[CCPC] 2017秦皇岛 NumbersI | Java BigInteger | 贪心

Posted 2021-10-03 PushyTao

题目描述

DreamGrid has a nonnegative integer n. He would like to divide n into m nonnegative integers a1,a2,…,am and minimizes their bitwise or (i.e. n=a1+a2+…+am and a1 OR a2 OR … OR am should be as small as possible).

input

There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case:
The first line contains two integers n and m (0≤n＜10¹⁰⁰⁰,1≤m＜10¹⁰⁰).
It is guaranteed that the sum of the length of n does not exceed 20000.

output

For each test case, output an integer denoting the minimum value of their bitwise or.

样例输入

5
3 1
3 2
3 3
10000 5
1244 10

样例输出 Copy

3
3
1
2000
125

题意：
给出一个数$n$，构造出一个数列 $a_1...a_m$，使得所有数之和$a_1+...+a_m$为$n$，并且还要使得$a_1|...|a_m$
尽可能的小，问最小的$or$运算之后的值是多少

首先可以看到题目的数据范围很大，考虑使用Java大数$BigInteger$

很容易想到这是个贪心，并且涉及位运算我们可以按照每一位进行考虑，首先将2的次方数预处理出来，存放在一个$BigInteger$数组里面
然后考虑从小往大里面进行相加，每次加入$m$个二的次方数，一直到$sum\ge n$ ，然后我们记录这个时候的$pos$
然后我们这个时候再从后向前$pos->0$ 依次求出$a[i]-1\ast m$ 是否 $\ge n$，如果说$\ge n$就跳过，反之求出现在的$n$还能凑出多少个$a[i]$，设个数为$cnt$，$cnt=min(cnt,m)$那么就将$ans$中加入$a[i]$，然后将$n$减去$cnt$个$a[i]$

main_code:

import java.math.BigInteger;
import java.util.Scanner;
 
public class Main {
    public static BigInteger a[] = new BigInteger[5007];///2 ** x
 
    public static void main(String[] args) {
        Scanner cin =  new Scanner(System.in);
        int pos = 0;
        a[0] = BigInteger.valueOf(1);
        for(int i=1;i<=5000;i++){
            a[i] = a[i-1].multiply(BigInteger.valueOf(2));
        }
//        System.out.println(a[3]);
        int T = cin.nextInt();
        BigInteger n,m;
        while(T > 0) {
            T -= 1;
            n = cin.nextBigInteger();
            m = cin.nextBigInteger();
            if (m.equals(BigInteger.valueOf(1))) {/// m == 1
                System.out.println(n);
                continue;
            }
            BigInteger tn = n;
            BigInteger sum = BigInteger.valueOf(0), ans = BigInteger.valueOf(0);
            for (int i = 0; ; i++) {
                if(sum.compareTo(n) >= 0) break;
                sum = sum.add(m.multiply(a[i]));
                pos = i;
            }
            for (int i = pos; i >= 0; i--) {
                BigInteger t = a[i].subtract(BigInteger.valueOf(1));
                if (t.multiply(m).compareTo(tn) >= 0) {
                    continue;
                }
                BigInteger cnt = tn.divide(a[i]);
                cnt = cnt.min(m);
                ans = ans.add(a[i]);
                tn = tn.subtract(cnt.multiply(a[i]));
            }
            System.out.println(ans);
        }
    }
}
/**
 5
 3 1
 3 2
 3 3
 10000 5
 1244 10
 * **/
 
/**************************************************************
    Problem: 4827
    Language: Java
    Result: 正确
    Time:1058 ms
    Memory:85000 kb
****************************************************************/