From d7910527f55320df292a86a1338385cf3f6fcacb Mon Sep 17 00:00:00 2001 From: Zengtudor Date: Fri, 9 Aug 2024 12:02:40 +0800 Subject: [PATCH] learn segment tree --- day7/P5431/1.in | 2 + day7/P5431/1.out | 1 + day7/P5431/P5431.cpp | 89 +++++++++ day7/P5431/chat.md | 84 +++++++++ day7/SegmentTree/SegmentTree.cpp | 8 + day7/SegmentTree/SegmentTree.md | 305 +++++++++++++++++++++++++++++++ day7/SegmentTree/SegmentTree.pdf | Bin 0 -> 632856 bytes xmake.lua | 17 +- 8 files changed, 504 insertions(+), 2 deletions(-) create mode 100644 day7/P5431/1.in create mode 100644 day7/P5431/1.out create mode 100644 day7/P5431/P5431.cpp create mode 100644 day7/P5431/chat.md create mode 100644 day7/SegmentTree/SegmentTree.cpp create mode 100644 day7/SegmentTree/SegmentTree.md create mode 100644 day7/SegmentTree/SegmentTree.pdf diff --git a/day7/P5431/1.in b/day7/P5431/1.in new file mode 100644 index 0000000..f31c00a --- /dev/null +++ b/day7/P5431/1.in @@ -0,0 +1,2 @@ +6 233 42 +1 4 2 8 5 7 \ No newline at end of file diff --git a/day7/P5431/1.out b/day7/P5431/1.out new file mode 100644 index 0000000..a46c9d2 --- /dev/null +++ b/day7/P5431/1.out @@ -0,0 +1 @@ +91 \ No newline at end of file diff --git a/day7/P5431/P5431.cpp b/day7/P5431/P5431.cpp new file mode 100644 index 0000000..2f89661 --- /dev/null +++ b/day7/P5431/P5431.cpp @@ -0,0 +1,89 @@ +//TLE 30 + +#include +using namespace std; +#define int long long + +const int MAX_N =5e6+5; + +int binExp(int b,int e,int p){ + int r=1; + while(e>0){ + if(e%2==1)r=(r*b)%p; + b=(b*b)%p; + e=e>>1; + } + return r; +} + +int inverse(int b,int p){ + return binExp(b, p-2,p); +} + +#ifdef OITEST +#endif +#ifndef OITEST +#endif + +#ifndef OITEST +int read(){ + int x=0,w=1; + char ch=0; + while(!isdigit(ch)){ + if(ch=='-')w=-1; + ch=getchar(); + } + while(isdigit(ch)){ + x=x*10+(ch-'0'); + ch=getchar(); + } + return x*w; +} +#endif + +signed main(signed argc,char *argv[]){ + ios::sync_with_stdio(false); + cin.tie(0); + #ifdef OITEST + assert(argc==3); + cout<>n>>p>>k; + #endif + + #ifndef OITEST + n=read(); + p=read(); + k=read(); + #endif + int ans=0; + int kp=k; + for(int i=1;i<=n;i++){ + int a; + #ifdef OITEST + cin>>a; + #endif + #ifndef OITEST + a=read(); + #endif + ans=((int)(kp*inverse(a, p))%p+ans)%p; + kp=(kp*k)%p; + } + cout<>my; + afile>>anser; + assert(my==anser); + #endif +} \ No newline at end of file diff --git a/day7/P5431/chat.md b/day7/P5431/chat.md new file mode 100644 index 0000000..f3d15c9 --- /dev/null +++ b/day7/P5431/chat.md @@ -0,0 +1,84 @@ +以下是上述问题的 C++ 实现代码,使用了快速幂计算逆元和高效的 IO 处理方式,以应对可能的大规模数据输入。 + +### C++ 实现 + +```cpp +#include +using namespace std; + +// 快速幂计算 (base^exp) % mod +long long mod_exp(long long base, long long exp, long long mod) { + long long result = 1; + while (exp > 0) { + if (exp % 2 == 1) { + result = (result * base) % mod; + } + base = (base * base) % mod; + exp /= 2; + } + return result; +} + +// 模逆元计算: a^(p-2) % p +long long mod_inverse(long long a, long long p) { + return mod_exp(a, p - 2, p); +} + +int main() { + ios::sync_with_stdio(false); // 提高cin/cout的效率 + cin.tie(0); // 取消cin与cout的关联 + + int n; + long long p, k; + cin >> n >> p >> k; + + long long result = 0; + long long k_power = 1; // k^i + + for (int i = 0; i < n; ++i) { + long long a_i; + cin >> a_i; + + long long inv_a = mod_inverse(a_i, p); + long long term = (k_power * inv_a) % p; + result = (result + term) % p; + + // 更新k_power + k_power = (k_power * k) % p; + } + + cout << result << endl; + return 0; +} +``` + +### 代码解释 + +1. **快速幂函数 `mod_exp`**:通过二进制分解指数,实现快速计算幂次取模。 +2. **模逆元函数 `mod_inverse`**:利用费马小定理,通过快速幂计算模逆元。 +3. **高效的输入输出**: + - `ios::sync_with_stdio(false)` 关闭 C++ 的 IO 与 C 的 IO 之间的同步,提高 IO 效率。 + - `cin.tie(0)` 解除 `cin` 与 `cout` 的关联,避免在每次 `cin` 之前执行 `cout`,进一步提升效率。 +4. **主函数逻辑**: + - 读取输入的 `n`, `p`, `k`,以及数组 `a_i`。 + - 逐步计算每个 $k^i \times a_i^{-1} \mod p$ 并累加到结果中。 + - 最终输出累加结果。 + +### 样例测试 + +使用样例输入: + +``` +6 233 42 +1 4 2 8 5 7 +``` + +程序将输出: + +``` +91 +``` + +### 性能优化 + +在 `n` 非常大的情况下,使用上述 C++ 实现,可以确保在合理时间内完成计算。此实现充分利用了快速幂和费马小定理的性质,以及 C++ 的高效输入输出特性。 \ No newline at end of file diff --git a/day7/SegmentTree/SegmentTree.cpp b/day7/SegmentTree/SegmentTree.cpp new file mode 100644 index 0000000..15fe60d --- /dev/null +++ b/day7/SegmentTree/SegmentTree.cpp @@ -0,0 +1,8 @@ +#include +using namespace std; + + + +int main(){ + +} \ No newline at end of file diff --git a/day7/SegmentTree/SegmentTree.md b/day7/SegmentTree/SegmentTree.md new file mode 100644 index 0000000..6fbb1a3 --- /dev/null +++ b/day7/SegmentTree/SegmentTree.md @@ -0,0 +1,305 @@ +线段树(Segment Tree)是一种用于处理区间查询与更新问题的数据结构。它在处理动态数组或序列时特别有效,能够在对区间进行查询(如求和、最小值、最大值)以及更新(如修改数组中的某个元素)时,提供高效的时间复杂度。 + +## 线段树的特点 + +1. **时间复杂度**: + - **构建**: 构建线段树的时间复杂度是 \(O(n)\),其中 \(n\) 是数组的长度。 + - **查询**: 线段树可以在 \(O(\log n)\) 时间内完成区间查询操作。 + - **更新**: 单点更新也可以在 \(O(\log n)\) 时间内完成。 + +2. **存储结构**: + - 线段树通常使用一个基于数组的完全二叉树来存储,每个节点表示一个区间的某种属性(如和、最大值、最小值等)。 + - 如果原数组的大小是 \(n\),则线段树的大小约为 \(4n\),但实际应用中,大部分实现都使用 \(2n\) 的大小。 + +## 线段树的构建 + +假设我们有一个数组 `arr`,它的长度为 `n`,我们需要构建一棵线段树来支持区间查询和更新。 + +### 基本思想 + +- 线段树的每个叶子节点表示数组中的一个元素。 +- 非叶子节点表示对应区间中的某个属性(如区间和、区间最小值等)。 + +### 构建过程 + +1. **递归分治**: + - 将区间 `[l, r]` 分成两个子区间 `[l, mid]` 和 `[mid+1, r]`。 + - 递归构建左右子树,直到区间长度为 1(即叶子节点)。 + +2. **合并**: + - 在返回的过程中,将左右子区间的值合并为当前区间的值。 + +### 示例 + +假设我们要构建一个表示区间和的线段树,对于数组 `arr = [1, 3, 5, 7, 9, 11]`: + +1. 构建区间 `[0, 5]` 的线段树,将其分为 `[0, 2]` 和 `[3, 5]` 两部分。 +2. 继续递归分治,直到处理单个元素。 +3. 然后,从底向上合并,构建完整的线段树。 + +## 线段树的查询与更新 + +### 查询 + +线段树的查询过程也是基于递归的。对于查询区间 `[l, r]`: + +1. 如果当前区间与查询区间没有交集,返回一个默认值(如 0 对于求和)。 +2. 如果当前区间完全包含在查询区间中,返回当前节点的值。 +3. 否则,将当前区间分成两个子区间,递归查询,并将结果合并。 + +### 更新 + +更新过程类似: + +1. 找到对应叶子节点,更新值。 +2. 向上递归更新父节点,直到根节点。 + +## 总结 + +线段树是处理区间操作的强大工具,特别适用于需要频繁查询和更新的大规模数据场景。它的高效性来自于其树形结构,使得每次查询和更新都只需要访问 \(O(\log n)\) 个节点。 + +在实际应用中,线段树可以处理很多种类的区间查询问题,包括区间和、区间最小值、区间最大值等,并且可以通过修改合并操作来适应不同的需求。 + +下面是一段使用C++实现线段树来处理区间最小值查询问题的代码。我们将逐行解释代码的原理和操作。 + +```cpp +#include +#include +#include // 用于 std::min 函数 + +using namespace std; + +class SegmentTree { +private: + vector tree; + int n; + + // 构建线段树,初始化区间 [start, end] 的节点 idx + void buildTree(const vector& arr, int start, int end, int idx) { + if (start == end) { + // 叶子节点表示单个元素 + tree[idx] = arr[start]; + } else { + int mid = start + (end - start) / 2; + // 递归构建左右子树 + buildTree(arr, start, mid, 2 * idx + 1); + buildTree(arr, mid + 1, end, 2 * idx + 2); + // 当前节点存储左右子树的最小值 + tree[idx] = min(tree[2 * idx + 1], tree[2 * idx + 2]); + } + } + + // 递归查询区间 [l, r] 的最小值 + int queryMin(int start, int end, int l, int r, int idx) { + if (l <= start && r >= end) { + // 完全覆盖,直接返回当前区间的最小值 + return tree[idx]; + } + if (end < l || start > r) { + // 没有交集,返回一个无穷大的值 + return INT_MAX; + } + // 部分覆盖,查询左右子区间 + int mid = start + (end - start) / 2; + return min(queryMin(start, mid, l, r, 2 * idx + 1), + queryMin(mid + 1, end, l, r, 2 * idx + 2)); + } + + // 递归更新节点,修改位置 pos 的值为 newVal + void updateValue(int start, int end, int pos, int newVal, int idx) { + if (start == end) { + // 更新叶子节点 + tree[idx] = newVal; + } else { + int mid = start + (end - start) / 2; + if (pos <= mid) { + // 更新左子树 + updateValue(start, mid, pos, newVal, 2 * idx + 1); + } else { + // 更新右子树 + updateValue(mid + 1, end, pos, newVal, 2 * idx + 2); + } + // 更新当前节点为左右子树的最小值 + tree[idx] = min(tree[2 * idx + 1], tree[2 * idx + 2]); + } + } + +public: + // 构造函数 + SegmentTree(const vector& arr) { + n = arr.size(); + tree.resize(4 * n); // 预留足够的空间 + buildTree(arr, 0, n - 1, 0); + } + + // 对外暴露的查询接口,查询区间 [l, r] 的最小值 + int queryMin(int l, int r) { + return queryMin(0, n - 1, l, r, 0); + } + + // 对外暴露的更新接口,将位置 pos 的值更新为 newVal + void updateValue(int pos, int newVal) { + updateValue(0, n - 1, pos, newVal, 0); + } +}; + +int main() { + vector arr = {2, 5, 1, 4, 9, 3}; + SegmentTree segTree(arr); + + cout << "Initial minimum in range [1, 4]: " << segTree.queryMin(1, 4) << endl; // 输出 1 + + segTree.updateValue(3, 0); // 更新 arr[3] 为 0 + cout << "Minimum in range [1, 4] after update: " << segTree.queryMin(1, 4) << endl; // 输出 0 + + return 0; +} +``` + +## 代码解释 + +### 1. 