PAT甲级1103 Integer Factorization (30分)

Posted 2020-11-25 musecho

1103 Integer Factorization (30分)

The K?P factorization of a positive integer N is to write N as the sum of the P-th power of K positive integers. You are supposed to write a program to find the K?P factorization of N for any positive integers N, K and P.

Input Specification:

Each input file contains one test case which gives in a line the three positive integers N (≤400), K (≤N) and P (1<P≤7). The numbers in a line are separated by a space.

Output Specification:

For each case, if the solution exists, output in the format:

N = n[1]^P + ... n[K]^P

      
    

where n[i] (i = 1, ..., K) is the i-th factor. All the factors must be printed in non-increasing order.

Note: the solution may not be unique. For example, the 5-2 factorization of 169 has 9 solutions, such as 122+42+22+22+12, or 112+62+22+22+22, or more. You must output the one with the maximum sum of the factors. If there is a tie, the largest factor sequence must be chosen -- sequence { a1,a2,?,*aK* } is said to be larger** than { b1,b2,?,*b**K* } if there exists 1≤L≤K such that ai=bi for i<L and aL>bL.

If there is no solution, simple output Impossible.

Sample Input 1:

169 5 2

      
    

Sample Output 1:

169 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2

      
    

Sample Input 2:

169 167 3

      
    

Sample Output 2:

Impossible

分析

将正整数N分解为K项数字（因子）的P次方之和，因子以非增顺序排列。多个解中取因子之和最大的，若和相同则取因子的字典序在前的。没解输出Impossible。

• 设置数组a：将index=0开始的数字的P次方存到数组a中，直到a[index]>n。

• 全局变量：设置数组tempAns和ans，变量facSum和maxFacSum

  数组tempAns存放当前解的临时因子们，ans存放最优解的因子们。

  maxFacSum最优解的因子之和，初值设置为-1。

• 开始dfs：

  • tempK表示当前K的个数，当tempK==K时，判断tempSum当前总和==N且（是否为最优解）facSum当前解的因子之和>maxFacSum：更新最优结果变量ans和maxFacSum。

  • index从高后前遍历数组a

    • 判断当前项a[index]是否加入解：tempSum+a[index]<=N，满足则更新tempAns，并递归调用dfs确定下一项

    • 判断是否完成一个解：index==1

    • 若不满足tempSum+a[index]<=N，则index--，因子变小再判断。

代码（参考柳神）

#include<iostream>
#include<vector>
#include<cmath>

using namespace std;

int K,N,P;
vector<int> a,tempAns,ans;
int maxFacSum=-1;

/*初始化：计算a数组*/
void init() {
    int temp=0;
    for(int i=1; temp<=N; i++) {//注意；i从1开始
        a.push_back(temp);
        temp=pow(i,P);
    }
}
/*dfs*/
void dfs(int index,int tempK,int tempSum,int facSum) {
    if(tempK==K) {
        if(tempSum==N&&facSum>maxFacSum) {
            ans=tempAns;
            maxFacSum=facSum;
        }
        return;
    }

    while(index>=1) {
        if(tempSum+a[index]<=N) {
            tempAns[tempK]=index;//使用数组访问必须对vector进行resize
            dfs(index,tempK+1,tempSum+a[index],facSum+index);
        }
        if(index==1) return;
        index--;
    }

}
int main() {
    scanf("%d %d %d",&N,&K,&P);

    init();
    tempAns.resize(K);
    dfs(a.size()-1,0,0,0);
    if(maxFacSum==-1) {
        printf("Impossible");
        return 0;
    }

    printf("%d = ",N);
    for(int i=0; i<ans.size(); i++) {
        if(i!=0) printf(" + ");
        printf("%d^%d",ans[i],P);
    }

    return 0;
}

单词

• factorization：因式分解

  factor：因子

• If there is a tie , ..:平手（相等）