Cartesian tree (#124)

* Add cartesian tree

* [auto-verifier] verify commit a5ff3e7

Co-authored-by: GitHub <noreply@github.com>

Author: hitonanode

.verify-helper/timestamps.remote.json

@@ -175,6 +175,7 @@
 "string/test/sa_count_keyword.test.cpp": "2021-01-02 01:50:58 +0900",
 "string/test/suffix_array.test.cpp": "2021-01-02 00:51:41 +0900",
 "string/test/z_algorithm.test.cpp": "2021-01-01 16:52:32 +0900",
+"tree/test/cartesian_tree.test.cpp": "2021-10-01 00:13:15 +0900",
 "tree/test/diameter.test.cpp": "2021-06-09 00:31:07 +0900",
 "tree/test/frequency_table_of_tree_distance.stress.test.cpp": "2021-09-07 01:07:11 +0900",
 "tree/test/frequency_table_of_tree_distance.test.cpp": "2021-09-04 01:18:16 +0900",

tree/cartesian_tree.hpp

@@ -0,0 +1,30 @@
+#pragma once
+#include <functional>
+#include <vector>
+
+// Cartesian tree
+// Complexity: O(n)
+// By default, the smaller node is nearer to the root
+// Return : -1 (root), parent_idx (otherwise)
+// Example: [1, 0, 2] => [1, -1, 1]
+// Verified: https://judge.yosupo.jp/problem/cartesian_tree
+template <class T, class Cmp = std::less<T>>
+std::vector<int> cartesian_tree(const std::vector<T> &X) {
+    const int n = X.size();
+    Cmp comp;
+    std::vector<int> st(n);
+    int sz = 0;
+
+    std::vector<int> par(n, -1);
+
+    for (int i = 0; i < n; ++i) {
+        while (sz >= 2 and comp(X[i], X[st[sz - 1]])) {
+            par[st[sz - 1]] = comp(X[i], X[st[sz - 2]]) ? st[sz - 2] : i;
+            --sz;
+        }
+        if (sz == 1 and comp(X[i], X[st[sz - 1]])) par[st[--sz]] = i;
+        st[sz++] = i;
+    }
+    for (; sz > 1; --sz) par[st[sz - 1]] = st[sz - 2];
+    return par;
+};

tree/cartesian_tree.md

@@ -0,0 +1,20 @@
+---
+title: Cartesian tree
+documentation_of: ./cartesian_tree.hpp
+---
+
+比較可能な要素の列に対し，Cartesian tree を計算．各要素の親となる要素の番号を持った長さ $N$ の `std::vector<int>` を返す．$O(N)$．デフォルトでは小さい要素がより根に近くなるが，テンプレート引数に `std::greater<T>` を与えてやることで逆転可能．
+
+## 使用方法
+
+```cpp
+std::vector<int> A(N);
+for (auto &x : Ainv) x = -x;
+auto ret = cartesian_tree(A);
+auto ret2 = cartesian_tree<int, std::greater<int>>(Ainv);
+```
+
+## 問題例
+
+- [Library Checker: Cartesian Tree](https://judge.yosupo.jp/problem/cartesian_tree)
+- [Codeforces Round #745 (Div. 1) D. Subsequence](https://codeforces.com/contest/1580/problem/D)

tree/test/cartesian_tree.test.cpp

@@ -0,0 +1,19 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/cartesian_tree"
+#include "../cartesian_tree.hpp"
+#include <cassert>
+#include <iostream>
+using namespace std;
+
+int main() {
+    cin.tie(nullptr), ios::sync_with_stdio(false);
+    int N;
+    cin >> N;
+    std::vector<int> A(N);
+    for (auto &x : A) cin >> x;
+    auto Ainv = A;
+    for (auto &x : Ainv) x = -x;
+    auto ret = cartesian_tree(A);
+    auto ret2 = cartesian_tree<int, std::greater<int>>(Ainv);
+    assert(ret == ret2);
+    for (int i = 0; i < N; ++i) cout << (ret[i] < 0 ? i : ret[i]) << (i + 1 == N ? '\n' : ' ');
+}