导入库与定义 + +```cpp +#include +#include +#include // 用于 std::min 函数 +``` +- 导入标准库,用于输入输出、容器操作以及求最小值函数 `std::min`。 + +### 2. 线段树类定义 + +```cpp +class SegmentTree { +private: + vector tree; + int n; +``` +- 定义一个 `SegmentTree` 类,私有成员包括存储线段树的 `tree` 数组和原始数组大小 `n`。 + +### 3. 线段树的构建 + +```cpp +void buildTree(const vector& arr, int start, int end, int idx) { + if (start == end) { + tree[idx] = arr[start]; + } else { + int mid = start + (end - start) / 2; + buildTree(arr, start, mid, 2 * idx + 1); + buildTree(arr, mid + 1, end, 2 * idx + 2); + tree[idx] = min(tree[2 * idx + 1], tree[2 * idx + 2]); + } +} +``` +- `buildTree` 函数递归地构建线段树。 +- 如果区间 `[start, end]` 只包含一个元素(`start == end`),则将该元素存入树的对应位置。 +- 如果区间大于1,则将区间分为两部分,递归构建左、右子树,并将两个子区间的最小值存入当前节点。 + +### 4. 区间最小值查询 + +```cpp +int queryMin(int start, int end, int l, int r, int idx) { + if (l <= start && r >= end) { + return tree[idx]; + } + if (end < l || start > r) { + return INT_MAX; + } + int mid = start + (end - start) / 2; + return min(queryMin(start, mid, l, r, 2 * idx + 1), + queryMin(mid + 1, end, l, r, 2 * idx + 2)); +} +``` +- `queryMin` 函数在 `[l, r]` 区间查询最小值。 +- 如果当前区间完全在查询区间内,直接返回当前节点的值。 +- 如果没有交集,返回 `INT_MAX` 作为无效值。 +- 如果部分重叠,递归查询左、右子区间,合并结果。 + +### 5. 更新操作 + +```cpp +void updateValue(int start, int end, int pos, int newVal, int idx) { + if (start == end) { + tree[idx] = newVal; + } else { + int mid = start + (end - start) / 2; + if (pos <= mid) { + updateValue(start, mid, pos, newVal, 2 * idx + 1); + } else { + updateValue(mid + 1, end, pos, newVal, 2 * idx + 2); + } + tree[idx] = min(tree[2 * idx + 1], tree[2 * idx + 2]); + } +} +``` +- `updateValue` 函数递归更新某个位置的值。 +- 通过递归找到需要更新的叶子节点并修改其值。 +- 更新完成后,向上递归更新其父节点,确保父节点值的正确性。 + +### 6. 构造函数 + +```cpp +SegmentTree(const vector& arr) { + n = arr.size(); + tree.resize(4 * n); + buildTree(arr, 0, n - 1, 0); +} +``` +- 构造函数初始化线段树。 +- `n` 为数组大小,`tree` 的大小设置为 `4 * n` 以确保足够的空间构建完全二叉树。 +- 调用 `buildTree` 函数从原始数组构建线段树。 + +### 7. 对外接口 + +```cpp +int queryMin(int l, int r) { + return queryMin(0, n - 1, l, r, 0); +} + +void updateValue(int pos, int newVal) { + updateValue(0, n - 1, pos, newVal, 0); +} +``` +- `queryMin` 和 `updateValue` 是对外暴露的接口,用于查询最小值和更新数组值。 + +### 8. 测试代码 + +```cpp +int main() { + vector arr = {2, 5, 1, 4, 9, 3}; + SegmentTree segTree(arr); + + cout << "Initial minimum in range [1, 4]: " << segTree.queryMin(1, 4) << endl; // 输出 1 + + segTree.updateValue(3, 0); // 更新 arr[3] 为 0 + cout << "Minimum in range [1, + +4] after update: " << segTree.queryMin(1, 4) << endl; // 输出 0 + + return 0; +} +``` +- 创建一个数组,并构建相应的线段树。 +- 查询区间 `[1, 4]` 的最小值,初始查询得到 1。 +- 更新数组中的一个值(将 `arr[3]` 从 4 更新为 0),再次查询该区间,结果变为 0。 +在这段代码中,`idx` 是一个参数,用于表示线段树中某个节点的索引。它指向线段树中某个节点的位置,用来追踪当前递归所处理的节点在 `tree` 数组中的位置。 + +### `idx` 的作用 + +- **在构建线段树时**: + - `idx` 用于标识当前节点在 `tree` 数组中的位置。 + - 在递归过程中,通过计算左子节点和右子节点的索引 (`2 * idx + 1` 和 `2 * idx + 2`),来构建整棵树。 + +- **在查询和更新操作中**: + - `idx` 也是用来指示当前节点的位置。通过递归传递 `idx` 参数,程序能够定位到需要操作的具体节点。 + - 对于查询操作,`idx` 表示当前递归所在的节点;对于更新操作,`idx` 表示需要更新的节点。 + +### 具体示例 + +假设有一个包含 6 个元素的数组 `arr = {2, 5, 1, 4, 9, 3}`,在构建线段树时: + +- `idx = 0` 表示根节点,负责管理整个数组 `[0, 5]`。 +- 递归过程中,`idx = 1` 可能对应左子树,管理数组的前半部分 `[0, 2]`,而 `idx = 2` 可能对应右子树,管理数组的后半部分 `[3, 5]`。 +- 继续递归,`idx = 3` 可能表示 `[0, 1]`,`idx = 4` 表示 `[2, 2]` 等等。 + +因此,`idx` 就是线段树中节点的唯一标识符,通过它可以在 `tree` 数组中找到相应节点的位置。 \ No newline at end of file diff --git a/day7/SegmentTree/SegmentTree.pdf b/day7/SegmentTree/SegmentTree.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e869c0600979b207639bc4b6164c5d06f2a3493b GIT binary patch literal 632856 zcmaI7V{9&5&@NotHg|2?wtLs+UEAL6uDfg7wr$(iu5GvHd4GI)^Cc%ISy?}>T$weq zk~Nu`HAA5+F3G^c$PP!bcyYE5$3ntP;$Q-VZJP z5_2{;c6D$jp%$}pcCfQ{v-=NIc6KmxGd2GYRI{-*X8KPci5&|o*Z;@1cCZ&WcKuJ1 zI4>(RD?2kcGY>O6Gb=MY9Wyi4f9pS#auY#t=BrI@Dq6#ET ziVn_p#w{$~J_r0sv2ikq7{n3=;d$(!3-x>}L2bFp*q z2nv$8xc+Cm9h_%YR`+@&!*~M4>$PEhF8+q9hHKcbeFRh)Aq@86&~K318ib51M2)YX 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z`jWpt$so#2>KP@xxEI9rLnW+evr4c1i?MaUsMb#^l@wzc(x)KAnrOffsQ7nMtMC1b%VlC-n*nXEq+kMCMr>htQ@#Gb(n_8YqrK z_)j>S@c0mY2Nf{-<_48%{;*?2MTD823@`UWUl!lDYaAjKly3mAukR~Uz`7}36h z*F)dn0gS%Qi!vj8u}GK?SupXe!Ls7 zez&uyc{`8yaPu~4*RJir@&0b6s7%+Gv>o8(HO&;e0Sjn4z@%+YPj@dt=6j^1lFW3KnPp literal 0 HcmV?d00001 diff --git a/xmake.lua b/xmake.lua index cefecc2..acae5a4 100644 --- a/xmake.lua +++ b/xmake.lua @@ -32,7 +32,7 @@ target("fperfect") set_rundir("./day5/perfect/") for val=1,4 do local strval = tostring(val) - add_tests(strval,{files="./day5/perfect/fix.cpp",defines="OITEST",runargs=strval,run_timeout=2000}) + add_tests(strval,{files="./day5/perfect/fix.cpp",defines="OITEST",runargs=strval,run_timeout=1000}) end target("pre88") @@ -46,4 +46,17 @@ target("inverse") for v=1,2 do local strname = tostring(v) add_tests(strname,{files="./day7/inverse/*.cpp",defines="OITEST",runargs=strname,run_timeout=1000}) - end \ No newline at end of file + end + +target("P5431") + set_rundir("./day7/P5431") + add_files("./day7/P5431/*.cpp") + for v=1,1 do + local s=tostring(v) + add_tests(s,{files="./day7/P5431/*.cpp",defines="OITEST",runargs={s..".in",s..".out"},run_timeout=1000}) + end + +target("segtree") + set_rundir("./day7/SegmentTree") + add_files("./day7/SegmentTree/*.cpp") + \ No newline at end of file