2 条题解
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1
前置知识:简要了解 CRT 和高斯消元
题意简述:给定一些系数,求 元线性同余方程组 的解。
注意到 ,而且他们都是质数,这引导着我们思考先分别求出模 意义下的解,然后使用 CRT 来合并答案。
对固定的质数求解 元线性同余方程组的具体步骤,首先我们考虑把所有数字都放在模意义下,我们发现这个问题就变成了求解线性方程组的问题,用高斯消元法求解即可。
换言之:求解 元线性同余方程组,就是模意义下的高斯消元。
下面说一些实现细节。- 1.我们读取日期的时候需要注意每个月的日子各不相同,开个数组存一下,然后我个人比较喜欢封装的写法,因此有了下面的代码:
int days[12] = {31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31}; int date() { int d, m; scanf("%d %d", &d, &m); for (int i = 0; i < m - 1; ++i) d += days[i]; return d - 1; }- 2.因为模数固定且质因子较少,我们直接手算式子就行。代码如下:
for (int i = 0; i < M; ++i) { printf("%d\n", (146 * Sol5[i] + 220 * Sol73[i] + 364) % 365 + 1); }- 3.做除法的时候别忘了,其实是要乘以逆元的。
差不多就这些了,下面贴个代码:
#include <bits/stdc++.h> using namespace std; const int maxN = 205, maxM = 205; int N, M; int mts[12] = {31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31}; int Init[maxN][maxM + 1], tmp[maxN][maxM + 1]; int date() { int d, m; scanf("%d %d", &d, &m); for (int i = 0; i < m - 1; ++i) d += mts[i]; return d - 1; } int Sol5[maxM], Sol73[maxM], inv[100]; void get_inv(int p) { inv[0] = 0; for (int i = 1; i < p; ++i) for (int j = 1; j < p; ++j) if ((i * j) % p == 1) inv[i] = j; } void Gauss(int p, int* Sol) { for (int i = 0; i < N; ++i) for (int j = 0; j <= M; ++j) tmp[i][j] = Init[i][j] % p; get_inv(p); int valid = 0; for (int s = 0; s < M; ++s) { int ff = -1; for (int i = valid; ff == -1 && i < N; ++i) if (tmp[i][s] != 0) ff = i; if (ff == -1) continue; if (valid != ff) for (int i = 0; i <= M; ++i) swap(tmp[valid][i], tmp[ff][i]); int rev = inv[tmp[valid][s]]; for (int i = 0; i <= M; ++i) tmp[valid][i] = (tmp[valid][i] * rev) % p; for (int i = 0; i < N; ++i) if (i != valid) { int cf = tmp[i][s]; for (int j = 0; j <= M; ++j) { tmp[i][j] -= cf * tmp[valid][j]; tmp[i][j] %= p; if (tmp[i][j] < 0) tmp[i][j] += p; } } ++valid; } for (int i = 0; i < N; ++i) { int first = -1; for (int j = 0; j < M; ++j) if (tmp[i][j] != 0) { first = j; break; } if (first == -1) { if (tmp[i][M] != 0) { cout << "-1\n"; exit(0); } continue; } Sol[first] = tmp[i][M]; } } int main() { cin >> N >> M; for (int i = 0; i < N; ++i) { int a = date(), b = date(); for (int j = 0; j < M; ++j) scanf("%d", Init[i] + j); Init[i][M] = ((b - a) % 365 + 365) % 365; } Gauss(5, Sol5); Gauss(73, Sol73); for (int i = 0; i < M; ++i) { cout << (146 * Sol5[i] + 220 * Sol73[i] + 364) % 365 + 1 << "\n"; } return 0; } -
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#include <bits/stdc++.h> #define lg2 std::__lg using std::cin; using std::cout; const int N = 400054, LN = 18, TH = 18; int n, m, q; int len[N], leap[N]; char s[N], t[N]; int pre[N], suf[N], bel[N]; int L, st[LN][N / TH], *lay = *st; inline int max(const int x, const int y) {return x < y ? y : x;} namespace SAM { int p, np = 1, cnt = 1; int pa[N], d[N][2], val[N]; #define q d[p][x] int extend(int x) { for (p = np, val[np = ++cnt] = val[p] + 1; p && !q; q = np, p = pa[p]); if (!p) pa[np] = 1; else if (val[p] + 1 == val[q]) pa[np] = q; else { int nq = ++cnt; val[nq] = val[p] + 1, memcpy(d[nq], d[q], 8); pa[nq] = pa[q], pa[np] = pa[q] = nq; for (int Q = q; p && q == Q; q = nq, p = pa[p]); } return np; } #undef q } void build_sparse_table(int *a) { int i, j, k, I, *f, *g = lay; for (I = i = 0; I < n; ++i, I += TH) { pre[I] = a[I], bel[I] = i; for (j = 1; j < TH && I + j < n; ++j) pre[I + j] = max(pre[I + j - 1], a[I + j]), bel[I + j] = i; for (--j, suf[I + j] = a[I + j]; --j >= 0; suf[I + j] = max(suf[I + j + 1], a[I + j])); lay[i] = suf[I]; } for (k = L = i, j = 0; 1 << (j + 1) <= L; ++j) for (f = g, g = st[j + 1], k -= 1 << j, i = 0; i < k; ++i) g[i] = max(f[i], f[i + (1 << j)]); } inline int RMQ(int x, int y) { if (y < x + TH) return *std::max_element(len + x, len + (y + 1)); int xb = bel[x] + 1, yb = bel[y], c = lg2(yb - xb); return max(max(suf[x], pre[y]), xb < yb ? max(st[c][xb], st[c][yb - (1 << c)]) : 0); } int main() { int i, j = 0, l, r, x, y, id; std::ios::sync_with_stdio(false), cin.tie(NULL); cin >> s >> t >> q, n = strlen(s), m = strlen(t); for (i = 0; i < m; ++i) SAM::extend(t[i] - 97); x = 1, y = 0; for (i = 0; i < n; ++i) { for (id = s[i] - 97; x && !SAM::d[x][id]; x = SAM::pa[x], y = SAM::val[x]); len[i] = (x ? (x = SAM::d[x][id], ++y) : (x = 1, y = 0)); } for (*leap = -1, j = i = n - 1; i >= 0; --i) for (; j > i - len[i]; --j) leap[j] = i; build_sparse_table(len); for (; q; --q) cin >> l >> r, cout << (leap[--l] < --r ? max(leap[l] - l + 1, RMQ(leap[l] + 1, r)) : r - l + 1) << '\n'; return 0; }
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信息
- ID
- 1946
- 时间
- 1000ms
- 内存
- 256MiB
- 难度
- 10
- 标签
- 递交数
- 5
- 已通过
- 3
- 上传者