算法设计与分析: 2-5 半数集问题

2-5 半数集问题

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问题描述

给定一个自然数n，由n开始可以依次产生半数集set(n)中的数如下：
（1）set(n)中包含n；
（2）在n的左边加上一个自然数，但该自然数不能超过最近添加的数的一半；
（3）按此规则进行处理，直至不能添加自然数为止。
例如，set(6)={6,16,26,126,36,136}。
注意：该半数集是多重集。

编程任务：对于给定的自然数n，编程计算半数集set(n)中元素的个数。

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分析

设$set(n)$中的元素个数为$f(n)$，则显然有：
$f(n) = 1 + \sum_{i=1}^{n/2}f(i)$

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Java: version-1

import java.util.Scanner;

public class Main {

    public static void main(String[] args) {
        Scanner input = new Scanner(System.in);
        int number;
        int count;

        while (true){
            System.out.println("Please input a number: ");

            number = input.nextInt();
            count = halfNumber(number);

            System.out.println("There are "+count+" elements!");
            System.out.println("----------------------------");
        }
    }

    private static int halfNumber(int number){
        int count = 1;
        if(number > 1){
            for(int i=1; i<=number/2; i++){
                count += halfNumber(i);
            }
        }

        return count;
    }
}

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Java: version-2

import java.util.Scanner;

public class Main {
    private static int[] countArray = new int[1001];

    public static void main(String[] args) {
        Scanner input = new Scanner(System.in);
        int number;
        int count;
        countArray[1] = 1;

        while (true){
            System.out.println("Please input a number: ");

            number = input.nextInt();
            count = halfNumber(number);

            System.out.println("There are "+count+" elements!");
            System.out.println("----------------------------");
        }
    }

    private static int halfNumber(int number){
        int count = 1;
        if(countArray[number] > 0)
            return countArray[number];

        for(int i=1; i<=number/2; i++){
            count += halfNumber(i);
        }
        countArray[number] = count;

        return count;
    }
}

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Output

Please input a number: 
1
There are 1 elements!
----------------------------
Please input a number: 
2
There are 2 elements!
----------------------------
Please input a number: 
3
There are 2 elements!
----------------------------
Please input a number: 
4
There are 4 elements!
----------------------------
Please input a number: 
5
There are 4 elements!
----------------------------
Please input a number: 
6
There are 6 elements!
----------------------------
Please input a number: 
34
There are 238 elements!
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Please input a number: 
23
There are 74 elements!
----------------------------
Please input a number: 
999
There are 1955133450 elements!
----------------------------
Please input a number: 
1000
There are 1981471878 elements!
----------------------------
Please input a number: 

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Reference

王晓东《计算机算法设计与分析》（第3版）P43