数列极差问题

题目描述：

在黑板上写了N个正整数作成的一个数列，进行如下操作：每一次擦去其中的两个数a和b，然后在数列中加入一个数a*b+1，如此下去直至黑板上剩下一个数，在所有按这种操作方式最后得到的数中，最大的max，最小的为min，则该数列的极差定义为M=max-min。

题目分析:

当看到此题时，我们会发现求max与求min是两个相似的过程。若我们把求解max与min的过程分开，着重探讨求max的问题。下面我们以求max为例来讨论此题用贪心策略求解的合理性。
讨论：

假设经(N-3)次变换后得到3个数：a, b, max＇ (max＇>= a >= b)， 其中max＇是 (N-2)个数经 (N-3) 次ｆ变换后所得的最大值，此时有两种求值方式，设其所求值分别为Z1 ，Z2 ，则有：Z1＝(a*b+1)*max＇+1，Z2 ＝(a*max＇+1)*b+1　所以 Z1-Z2 ＝max＇-b>=0 若经 (N-2) 次变换后所得的3个数为：m，a，b   (m >= a >= b)且m不为 (N-2) 次变换后的最大值，即ｍ＜max＇则此时所求得的最大值为： Z＝(a*b+1)*m+1　此时 Z1-Z2 ＝(a*b+1)(max＇-m) > 0　所以此时不为最优解。

所以若使第ｋ(1 <= k <= N-1) 次变换后所得值最大，必使 (k-1) 次变换后所得值最大（符合贪心策略的特点2），在进行第ｋ次变换时，只需取在进行(k-1)次变换后所得数列中的两最小数ｐ，ｑ施加ｆ操作：p←p*q+1，q←∞即可（符合贪心策略特点1），因此此题可用贪心策略求解。讨论完毕。

在求min时，我们只需在每次变换的数列中找到两个最大数p, q施加作用f：p ←p*q+1，q← -∞即可．原理同上

 

 

public class Array_1 {
 public static int[] maxarray = null;
 public static int[] minarray = null;
 public static int num;
 public static void handle() { //等待调用；
  int min, max, i, j, k, temp, a = 0, b;
  String text_temp=new String("");
  for (i = 0; i < num; i++) //minarray数组从大到小排序
  {
   for (k = i, j = i + 1; j < num; j++)
    if (minarray[k] < minarray[j])
     k = j;
   temp = minarray[k];
   minarray[k] = minarray[i];
   minarray[i] = temp;
  }
  for (i = 0; i < num; i++){
   maxarray[i] = minarray[num - i - 1]; //maxarray数组从小到大排序
  }
  for (i = 1; i < num; i++){
   minarray[i] = minarray[i] * minarray[i - 1] + 1;
   System.out.print(minarray[i] + " ");
   text_temp = text_temp + Integer.toString(minarray[i]) + "  ";
   for (int iner= i+1;iner<num;iner++){ 
    System.out.print(minarray[iner] + " ");
    text_temp = text_temp + Integer.toString(minarray[iner])+"  ";
   }text_temp = text_temp + "  ";
   System.out.print("\n");
   text_temp = text_temp + "\n";
  }
  text_temp = text_temp + "\n";
  System.out.print("\n");
  
  min = minarray[num - 1];
  for (i = 0; i < num - 1; i++) //还有问题需要修改；
  {
   a = 0;
   b = 1;
   for (j = 2; j < num; j++) //a,b存放当前数组中最小的2个数
   {
    if (maxarray[j] < maxarray[a] || maxarray[j] < maxarray[b])
     if (maxarray[a] < maxarray[b])
      b = j;
     else
      a = j;
   }
   maxarray[a] = maxarray[a] * maxarray[b] + 1;
   maxarray[b] = 999999999;
//  System.out.print(maxarray[a]+" ");
   
  for (int outer = 0; outer < num; outer++){
   if(maxarray[outer]!=999999999){
   System.out.print(maxarray[outer] + " ");
   text_temp = text_temp + Integer.toString(maxarray[outer])+"  ";
   }
  } 
  text_temp = text_temp + "\n"; 
   System.out.print("\n");  }System.out.print("\n");
  max = maxarray[a];
  int m = max -min;
  text_temp = text_temp + "\n" +"M="+max+"-"+min+"="+m;
  Graph.textArea3.setText(text_temp);
  System.out.println(m);
 }
}
import java.awt.EventQueue;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.border.EmptyBorder;
import javax.swing.JLabel;
import javax.swing.JTextField;
import javax.swing.JTextArea;
import javax.swing.JButton;
import java.awt.Font;
import javax.swing.JScrollPane;
import java.awt.event.ActionListener;
import java.awt.event.ActionEvent;
import java.io.IOException;
import java.net.URL;
public class Graph extends JFrame {
 private static final long serialVersionUID = 1L;
 public static String number = null;
 public static int no;
 private JPanel contentPane;
 public static JTextField textField = new JTextField();
 public static JTextArea textArea1 = new JTextArea();
 public static JTextArea textArea3 = new JTextArea();
 private JLabel label_1;
 public static void main(String[] args) {
  EventQueue.invokeLater(new Runnable() {
   public void run() {
    try {
     Graph frame = new Graph();
     frame.setVisible(true);
    } catch (Exception e) {
     e.printStackTrace();
    }
   }
  });
 }
 
