From 06e93e15d85033ba14bbe24f8d6b423a124fa52d Mon Sep 17 00:00:00 2001 From: Zengtudor Date: Wed, 7 Aug 2024 13:33:35 +0800 Subject: [PATCH] update RMQ learning by chat --- README.md | 2 + day5/RMQ_by_chat.md | 272 +++++++++++++++++++++++++++++++++++++++++++ day5/RMQ_by_chat.pdf | Bin 0 -> 574525 bytes 3 files changed, 274 insertions(+) create mode 100644 day5/RMQ_by_chat.md create mode 100644 day5/RMQ_by_chat.pdf diff --git a/README.md b/README.md index 7b0f5ec..81495d3 100644 --- a/README.md +++ b/README.md @@ -69,6 +69,8 @@ ll ksm(ll a,ll b,ll M){ #### [公平的竞争(fair)](./day5/statement.pdf) ##### $x=2+ {{x}\over{4}}$ => $x = {{8} \over 3}$是怎么得出来的 +# [RMQ 区间最值问题](./day5/RMQ_by_chat.md) +>点击跳转 # 排序 ## 稳定性 diff --git a/day5/RMQ_by_chat.md b/day5/RMQ_by_chat.md new file mode 100644 index 0000000..2dd7bb7 --- /dev/null +++ b/day5/RMQ_by_chat.md @@ -0,0 +1,272 @@ +稀疏表(Sparse Table)是一种高效的数据结构,主要用于解决静态数组上的区间查询问题,特别是最值查询(最大值、最小值等)。它的主要优势在于预处理时间和查询时间都非常高效,适用于数据不变的情况。 + +### 稀疏表的基本思路 + +1. **预处理阶段**: + - 构建一个二维数组 `st`,其中 `st[i][j]` 表示从位置 `i` 开始长度为 `2^j` 的区间的最值。 + - 利用动态规划的思想,逐步计算不同长度的区间最值,利用前面计算的结果来推导更长区间的结果。 + +2. **查询阶段**: + - 对于任意区间 `[L, R]`,将其拆分为两个可以重叠的最大幂次长度的区间,然后利用预处理的结果快速查询。 + +### 具体步骤 + +#### 1. 初始化与预处理 + +假设有一个数组 `arr`,其长度为 `n`。首先,初始化一个二维数组 `st` 和一个一维数组 `log`。`log` 数组用于快速计算区间长度的对数值。 + +```cpp +int n = arr.size(); +int K = log2(n) + 1; +vector> st(n, vector(K)); +vector log(n + 1); + +// 初始化 log 数组 +log[1] = 0; +for (int i = 2; i <= n; ++i) + log[i] = log[i / 2] + 1; + +// 初始化 st 数组 +for (int i = 0; i < n; ++i) + st[i][0] = arr[i]; +``` + +#### 2. 动态规划预处理 + +使用动态规划来填充 `st` 数组。`st[i][j]` 表示从 `i` 开始长度为 `2^j` 的区间的最值。可以通过两个长度为 `2^(j-1)` 的子区间的最值来得到。 + +```cpp +for (int j = 1; j <= K; ++j) { + for (int i = 0; (i + (1 << j)) <= n; ++i) { + st[i][j] = min(st[i][j - 1], st[i + (1 << (j - 1))][j - 1]); + } +} +``` + +#### 3. 查询阶段 + +对于任意区间 `[L, R]`,可以利用预处理的结果快速查询。将 `[L, R]` 拆分为两个可以重叠的长度为 `2^k` 的区间,其中 `k` 为 `log[R - L + 1]`。 + +```cpp +int query(int L, int R) { + int j = log[R - L + 1]; + return min(st[L][j], st[R - (1 << j) + 1][j]); +} +``` + +### 代码示例 + +以下是完整的 C++ 代码示例: + +```cpp +#include +#include +#include +using namespace std; + +class SparseTable { +private: + vector> st; + vector log; + int n; + +public: + SparseTable(const vector& arr) { + n = arr.size(); + int K = log2(n) + 1; + st.assign(n, vector(K)); + log.assign(n + 1, 0); + + // 初始化 log 数组 + log[1] = 0; + for (int i = 2; i <= n; ++i) { + log[i] = log[i / 2] + 1; + } + + // 初始化 st 数组 + for (int i = 0; i < n; ++i) { + st[i][0] = arr[i]; + } + + // 动态规划填充 st 数组 + for (int j = 1; j <= K; ++j) { + for (int i = 0; (i + (1 << j)) <= n; ++i) { + st[i][j] = min(st[i][j - 1], st[i + (1 << (j - 1))][j - 1]); + } + } + } + + int query(int L, int R) { + int j = log[R - L + 1]; + return min(st[L][j], st[R - (1 << j) + 1][j]); + } +}; + +int main() { + vector arr = {1, 3, 5, 7, 9, 11}; + SparseTable sparseTable(arr); + cout << "Min value in range [1, 4]: " << sparseTable.query(1, 4) << endl; + return 0; +} +``` + +### 总结 +- 稀疏表的预处理时间为 \(O(n \log n)\),查询时间为 \(O(1)\),适合静态数据场景。 +- 预处理过程通过动态规划构建不同区间长度的最值,查询通过预处理结果快速得到答案。 + +这个方法对静态数组的多次区间查询效率非常高,但不适用于需要频繁更新的场景。 + +好的,下面我们详细讲解稀疏表的动态规划预处理过程。 + +### 动态规划预处理详细解释 + +稀疏表的核心在于利用动态规划的思想预处理数组中的区间最值,从而实现高效的查询。预处理过程中,我们会构建一个二维数组 `st`,其中 `st[i][j]` 表示从索引 `i` 开始长度为 `2^j` 的区间的最小值(当然你也可以用来处理最大值问题,只需要把 `min` 改成 `max`)。 + +#### 1. 初始化 + +我们需要初始化一个二维数组 `st`,并且初始化一个 `log` 数组,`log[i]` 表示 `i` 的对数值(向下取整)。 + +```cpp +int n = arr.size(); +int K = log2(n) + 1; // K 是最大区间长度的对数 +vector> st(n, vector(K)); +vector log(n + 1); + +// 初始化 log 数组 +log[1] = 0; +for (int i = 2; i <= n; ++i) + log[i] = log[i / 2] + 1; +``` + +在这个过程中,`log[i]` 记录的是 `i` 的对数值(向下取整)。例如,`log[16]` 是 4,因为 \( 2^4 = 16 \),而 `log[10]` 是 3,因为 \( 2^3 = 8 \)。 + +#### 2. 初始化稀疏表的第一列 + +`st[i][0]` 表示从索引 `i` 开始长度为 \(2^0 = 1\) 的区间的最小值,也就是元素本身。 + +```cpp +for (int i = 0; i < n; ++i) + st[i][0] = arr[i]; +``` + +#### 3. 动态规划填充稀疏表 + +为了填充 `st` 数组,我们使用动态规划的方法。对于每个 `j > 0`,我们用 `st[i][j-1]` 和 `st[i + (1 << (j-1))][j-1]` 来计算 `st[i][j]`。 + +```cpp +for (int j = 1; j <= K; ++j) { + for (int i = 0; (i + (1 << j)) <= n; ++i) { + st[i][j] = min(st[i][j - 1], st[i + (1 << (j - 1))][j - 1]); + } +} +``` + +这里的 `st[i][j]` 表示从 `i` 开始长度为 `2^j` 的区间的最小值。我们把长度为 `2^j` 的区间分成两半,每一半长度为 `2^(j-1)`,然后分别计算这两半的最小值,并取这两个最小值中的较小者。 + +#### 例子说明 + +假设我们有数组 `arr = [1, 3, 5, 7, 9, 11]`。 + +1. 初始化 `st` 的第一列: + + ``` + st[i][0] = arr[i] + ``` + + 得到: + + ``` + st[0][0] = 1 + st[1][0] = 3 + st[2][0] = 5 + st[3][0] = 7 + st[4][0] = 9 + st[5][0] = 11 + ``` + +2. 