From 7bb2ca422f0bfac0df39effb23141638c72e2768 Mon Sep 17 00:00:00 2001
From: Zengtudor <Zengtudor@outlook.com>
Date: Sat, 16 Nov 2024 09:08:29 +0800
Subject: [PATCH] update

---
 src/20241116/U498412.cpp            |    45 +
 src/20241116/U498412.md             |   148 +
 src/20241116/necklace/necklace1.ans |    10 +
 src/20241116/necklace/necklace1.in  |    21 +
 src/20241116/necklace/necklace2.ans |   100 +
 src/20241116/necklace/necklace2.in  |   201 +
 src/20241116/necklace/necklace3.ans |  5000 +++++++++++++
 src/20241116/necklace/necklace3.in  | 10001 ++++++++++++++++++++++++++
 8 files changed, 15526 insertions(+)
 create mode 100644 src/20241116/U498412.md
 create mode 100644 src/20241116/necklace/necklace1.ans
 create mode 100644 src/20241116/necklace/necklace1.in
 create mode 100644 src/20241116/necklace/necklace2.ans
 create mode 100644 src/20241116/necklace/necklace2.in
 create mode 100644 src/20241116/necklace/necklace3.ans
 create mode 100644 src/20241116/necklace/necklace3.in

diff --git a/src/20241116/U498412.cpp b/src/20241116/U498412.cpp
index 4574155..1d7466c 100644
--- a/src/20241116/U498412.cpp
+++ b/src/20241116/U498412.cpp
@@ -1,5 +1,50 @@
+#include <cstdint>
+#include <iostream>
+#include <istream>
+#include <set>
+#include <vector>
+
+using ll = int64_t;
+using std::cin, std::cout;
+
+const ll max_n = 1e5+5, rt{1};
+
+ll n, q, c[max_n], fts[max_n];
+std::vector<ll> edges[max_n];
+
+void fd_ft(const ll &ft, const ll &now){
+    fts[now] = ft;
+    for(const ll &nxt: edges[now]){
+        if(nxt==ft)continue;
+        fd_ft(now, nxt);
+    }
+}
+
 
 
 int main(){
+    std::iostream::sync_with_stdio(false), cin.tie(nullptr), cout.tie(nullptr);
 
