 *  
 *  % java Shell < words3.txt
 *  all bad bed bug dad ... yes yet zoo    [ one string per line ]
 *
 *
 *************************************************************************/

/**
 *  The Shell class provides static methods for sorting an
 *  array using Shellsort with Knuth's increment sequence (1, 4, 13, 40, ...).
 *  

 *  For additional documentation, see Section 2.1 of
 *  Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.
 *  
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class Shell {

    // This class should not be instantiated.
    private Shell() { }

    /**
     * Rearranges the array in ascending order, using the natural order.
     * @param a the array to be sorted
     */
    public static void sort(Comparable[] a) {
        int N = a.length;

        // 3x+1 increment sequence:  1, 4, 13, 40, 121, 364, 1093, ... 
        int h = 1;
        while (h < N/3) h = 3*h + 1; 

        while (h >= 1) {
            // h-sort the array
            for (int i = h; i < N; i++) {
                for (int j = i; j >= h && less(a[j], a[j-h]); j -= h) {
                    exch(a, j, j-h);
                }
            }
            assert isHsorted(a, h); 
            h /= 3;
        }
        assert isSorted(a);
    }



   /***********************************************************************
    *  Helper sorting functions
    ***********************************************************************/
    
    // is v < w ?
    private static boolean less(Comparable v, Comparable w) {
        return (v.compareTo(w) < 0);
    }
        
    // exchange a[i] and a[j]
    private static void exch(Object[] a, int i, int j) {
        Object swap = a[i];
        a[i] = a[j];
        a[j] = swap;
    }


   /***********************************************************************
    *  Check if array is sorted - useful for debugging
    ***********************************************************************/
    private static boolean isSorted(Comparable[] a) {
        for (int i = 1; i < a.length; i++)
            if (less(a[i], a[i-1])) return false;
        return true;
    }

    // is the array h-sorted?
    private static boolean isHsorted(Comparable[] a, int h) {
        for (int i = h; i < a.length; i++)
            if (less(a[i], a[i-h])) return false;
        return true;
    }

    // print array to standard output
    private static void show(Comparable[] a) {
        for (int i = 0; i < a.length; i++) {
            StdOut.println(a[i]);
        }
    }

    /**
     * Reads in a sequence of strings from standard input; Shellsorts them; 
     * and prints them to standard output in ascending order. 
     */
    public static void main(String[] args) {
        String[] a = StdIn.readAllStrings();
        Shell.sort(a);
        show(a);
    }

}

Property. The number of compares used by shellsort with the increments 1, 4, 13, 40, 121, 364, ... is bounded by a small multiple of N times the number of increments used.
Proposition. The number of compares used by shellsort with the increments 1, 4, 13, 40, 121, 364, ... is O(N^(3/2)).

Classwork:

1. Trace by showing the array contents after each pass:

E A S Y S H E L L S O R T Q U E S T I O N

Note: you do not have to trace it by hand
2. The shell sort uses as a helper another sort. Which one? Why?
3. Tell the story of the Shell sort.