class MyEvent implements ActionListener
{
  public void actionPerformed(ActionEvent e)
   {
    if (e.getActionCommand().equals("问题描述"))
     {
      URL url=null;
      try {
      url = new URL(" http://202.202.111.145/~zhousir");
      System.out.println(url.getContent());
      System.out.println(url.getAuthority());
      } catch (Exception e1) {
       e1.printStackTrace();
      }
      try {
       Runtime.getRuntime().exec("rundll32 url.dll,FileProtocolHandler "+" http://202.202.111.145/~zhousir" );
       } catch (IOException e2) {
       e2.printStackTrace();
      }
     }

   }
}
 public Graph() {
  setTitle("数列极差_贪心算法");
  setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
  setBounds(100, 100, 604, 380);
  contentPane = new JPanel();
  contentPane.setBorder(new EmptyBorder(5, 5, 5, 5));
  setContentPane(contentPane);
  contentPane.setLayout(null);
  {
   JLabel label = new JLabel("\u8BF7\u8F93\u5165\u6570\u636E\u4E2A\u6570\uFF1A");
   label.setFont(new Font("微软雅黑", Font.BOLD, 15));
   label.setBounds(23, 10, 166, 34);
   contentPane.add(label);
  }
  {
//   textField = new JTextField();
   textField.setBounds(23, 42, 192, 34);
   textField.setText("5");
//   no = Integer.parseInt(textField.getText());
   contentPane.add(textField);
   textField.setColumns(10);
  }
  {
   label_1 = new JLabel(
     "\u8BF7\u4F9D\u6B21\u8F93\u5165\u6570\u636E\uFF1A");
   label_1.setFont(new Font("微软雅黑", Font.BOLD, 15));
   label_1.setBounds(23, 86, 166, 30);
   contentPane.add(label_1);
  }
  {
   JButton button = new JButton("\u95EE\u9898\u63CF\u8FF0");
   button.setFont(new Font("微软雅黑", Font.BOLD, 14));
   button.setBounds(23, 282, 107, 34);
   button.addActionListener(new MyEvent());
   contentPane.add(button);
  }
  {
   JScrollPane scrollPane = new JScrollPane();
   scrollPane.setBounds(23, 125, 192, 147);
   contentPane.add(scrollPane);
   {
//    JTextArea textArea1 = new JTextArea();
    textArea1.setText("1 2 3 4 5");
//    number = textArea1.getText();
    scrollPane.setViewportView(textArea1);
   }
  }
  {
   JButton button = new JButton("\u786E\u5B9A");
   button.addActionListener(new ActionListener() {
    public void actionPerformed(ActionEvent e) {
     if (e.getActionCommand().equals("确定")) {
      // String number = textField.getText();
      Array_1.num = Integer.parseInt(textField.getText());
      System.out.println(Array_1.num);
      
      String[] data = textArea1.getText().split(" ");
//      char[] data = number.toCharArray(); // 数组长度出错;
      // System.out.println('a');
//      System.out.println(data.length);
      System.out.println(textArea1.getText());
      Array_1.maxarray = new int[Array_1.num];
      Array_1.minarray = new int[Array_1.num];
      
      for (int i = 0, j = 0; i < data.length; i++, j++) {
       Array_1.maxarray[j] = Array_1.minarray[j] = Integer.parseInt(data[i]);
      }
      Array_1.handle();
     }
    }
   });
   button.setFont(new Font("微软雅黑", Font.BOLD, 14));
   button.setBounds(140, 282, 75, 34);
   contentPane.add(button);
  }
  {
   JLabel label = new JLabel("\u6267\u884C\u8FC7\u7A0B");
   label.setFont(new Font("微软雅黑", Font.BOLD, 15));
   label.setBounds(369, 20, 84, 15);
   contentPane.add(label);
  }
  {
   JScrollPane scrollPane = new JScrollPane();
   scrollPane.setBounds(248, 42, 321, 273);
   contentPane.add(scrollPane);
   {
    scrollPane.setViewportView(textArea3);
   }
  }
 }
}