计算 `st[i][1]`,区间长度为 `2^1 = 2`: + + ``` + st[0][1] = min(st[0][0], st[1][0]) = min(1, 3) = 1 + st[1][1] = min(st[1][0], st[2][0]) = min(3, 5) = 3 + st[2][1] = min(st[2][0], st[3][0]) = min(5, 7) = 5 + st[3][1] = min(st[3][0], st[4][0]) = min(7, 9) = 7 + st[4][1] = min(st[4][0], st[5][0]) = min(9, 11) = 9 + ``` + +3. 计算 `st[i][2]`,区间长度为 `2^2 = 4`: + + ``` + st[0][2] = min(st[0][1], st[2][1]) = min(1, 5) = 1 + st[1][2] = min(st[1][1], st[3][1]) = min(3, 7) = 3 + st[2][2] = min(st[2][1], st[4][1]) = min(5, 9) = 5 + ``` + + 注意:此时 `i + 2^2` 必须小于等于 `n`,所以我们只计算到 `st[2][2]`。 + +### 查询过程 + +预处理完成后,我们可以在 \(O(1)\) 时间内完成区间最值查询。具体方法是,对于查询区间 `[L, R]`,找到 `k = log[R - L + 1]`,然后查询 `st[L][k]` 和 `st[R - (1 << k) + 1][k]` 的最小值。 + +```cpp +int query(int L, int R) { + int j = log[R - L + 1]; + return min(st[L][j], st[R - (1 << j) + 1][j]); +} +``` + +### 完整代码示例 + +```cpp +#include +#include +#include +using namespace std; + +class SparseTable { +private: + vector> st; + vector log; + int n; + +public: + SparseTable(const vector& arr) { + n = arr.size(); + int K = log2(n) + 1; + st.assign(n, vector(K)); + log.assign(n + 1, 0); + + // 初始化 log 数组 + log[1] = 0; + for (int i = 2; i <= n; ++i) { + log[i] = log[i / 2] + 1; + } + + // 初始化 st 数组 + for (int i = 0; i < n; ++i) { + st[i][0] = arr[i]; + } + + // 动态规划填充 st 数组 + for (int j = 1; j <= K; ++j) { + for (int i = 0; (i + (1 << j)) <= n; ++i) { + st[i][j] = min(st[i][j - 1], st[i + (1 << (j - 1))][j - 1]); + } + } + } + + int query(int L, int R) { + int j = log[R - L + 1]; + return min(st[L][j], st[R - (1 << j) + 1][j]); + } +}; + +int main() { + vector arr = {1, 3, 5, 7, 9, 11}; + SparseTable sparseTable(arr); + cout << "Min value in range [1, 4]: " << sparseTable.query(1, 4) << endl; + return 0; +} +``` + +这个例子中,`SparseTable` 类在构造函数中完成了预处理,并且提供了 `query` 方法用于快速查询任意区间的最小值。通过这种预处理,我们可以在 \(O(1)\) 时间内完成区间查询,非常高效。 \ No newline at end of file diff --git a/day5/RMQ_by_chat.pdf b/day5/RMQ_by_chat.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2130f5619c748549dd75e4a1e7924db429048f5f GIT binary patch literal 574525 zcmaHyV~l9QvZ%+lZQHhO+qONk$F@CtY}>YN+xE=e=f}&*OKxt`-D|DNm+GooNq48x zMXDeoM$1UY0!6xbdA<+DNWef~Z)630)C_K%pe7sBh${Z){=cLJ37L>||=_ zV(&yiA#CAfZ)@pl`%k9eWN+eXZ2C{AY;9>s|4$)-Eh7{A|D!GK?L-V+{%I27Vq#!o zVc=k3XJBP!X5^%1U?Bh3{&BMQCjU3W#|LF^$7qj_isfekuy@@Fly^N`yxr+q>3o`>ND<2<$v&%o< 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