From aef1c50fe44623c8b705ccd0d435ddb392762059 Mon Sep 17 00:00:00 2001 From: TooYoungTooSimp <6648049+TooYoungTooSimp@users.noreply.github.com> Date: Thu, 23 Feb 2017 16:57:45 +0800 Subject: [PATCH] Section 4.1 4.2 --- 省选准备/assets/day4/CF359D.cpp | 49 +++++++++++++++++++++++++++ 省选准备/assets/day4/FenwickTree.cpp | 13 +++++++ 省选准备/assets/day4/poj1990.cpp | 42 +++++++++++++++++++++++ 省选准备/assets/day4/poj2299.cpp | 42 +++++++++++++++++++++++ 省选准备/assets/day4/poj2299.jpg | Bin 0 -> 5351 bytes 省选准备/assets/day4/poj2352.cpp | 30 ++++++++++++++++ 省选准备/assets/day4/poj3264.cpp | 29 ++++++++++++++++ 省选准备/省选基础算法.pdf | Bin 761464 -> 793853 bytes 省选准备/省选基础算法.tex | 40 ++++++++++++++++++---- 9 files changed, 239 insertions(+), 6 deletions(-) create mode 100644 省选准备/assets/day4/CF359D.cpp create mode 100644 省选准备/assets/day4/FenwickTree.cpp create mode 100644 省选准备/assets/day4/poj1990.cpp create mode 100644 省选准备/assets/day4/poj2299.cpp create mode 100644 省选准备/assets/day4/poj2299.jpg create mode 100644 省选准备/assets/day4/poj2352.cpp create mode 100644 省选准备/assets/day4/poj3264.cpp diff --git a/省选准备/assets/day4/CF359D.cpp b/省选准备/assets/day4/CF359D.cpp new file mode 100644 index 0000000..c6db778 --- /dev/null +++ b/省选准备/assets/day4/CF359D.cpp @@ -0,0 +1,49 @@ +#include +#include +inline int min(int a, int b) { return a < b ? a : b; } +inline int gcd(int a, int b) { return b == 0 ? a : gcd(b, a%b); } +inline void read(int &x) +{ + int ch = x = 0; + while (!isdigit(ch = getchar())); + for (; isdigit(ch); ch = getchar()) x = (x << 3) + (x << 1) + ch - '0'; +} +int n, a[1 << 20 | 1], Log[1 << 20 | 1]; +int M[1 << 20 | 1][20], G[1 << 20 | 1][20]; +struct +{ + int c, a[1 << 20 | 1], r; +}ans; +inline bool can(int l, int r) +{ + int k = Log[r - l + 1]; + return min(M[l][k], M[r - (1 << k) + 1][k]) == gcd(G[l][k], G[r - (1 << k) + 1][k]); +} +bool check(int len) +{ + int cnt = 0; + for (int i = 0; i + len < n; i++) + if (can(i, i + len)) + ans.a[cnt++] = i; + if (cnt) ans.r = len, ans.c = cnt; + return cnt; +} +int main() +{ + read(n); + Log[0] = -1; + for (int i = 1; i <= n; i++) Log[i] = Log[i >> 1] + 1; + for (int i = 0; i < n; i++) read(a[i]); + for (int i = 0; i < n; i++) M[i][0] = G[i][0] = a[i]; + for (int j = 1; j <= Log[n]; j++) + for (int i = 0; i + (1 << j) - 1 < n; i++) + M[i][j] = min(M[i][j - 1], M[i + (1 << (j - 1))][j - 1]), + G[i][j] = gcd(G[i][j - 1], G[i + (1 << (j - 1))][j - 1]); + int L = 0, R = n, m; + while (L < R) + check(m = (L + R) >> 1) ? L = m + 1 : R = m; + printf("%d %d\n", ans.c, ans.r); + for (int i = 0; i < ans.c; i++) + printf("%d ", ans.a[i] + 1); + return 0; +} \ No newline at end of file diff --git a/省选准备/assets/day4/FenwickTree.cpp b/省选准备/assets/day4/FenwickTree.cpp new file mode 100644 index 0000000..dae196b --- /dev/null +++ b/省选准备/assets/day4/FenwickTree.cpp @@ -0,0 +1,13 @@ +void add(int x, int v) +{ + for (; x <= n; x += lowbit(x)) + A[x] += v; +} + +void sum(int x) +{ + int res = 0; + for (; x; x -= lowbit(x)) + res += A[x]; + return res; +} diff --git a/省选准备/assets/day4/poj1990.cpp b/省选准备/assets/day4/poj1990.cpp new file mode 100644 index 0000000..f6be133 --- /dev/null +++ b/省选准备/assets/day4/poj1990.cpp @@ -0,0 +1,42 @@ +#include +#include +#define lowbit(x) ((x) & -(x)) +const int N = 20005; +void add(int *arr, int pos, int val) +{ + for (; pos < N; pos += lowbit(pos)) + arr[pos] += val; +} +int query(int *arr, int pos) +{ + int ans = 0; + for (; pos; pos -= lowbit(pos)) + ans += arr[pos]; + return ans; +} +int sum[2][N]; +struct cow +{ + int pos, vol; + bool operator<(const cow &rhs) const { return vol < rhs.vol; } +} cows[N]; +int main() +{ + int n; + scanf("%d", &n); + for (int i = 1; i <= n; i++) + scanf("%d%d", &cows[i].vol, &cows[i].pos); + std::sort(cows + 1, cows + n + 1); + long long ans = 0; + for (int i = 1; i <= n; i++) + { + long long a = query(sum[0], cows[i].pos), b = query(sum[1], cows[i].pos); + ans += (cows[i].pos * a - b + query(sum[1], 20000) - b - + (i - 1 - a) * cows[i].pos) * + cows[i].vol; + add(sum[0], cows[i].pos, 1); + add(sum[1], cows[i].pos, cows[i].pos); + } + printf("%lld", ans); + return 0; +} \ No newline at end of file diff --git a/省选准备/assets/day4/poj2299.cpp b/省选准备/assets/day4/poj2299.cpp new file mode 100644 index 0000000..29a4434 --- /dev/null +++ b/省选准备/assets/day4/poj2299.cpp @@ -0,0 +1,42 @@ +#include +#include +#include +#include +#define lowbit(x) ((x) & -(x)) +const int N = 500005; +int sum[N]; +int query(int x) +{ + int ans = 0; + for 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= 1 << 15 | 1; +int a[maxn], n, m, level[maxn]; +void update(int pos, int x) +{ + for (; pos < maxn; pos += lowbit(pos)) a[pos] += x; +} +int query(int pos) +{ + int ans = 0; + for (; pos; pos -= lowbit(pos)) ans += a[pos]; + return ans; +} +int main() +{ + scanf("%d", &n); + memset(level, 0, sizeof(level)); + memset(a, 0, sizeof(a)); + for (int i = 0, x, y; i < n; ++i) + { + scanf("%d%d", &x, &y); + level[query(++x)]++; + update(x, 1); + } + for (int i = 0; i < n; ++i) + printf("%d\n", level[i]); + return 0; +} \ No newline at end of file diff --git a/省选准备/assets/day4/poj3264.cpp b/省选准备/assets/day4/poj3264.cpp new file mode 100644 index 0000000..305ee46 --- /dev/null +++ b/省选准备/assets/day4/poj3264.cpp @@ -0,0 +1,29 @@ +#include +inline int min(int a, int b) { return a < b ? a : b; } +inline int max(int a, int b) { return a > b ? a : b; } +const int N = 50010; +const int LogN = 16; +int minT[LogN][N], maxT[LogN][N], Log[N]; +int main() +{ + Log[0] = -1; + for (int i = 1; i < N; i++) 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z9v{}NhS9>q%V8P468@2q5?MYmPSi^)xG1bEt*$GM%!Xb0D!