A1103. Integer Factorization
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 = n1^P + ... nK^P
where ni (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, ... bK } 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
1 #include<cstdio> 2 #include<iostream> 3 #include<algorithm> 4 #include<vector> 5 using namespace std; 6 vector<int> ans, temp, fac; 7 int N, K, P, maxSumfac = -1; 8 void dfs(int index, int cnt, int sum, int sumfac){ 9 if(sum == N && cnt == K){ 10 if(sumfac > maxSumfac){ 11 maxSumfac = sumfac; 12 ans = temp; 13 } 14 return; 15 } 16 if(index <= 0 || sum > N || cnt > K) 17 return; 18 if(fac[index] + sum <= N){ 19 temp.push_back(index); 20 dfs(index, cnt + 1, sum + fac[index], sumfac + index); 21 temp.pop_back(); 22 } 23 dfs(index - 1, cnt, sum, sumfac); 24 } 25 int power(int n, int p){ 26 int bas = 1; 27 for(int i = 0; i < p; i++) 28 bas *= n; 29 return bas; 30 } 31 int main(){ 32 scanf("%d %d %d", &N, &K, &P); 33 int i, num; 34 for(i = 0; ; i++){ 35 num = power(i, P); 36 if(num > N) 37 break; 38 fac.push_back(num); 39 } 40 if(num > N) 41 dfs(i - 1, 0, 0, 0); 42 else dfs(i, 0, 0, 0); 43 if(ans.size() == 0){ 44 printf("Impossible"); 45 }else{ 46 printf("%d = %d^%d", N, ans[0], P); 47 int len = ans.size(); 48 for(int i = 1; i < len; i++){ 49 printf(" + %d^%d", ans[i], P); 50 } 51 } 52 cin >> N; 53 return 0; 54 }
总结:
1、本题题意:给出N、P、K,要求选出K个数,使得他们分别的P次方再求和等于N。按照降序输出序列,且若有多个答案,选择一次方和最大的一个输出。
2、预处理,计算中肯定需要反复用到一个数的P次方,如果每次用时都计算,显然太慢且重复。可以提前预计算一个数组,i的P次方为 fac[i] 。具体范围应该计算到 i 的P次方大于N的那个i。然后dfs从i - 1开始递减搜索。 在main函数开始的地方不要忘记写预处理的语句。 同样,当前序列的和应该在选择过程中计算,而不是每次都重复计算。
3、注意不要少写递归结束的语句。当结果符合要求时需要return,但不符合要求时也需要return。
4、vector<int> ans, temp;ans、temp一个存全局最优答案,一个存当前答案,两者可以直接赋值,其内容是拷贝。
5、注意index <= 0时也不符合条件,需要加到搜索结束的条件中去。