五、排序算法（选择排序、插入排序、希尔排序）

排序

初级排序算法

通用API：
less():比较 exch():交换 show():输出 sort:排序
为了验证是不是成功，加上 assert isSorted(a)
成本模型：
在研究排序算法时，我们需要计算比较和交换的数量，对于不交换元素的算法，会计算访问数组的次数

选择排序

找到最小的元素，和第一个交换，再在剩下的元素中找到最小的元素，与第二个交换，一次进行下去。
代码（含有多个重载方法）

public class Selection {
    // This class should not be instantiated.
    private Selection() { }
    public static void sort(Comparable[] a) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            int min = i;
            for (int j = i+1; j < N; j++) {
                if (less(a[j], a[min])) min = j;
            }
            exch(a, i, min);
            assert isSorted(a, 0, i);
        }
        assert isSorted(a);
    }
    public static void sort(Object[] a, Comparator c) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            int min = i;
            for (int j = i+1; j < N; j++) {
                if (less(c, a[j], a[min])) min = j;
            }
            exch(a, i, min);
            assert isSorted(a, c, 0, i);
        }
        assert isSorted(a, c);
    }
    // is v < w ?
    private static boolean less(Comparable v, Comparable w) {
        return v.compareTo(w) < 0;
    }

    // is v < w ?
    private static boolean less(Comparator c, Object v, Object w) {
        return c.compare(v, 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;
    }
    // is the array a[] sorted?
    private static boolean isSorted(Comparable[] a) {
        return isSorted(a, 0, a.length - 1);
    }  
    // is the array sorted from a[lo] to a[hi]
    private static boolean isSorted(Comparable[] a, int lo, int hi) {
        for (int i = lo + 1; i <= hi; i++)
            if (less(a[i], a[i-1])) return false;
        return true;
    }
    // is the array a[] sorted?
    private static boolean isSorted(Object[] a, Comparator c) {
        return isSorted(a, c, 0, a.length - 1);
    }
    // is the array sorted from a[lo] to a[hi]
    private static boolean isSorted(Object[] a, Comparator c, int lo, int hi) {
        for (int i = lo + 1; i <= hi; i++)
            if (less(c, a[i], a[i-1])) 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]);
        }
    }
    public static void main(String[] args) {
        String[] a = StdIn.readAllStrings();
        Selection.sort(a);
        show(a);
    }
}

两个鲜明的特点：1. 运行时间和输入无关；2.数据一定是最少的

插入排序

插入排序的基本操作就是将一个数据插入到已经排好序的有序数据中

public class Insertion {
    // This class should not be instantiated.
    private Insertion() { }
    public static void sort(Comparable[] a) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            for (int j = i; j > 0 && less(a[j], a[j-1]); j--) {
                exch(a, j, j-1);
            }
            assert isSorted(a, 0, i);
        }
        assert isSorted(a);
    }
    public static void sort(Comparable[] a, int lo, int hi) {
        for (int i = lo; i <= hi; i++) {
            for (int j = i; j > lo && less(a[j], a[j-1]); j--) {
                exch(a, j, j-1);
            }
        }
        assert isSorted(a, lo, hi);
    }
    public static void sort(Object[] a, Comparator comparator) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            for (int j = i; j > 0 && less(a[j], a[j-1], comparator); j--) {
                exch(a, j, j-1);
            }
            assert isSorted(a, 0, i, comparator);
        }
        assert isSorted(a, comparator);
    }
    public static void sort(Object[] a, int lo, int hi, Comparator comparator) {
        for (int i = lo; i <= hi; i++) {
            for (int j = i; j > lo && less(a[j], a[j-1], comparator); j--) {
                exch(a, j, j-1);
            }
        }
        assert isSorted(a, lo, hi, comparator);
    }
    public static int[] indexSort(Comparable[] a) {
        int N = a.length;
        int[] index = new int[N];
        for (int i = 0; i < N; i++)
            index[i] = i;

        for (int i = 0; i < N; i++)
            for (int j = i; j > 0 && less(a[index[j]], a[index[j-1]]); j--)
                exch(index, j, j-1);

        return index;
    }
    // is v < w ?
    private static boolean less(Comparable v, Comparable w) {
        return v.compareTo(w) < 0;
    }
    // is v < w ?
    private static boolean less(Object v, Object w, Comparator comparator) {
        return comparator.compare(v, 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;
    }
    // exchange a[i] and a[j] (for indirect sort)
    private static void exch(int[] a, int i, int j) {
        int swap = a[i];
        a[i] = a[j];
        a[j] = swap;
    }
    private static boolean isSorted(Comparable[] a) {
        return isSorted(a, 0, a.length - 1);
    }
    // is the array sorted from a[lo] to a[hi]
    private static boolean isSorted(Comparable[] a, int lo, int hi) {
        for (int i = lo+1; i <= hi; i++)
            if (less(a[i], a[i-1])) return false;
        return true;
    }
    private static boolean isSorted(Object[] a, Comparator comparator) {
        return isSorted(a, 0, a.length - 1, comparator);
    }
    // is the array sorted from a[lo] to a[hi]
    private static boolean isSorted(Object[] a, int lo, int hi, Comparator comparator) {
        for (int i = lo + 1; i <= hi; i++)
            if (less(a[i], a[i-1], comparator)) 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]);
        }
    }
    public static void main(String[] args) {
        String[] a = StdIn.readAllStrings();
        Insertion.sort(a);
        show(a);
    }
}

插入排序对于已经有序的数组运行时间是线性的（选择排序仍是平方级别的）

倒置（也就是逆序）：

插入排序的进一步提升： 只要在内循环中将较大的元素都向右移动而不总是交换两个元素（这样访问数组的次数就能减半（因为交换必然访问两个元素）） 无代码

排序算法的可视化

采用棒状图：
可以利用自带库StdDraw画出来

比较两种排序

希尔排序

基于插入排序，交换不相邻的元素已对数组的局部进行排序，并最终用插入排序将局部有序的数组排序。思想是使数组中任意间隔为h的元素都是有序的，这样的数组称为h有序数组.

public class Shell {
    // This class should not be instantiated.
    private Shell() { }
    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);
    }
    // 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;
    }
    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]);
        }
    }
    public static void main(String[] args) {
        String[] a = StdIn.readAllStrings();
        Shell.sort(a);
        show(a);
    }
}

轨迹示例：黑色的表示移动，灰色的表示不移动，红色的表示a[j]