}Hn%#O`4Bv>D+B_HXg z3OU*k;~E(y1e$4Yq1o7O+HP;%Z0!1NA@=9eQY3sKlu=F~D$ur@S=8eG{2x_wdR-m5a{}$Rh zK5TJQ7je$N7d^nNZ@GP-EW@a!d34(yp=5q1nq&JPsyUduxtqJU8`PRWOhi;%P?Ug` KRbEYj;J*N99=ApS diff --git a/省选准备/省选基础算法.tex b/省选准备/省选基础算法.tex index b85b029..8dadd76 100644 --- a/省选准备/省选基础算法.tex +++ b/省选准备/省选基础算法.tex @@ -284,15 +284,13 @@ Manacher 模板题 \paragraph{中国剩余定理} 若$m_1,m_2,m_3,\ldots,m_r$是两两互素的正整数,则同余方程组 \begin{equation} - \left\{ - \begin{aligned} + \left\{\begin{aligned} & x\equiv a_1\pmod{m_1}\\ & x\equiv a_2\pmod{m_2}\\ & x\equiv a_3\pmod{m_3}\\ & \cdots\\ & x\equiv a_r\pmod{m_r}\\ - \end{aligned} - \right. + \end{aligned}\right. \end{equation} 有$\mod M=\prod\limits_1^r m$的唯一解,即为中国剩余定理。\\ 若$n=pq$且$gcd(p,q)=1$,那么$x\mod p,x\mod q$的值确定后,$x\mod n$的值也会随之确定。 @@ -330,10 +328,40 @@ Manacher 模板题 $$gcd(a,n)=1\Rightarrow Ord_n(a)|\varphi(n)$$ $$\mbox{特别的,}p\in\mathbb{P},gcd(a,p)=1\Rightarrow Ord_p(a)|p-1\mbox{,显然}\varphi(p)=p-1$$ \paragraph{原根} $n\in\mathbb{N_+},a\in\mathbb{Z},Ord_n(a)=\varphi(n)$,则称$a$为模$n$的一个原根,由阶的定义可知原根和$n$必然互质。 -\paragraph{求质数$p$的原根算法} - 暴力从小到大枚举$g\in\mathbb{N_+}$,$\forall a\in\{x\;|\;x|p-1,x\in\mathbb{P}\},g^{\frac{p-1}{a}}\not\equiv1\pmod{p}$ +\paragraph{求质数$p$的原根算法} 暴力从小到大枚举$g\in\mathbb{N_+}$,$\forall a\in\{x\;|\;x|p-1,x\in\mathbb{P}\},g^{\frac{p-1}{a}}\not\equiv1\pmod{p}$ \codeinput[Primitive Root]{assets/day3/proot.cpp} \paragraph{练习题} \subparagraph{\href{http://poj.org/problem?id=1284}{POJ1284} - Primitive Roots} 如果$n\in\mathbb{N_+}$有一个原根,那么$n$一共有$\varphi(\varphi(n))$个不同余的原根 \codeinput[Primitive Roots]{assets/day3/poj1284.cpp} +\section{day4 数据结构} +\subsection{树状数组} 在$\lg{n}$的时间内更新或查询$\sum\limits_{i=1}^n a_i$ +\paragraph{lowbit} \verb|lowbit(x) = x & -x| +\paragraph{两个操作} $\lg{n}$更新或查询 +\codeinput[Fenwick tree]{assets/day4/FenwickTree.cpp} +\paragraph{练习题} +\subparagraph{\href{http://poj.org/problem?id=2352}{POJ2352} - Stars} 。。。 +\codeinput[Stars]{assets/day4/poj2352.cpp} +\subparagraph{\href{http://poj.org/problem?id=2299}{POJ2299} - Ultra-QuickSort} 求逆序对数量,不过配图是啥玩意?马桶橛子? +\codeinput[Ultra-QuickSort]{assets/day4/poj2299.cpp} +\begin{center}\includegraphics[height=10cm]{assets/day4/poj2299.jpg}\end{center} +\subparagraph{\href{http://poj.org/problem?id=1990}{POJ1990} - MooFest} 杀光奶牛问题就会得到解决 +\codeinput[MooFest]{assets/day4/poj1990.cpp} +\subsection{Sparse Table} 处理区间最值,即 RMQ(Range Minimum Query)问题。 +\paragraph{预处理} 计算一个数组$f$,使$f[i][j]=min[i,i+2^j)$,然后你就可以开始倍增了。 + \begin{eqnarray} + f[i][j]=\left\{\begin{array}{lr} + a[i] & j=0 \\ + min(f[i][j-1],f[i+2^{j-1}][j-1]) & j>0 + \end{array}\right. + \end{eqnarray} +\paragraph{询问} + 考虑一个询问区间$[x,y)$的最小值的询问操作。\\ + 我们可以求出满足$2^i\le y-x < 2^{i+1}$的$i$,即$\lfloor\log_2{y-x}\rfloor$,这样我们可以用两个长度为$2^i$的小区间覆盖询问的大区间。\\ + 而长度为$2^i$的小区间的最小值在预处理时已经求出,于是区间$[x,y)$的最小值为$min\{f[x][i],f[y-2^i][i]\}$,由于是求最值,区间有交集也没关系。 +\paragraph{练习题} +\subparagraph{\href{http://poj.org/problem?id=3264}{POJ3264} - Balanced Lineup} + 给你一个长度为$n$的序列$a[N]$,询问$Q$次,每次输出$[L,R]$区间最大值与最小值的差是多少,果真这种简化了的题面真是清晰易懂。 +\codeinput[Balanced Lineup]{assets/day4/poj3264.cpp} +\subparagraph{\href{http://codeforces.com/problemset/problem/359/D}{CF359D} - Pair of Numbers} 首先要知道gcd有“最值的性质”,然后二分暴力。 +\codeinput[Pair of Numbers]{assets/day4/CF359D.cpp} \end{document} \ No newline at end of file