codeforces 449D

本文探讨了如何高效计算一个包含1e6个元素的数组中所有子序列的值通过AND运算等于0的数量。利用动态规划和容斥原理,文章提出了一种优化方案来解决这一问题,避免了传统方法中的高复杂度。

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题目

给你1e6个数,你需要找出有多少子序列,他们的值and起来为0。

思路

f(s)表示状态为ss中为1的位一定是1,为0的位可能为1的可选数字的个数,g(s)表示状态s中为1的位的个数。那么可以由容斥原理得到ans=220s=0(1)g(s)(2f(s)1)
核心就在于计算f(s)的值,若暴力计算,需要枚举s的子集复杂度高达320,不可行。考虑dp[i][s]表示低i位为s0,高(19i)位为s1的数的个数,其中,s0中为1的一定为1,为0的可能为1,s1中为1的一定为1,为0的一定为0。
转移显然,
dp[i][s]={dp[i1][s], dp[i1][s]+dp[i1][s|1<<(i1)],si=1si=0
//
//  Created by Running Photon
//  Copyright (c) 2015 Running Photon. All rights reserved.
//
#include <algorithm>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <iomanip>
#include <cassert>
#include <iostream>
#include <map>
#include <queue>
#include <string>
#include <sstream>
#include <set>
#include <vector>
#include <stack>
#define ALL(x) x.begin(), x.end()
#define INS(x) inserter(x, x,begin())
#define ll long long
#define CLR(x) memset(x, 0, sizeof x)
using namespace std;
const int inf = 0x3f3f3f3f;
const ll MOD = 1e9 + 7;
const int maxn = 1025 * 1025;
const int maxv = 1e3 + 10;
const double eps = 1e-9;

ll dp[21][maxn];
ll Pow(ll a, ll n) {
    ll ret = 1;
    while(n) {
        if(n & 1) ret = ret * a % MOD;
        n >>= 1;
        a = a * a % MOD;
    }
    return ret;
}
int main() {
#ifdef LOCAL
    freopen("C:\\Users\\Administrator\\Desktop\\in.txt", "r", stdin);
    freopen("C:\\Users\\Administrator\\Desktop\\out.txt","w",stdout);
#endif
//  ios_base::sync_with_stdio(0);
    int n;
    scanf("%d", &n);
    for(int i = 0; i < n; i++) {
        int a;
        scanf("%d", &a);
        dp[0][a]++;
    }
    for(int i = 1; i <= 20; i++) {
        for(int s = 0; s < 1 << 20; s++) {
            dp[i][s] = dp[i-1][s];
            if((s >> (i - 1) & 1) == 0) dp[i][s] += dp[i-1][s | (1 << i - 1)];
            assert(dp[i][s] >= 0);
        }
    }
    ll ret = 0;
    for(int i = 0; i < 1 << 20; i++) {
        int num = __builtin_popcount(i);
        if(num & 1) ret = (ret + MOD - Pow(2, dp[20][i]) + 1) % MOD;
        else ret = (ret + Pow(2, dp[20][i]) - 1 + MOD) % MOD;
    }
    cout << ret << endl;
    return 0;
}

### Codeforces 1487D Problem Solution The problem described involves determining the maximum amount of a product that can be created from given quantities of ingredients under an idealized production process. For this specific case on Codeforces with problem number 1487D, while direct details about this exact question are not provided here, similar problems often involve resource allocation or limiting reagent type calculations. For instance, when faced with such constraints-based questions where multiple resources contribute to producing one unit of output but at different ratios, finding the bottleneck becomes crucial. In another context related to crafting items using various materials, it was determined that the formula `min(a[0],a[1],a[2]/2,a[3]/7,a[4]/4)` could represent how these limits interact[^1]. However, applying this directly without knowing specifics like what each array element represents in relation to the actual requirements for creating "philosophical stones" as mentioned would require adjustments based upon the precise conditions outlined within 1487D itself. To solve or discuss solutions effectively regarding Codeforces' challenge numbered 1487D: - Carefully read through all aspects presented by the contest organizers. - Identify which ingredient or component acts as the primary constraint towards achieving full capacity utilization. - Implement logic reflecting those relationships accurately; typically involving loops, conditionals, and possibly dynamic programming depending on complexity level required beyond simple minimum value determination across adjusted inputs. ```cpp #include <iostream> #include <vector> using namespace std; int main() { int n; cin >> n; vector<long long> a(n); for(int i=0;i<n;++i){ cin>>a[i]; } // Assuming indices correspond appropriately per problem statement's ratio requirement cout << min({a[0], a[1], a[2]/2LL, a[3]/7LL, a[4]/4LL}) << endl; } ``` --related questions-- 1. How does identifying bottlenecks help optimize algorithms solving constrained optimization problems? 2. What strategies should contestants adopt when translating mathematical formulas into code during competitive coding events? 3. Can you explain why understanding input-output relations is critical before implementing any algorithmic approach? 4. In what ways do prefix-suffix-middle frameworks enhance model training efficiency outside of just tokenization improvements? 5. Why might adjusting sample proportions specifically benefit models designed for tasks requiring both strong linguistic comprehension alongside logical reasoning skills?
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