[杂题] Codeforces #577B. Modulo Sum

本文探讨了模m子序列和的问题,通过抽屉原理分析得出当子序列长度大于等于m时,必然存在模m相等的情况。文章提供了一个O(m^2)复杂度的解决方案,并附带了完整的C++实现代码。

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简单题。
a1,a1+a2,a1+a2+a3,a1+a2+a3+a4...
抽屉原理,这些数模m一定会有相等的或为 0 的。容易想到,当 nm 时是一定有解的。
所以直接 O(m2) 搞搞就好了。

#include<cstdio>
#include<cstring> 
#include<algorithm>
using namespace std;
const int maxn=5005;
int n,m,a[maxn];
int f[2][maxn];
int main(){
    freopen("cf577B.in","r",stdin);
    freopen("cf577B.out","w",stdout);
    scanf("%d%d",&n,&m);
    if(n>=m) return puts("YES\n"),0;
    for(int i=1;i<=n;i++) scanf("%d",&a[i]), a[i]%=m;
    f[1][a[1]]=1;
    for(int i=1;i<=n-1;i++){
        memset(f[(i&1)^1],0,sizeof(f[(i&1)^1]));
        f[(i&1)^1][a[i+1]]=1;
        for(int j=0;j<=m-1;j++) if(f[(i&1)][j]) f[(i&1)^1][j]|=1, f[(i&1)^1][(j+a[i+1])%m]|=1; 
    }
    if(f[n&1][0]) printf("YES\n"); else printf("NO\n");
    return 0; 
}
### Codeforces Problem 797B Explanation The problem titled "Restoring the Permutation" requires reconstructing a permutation from its prefix sums modulo \( m \). Given an array of integers representing these prefix sums, one must determine whether it is possible to restore such a permutation. In this context, a **permutation** refers to an ordered set containing each integer exactly once within a specified range. The task involves checking if there exists any valid sequence that matches the provided conditions when performing operations as described in the problem statement[^1]. To solve this issue effectively: - Iterate through all elements while maintaining two variables: `current_sum` which tracks cumulative sum during iteration; and `min_value`, used later for adjustments. ```cpp int n, m; cin >> n >> m; vector<int> s(n); for (auto& x : s) cin >> x; ``` Calculate differences between consecutive terms after adjusting initial values appropriately by subtracting minimum value found so far at every step. This adjustment ensures non-negativity throughout calculations without altering relative order among elements. Check feasibility based on properties derived from constraints given in the question text. Specifically, ensure no duplicate residues appear under modulus operation since they would violate uniqueness required for permutations. Finally, construct answer using adjusted difference list obtained previously along with necessary checks ensuring correctness according to rules outlined above.
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