算法第四版学习笔记之初级排序算法

软件：DrJava

参考书：算法（第四版）

章节：2.1（以下截图是算法配套视频所讲内容截图）

1:rules of the game(游戏规则)

2:selection sort（选择排序）

3:insertion sort（插入排序）

4:shellsort（希尔排序）

5:shuffling（洗牌）

6:convex hull（凸包算法）

1:rules of the game(游戏规则)

排序算法的目标就是降所有元素的主键按照某种方式排列。在Java中，元素通常都是对象，对主键的抽象描述则是通过一种内置的机制（Comparable接口）来完成的。

2:selection sort（选择排序）

选择排序的两个鲜明的特点：1：运行时间和输入无关。2：数据移动是最少的。

/*************************************************************************
 *  Compilation:  javac Selection.java
 *  Execution:    java  Selection < 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 selection sort.
 *   
 *  % more tiny.txt
 *  S O R T E X A M P L E
 *
 *  % java Selection < tiny.txt
 *  S O R T E X A M P L E A               [ one string per line ]
 *    
 *  % more words3.txt
 *  bed bug dad yes zoo ... all bad yet
 *  
 *  % java Selection < words3.txt
 *  all bad bed bug dad ... yes yet zoo    [ one string per line ]
 *
 *************************************************************************/

import edu.princeton.cs.algs4.*;
import java.util.Comparator;

public class Selection {

    // selection sort
    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);
    }

    // use a custom order and Comparator interface - see Section 3.5
    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);
    }


   /***********************************************************************
    *  Helper sorting functions
    ***********************************************************************/
    
    // 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;
    }


   /***********************************************************************
    *  Check if array is sorted - useful for debugging
    ***********************************************************************/

    // 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]);
        }
    }

    // Read strings from standard input, sort them, and print.
    public static void main(String[] args) {
        String[] a = StdIn.readStrings();
        Selection.sort(a);
        show(a);
    }
}

3:insertion sort（插入排序）

插入排序：只需比较前后两个值，之后后边值比前边小，就交换位置。

/*************************************************************************
 *  Compilation:  javac Insertion.java
 *  Execution:    java Insertion < 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 insertion sort.
 *
 *  % more tiny.txt
 *  S O R T E X A M P L E
 *
 *  % java Insertion < tiny.txt
 *  S O R T E X A M P L E A               [ one string per line ]
 *
 *  % more words3.txt
 *  bed bug dad yes zoo ... all bad yet
 *
 *  % java Insertion < words3.txt
 *  all bad bed bug dad ... yes yet zoo   [ one string per line ]
 *
 *************************************************************************/
import edu.princeton.cs.algs4.*;
import java.util.Comparator;

public class Insertion {

    // use natural order and Comparable interface
    public static void sort(Comparable[] a) {
        //将a[]按升序排列
        int N = a.length;
        for (int i = 0; i < N; i++) {
            //将a[i]插入到a[i-1]、a[i-2]、a[i-3]...之中
            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);
    }

    // use a custom order and Comparator interface - see Section 3.5
    public static void sort(Object[] a, Comparator c) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            for (int j = i; j > 0 && less(c, a[j], a[j-1]); j--) {
                exch(a, j, j-1);
            }
            assert isSorted(a, c, 0, i);
        }
        assert isSorted(a, c);
    }

    // return a permutation that gives the elements in a[] in ascending order
    // do not change the original array a[]
    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;
    }

   /***********************************************************************
    *  Helper sorting functions
    ***********************************************************************/
    
    // 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;
    }

    // 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;
    }

   /***********************************************************************
    *  Check if array is sorted - useful for debugging
    ***********************************************************************/
    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 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]);
        }
    }

    // Read strings from standard input, sort them, and print.
    public static void main(String[] args) {
        String[] a = StdIn.readStrings();
        Insertion.sort(a);
        show(a);
    }
}

比较两种排序算法：对于随机排序的无重复主键的数组，插入排序和选择排序的运行时间是平方级别的，两者之比应该是一个较小的常数。

4:shellsort（希尔排序）

希尔排序的思想是使数组中任意间隔为h的元素都是有序的。

/*************************************************************************
 *  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
 *  S O R T E X A M P L E A               [ 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 ]
 *
 *
 *************************************************************************/
import edu.princeton.cs.algs4.*;
public class Shell {

    // sort the array a[] in ascending order using Shellsort
    public static void sort(Comparable[] a) {
        //将a[]按升序排列
        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 将数组变为h有序
            for (int i = h; i < N; i++) {
                //将a[i]插入到a[i-h]、a[i-2*h]。。。之中               
                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;
        }
        //查看是否已经排序成功，如果没成功，发出异常报警
        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]);
        }
    }

    // Read strings from standard input, sort them, and print.
    public static void main(String[] args) {
        String[] a = StdIn.readStrings();
        Shell.sort(a);
        show(a);
    }

}

希尔排序比插入排序以及选择排序快得多。

通过提升速度来解决其他方式无法解决的问题是研究算法的设计和性能的主要原因之一。