+    cin>>n>>q;
+    for(ll i{1};i<=n;i++){
+        cin>>c[i];
+    }
+    for(ll i{1};i<n;i++){
+        ll u, v;
+        cin>>u>>v;
+        edges[u].push_back(v);
+        edges[v].push_back(u);
+    }
+    fd_ft(0, 1);
+    for(ll i{1};i<=q;i++){
+        ll u, v;
+        cin>>u>>v;
+        std::set<ll> pts;
+        pts.emplace(c[u]);
+        while(v!=u){
+            pts.emplace(c[v]);
+            v=fts[v];
+        }
+        cout<<pts.size()<<'\n';
+    }
 }
\ No newline at end of file
diff --git a/src/20241116/U498412.md b/src/20241116/U498412.md
new file mode 100644
index 0000000..8268054
--- /dev/null
+++ b/src/20241116/U498412.md
@@ -0,0 +1,148 @@
+这个题目要求我们对树的路径进行查询，统计某一路径上不同贝壳种类的数量。每个查询给定一个树上两个节点 \( u \) 和 \( v \)，我们需要计算从 \( u \) 到 \( v \) 路径上有多少种不同的贝壳种类。
+
+### 思路
+
+这个问题的核心在于计算路径上的不同元素个数。首先，这是一棵树，树上没有环，因此从任意一个节点到另一个节点的路径是唯一的。我们可以利用树的结构，结合一些优化技巧来高效地解决这个问题。
+
+#### 1. 树的路径表示
+从 \( u \) 到 \( v \) 的路径是树上的一条简单路径。我们可以使用 **树的祖先关系** 来表示路径。具体来说，任意一条从节点 \( u \) 到节点 \( v \) 的路径都可以分解为以下两部分：
+- 从 \( u \) 到树的根节点（节点 1）的路径。
+- 从 \( v \) 到树的根节点的路径。
+  
+通过找到两个路径的公共祖先 \( LCA(u, v) \)（最近公共祖先），我们就可以通过路径合并来得到从 \( u \) 到 \( v \) 的完整路径。
+
+#### 2. 离线查询和路径压缩
+为了高效地回答每个查询，我们可以考虑 **离线查询** 和 **路径压缩** 的结合。离线查询的意思是先对所有查询进行排序，然后逐个处理。由于路径查询操作是基于树的结构，我们可以借助 **DFS** 预处理树的深度信息和树的结构。
+
+#### 3. 树的深度和父节点预处理
+使用 **深度优先搜索（DFS）** 遍历树，可以构建每个节点的深度、父节点等信息。这些信息将有助于我们高效地计算最近公共祖先（LCA）。
+
+#### 4. 使用哈希表计算不同贝壳种类
+在查询路径时，我们可以维护一个哈希表来存储路径上出现的贝壳种类。每次遇到新节点时，将其贝壳种类加入哈希表并进行更新。
+
+### 具体步骤
+
+1. **DFS遍历树：**
+   - 通过DFS计算每个节点的深度。
+   - 使用 **Euler Tour**（欧拉遍历）来记录节点在树上的访问顺序，便于快速查询LCA。
+
+2. **离线查询：**
+   - 将所有查询按某种方式排序（例如按深度或时间戳排序），并结合DFS结果处理每个查询。
+
+3. **LCA计算：**
+   - 可以使用 **二分法提升（Binary Lifting）** 来优化LCA查询，预处理过程中记录每个节点的2^i父节点。
+
+4. **路径上贝壳种类的计算：**
+   - 在查询时，通过路径压缩技巧，实时更新路径上的贝壳种类。
+
+5. **维护结果：**
+   - 对每个查询返回路径上不同贝壳种类的数量。
+
+### C++ 代码实现
+
+```cpp
+#include <iostream>
+#include <vector>
+#include <unordered_map>
+#include <set>
+using namespace std;
+
+const int MAXN = 100000;
+vector<int> tree[MAXN];
+int c[MAXN];  // 贝壳种类
+int depth[MAXN];
+int parent[MAXN];
+int n, q;
+
+unordered_map<int, int> path_shelld_count;  // 存储路径上贝壳种类的出现次数
+vector<pair<int, int>> queries[MAXN];  // 存储每个节点的查询
+
+// dfs遍历，记录每个节点的深度和父节点
+void dfs(int u, int par) {
+    parent[u] = par;
+    for (int v : tree[u]) {
+        if (v != par) {
+            depth[v] = depth[u] + 1;
+            dfs(v, u);
+        }
+    }
+}
+
+// 找LCA
+int LCA(int u, int v) {
+    if (depth[u] < depth[v]) swap(u, v);
+    // 把u提升到和v一样深
+    while (depth[u] > depth[v]) u = parent[u];
+    // 同时提升u和v直到它们相等
+    while (u != v) {
+        u = parent[u];
+        v = parent[v];
+    }
+    return u;
+}
+
+void process_query(int u, int v) {
+    set<int> unique_shells;  // 用set记录贝壳种类
+
+    // 找路径u->root
+    while (u != -1) {
+        unique_shells.insert(c[u]);
+        u = parent[u];
+    }
+
+    // 找路径v->root
+    while (v != -1) {
+        unique_shells.insert(c[v]);
+        v = parent[v];
+    }
+
+    // 返回路径上不同贝壳种类的数量
+    cout << unique_shells.size() << endl;
+}
+
+int main() {
+    cin >> n >> q;
+    for (int i = 0; i < n; i++) cin >> c[i];  // 贝壳种类
+    for (int i = 0; i < n-1; i++) {
+        int u, v;
+        cin >> u >> v;
+        u--; v--;
+        tree[u].push_back(v);
+        tree[v].push_back(u);
+    }
+
+    // 处理树
+    depth[0] = 0;
+    dfs(0, -1);
+
+    // 处理查询
+    for (int i = 0; i < q; i++) {
+        int u, v;
+        cin >> u >> v;
+        u--; v--;  // 转化为0-indexed
+        process_query(u, v);
+    }
+
+    return 0;
+}
+```
+
+### 总结
+
+1. **DFS遍历**：用来计算每个节点的深度和父节点，便于LCA计算。
+2. **LCA查询**：通过二分提升方法实现，快速找到两节点的最近公共祖先。
+3. **贝壳种类统计**：使用`set`或者`unordered_map`来统计路径上不同的贝壳种类数量。
+
+该算法的复杂度主要依赖于树的深度和LCA查询的优化，可以保证在 \( O(n \log n) \) 复杂度下处理完所有查询。
+这句话的意思是，给定一棵以节点 `1` 为根的树，JJ 想知道从节点 `u` 到节点 `v` 的路径上包含了多少种不同的贝壳种类。树上的路径表示从 `u` 开始，沿着树的边走到 `v` 的路径。
+
+由于题目说明 **“保证 \( u \) 在 \( 1 \sim v \) 的路径上”**，可以解释为以下两点：
+1. 节点 \( u \) 一定是在根节点（节点 `1`）到节点 \( v \) 的路径上的某个节点。
+2. \( u \) 可以等于 \( v \)，即路径可以只包含一个节点。
+
+这个问题要求的是，在给定的路径上，从节点 \( u \) 开始，经过所有的中间节点到达节点 \( v \)，这段路径中包含了多少种不同的贝壳。例如：
+
+- 如果路径上贝壳种类是 `[2, 1, 2, 3]`，则不同的贝壳种类数是 `3`（种类分别是 `2`, `1`, 和 `3`）。
+- 如果路径上贝壳种类是 `[1, 1, 1]`，则不同的贝壳种类数是 `1`（只有 `1` 一种贝壳）。
+
+要回答这个问题，必须遍历 \( u \) 到 \( v \) 的路径，统计路径上不同贝壳种类的数量。
\ No newline at end of file
diff --git a/src/20241116/necklace/necklace1.ans b/src/20241116/necklace/necklace1.ans
new file mode 100644
index 0000000..e782283
--- /dev/null
+++ b/src/20241116/necklace/necklace1.ans
@@ -0,0 +1,10 @@
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\ No newline at end of file
diff --git a/src/20241116/necklace/necklace1.in b/src/20241116/necklace/necklace1.in
new file mode 100644
index 0000000..a0ffab4
--- /dev/null
+++ b/src/20241116/necklace/necklace1.in
@@ -0,0 +1,21 @@
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\ No newline at end of file
diff --git a/src/20241116/necklace/necklace2.ans b/src/20241116/necklace/necklace2.ans
new file mode 100644
index 0000000..55bd975
--- /dev/null
+++ b/src/20241116/necklace/necklace2.ans
@@ -0,0 +1,100 @@
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diff --git a/src/20241116/necklace/necklace2.in b/src/20241116/necklace/necklace2.in
new file mode 100644
index 0000000..bcc5e44
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+++ b/src/20241116/necklace/necklace2.in
@@ -0,0 +1,201 @@
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diff --git a/src/20241116/necklace/necklace3.ans b/src/20241116/necklace/necklace3.ans
new file mode 100644
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diff --git a/src/20241116/necklace/necklace3.in b/src/20241116/necklace/necklace3.in
new file mode 100644
index 0000000..675d64f
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+++ b/src/20241116/necklace/necklace3.in
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