cartesian and its document.

Author: a1fredbao

documents/alfred/data_structure/cartesian.md

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+---
+title: Cartesian Tree Construction.
+documentation_of: ./src/alfred/data_structure/cartesian.hpp
+---
+
+## Description
+
+This module provides a function to build a Cartesian tree from a given vector in O(n) time. A Cartesian tree is a binary tree where each node's value is greater than the values of its children, and it is constructed based on the order of elements in the input vector.
+
+## Usage Example
+
+```cpp
+#include "cartesian.hpp"
+std::vector<int> a = {3, 1, 2};
+auto res = cartesian(a); // returns a vector of pairs representing ls and rs of each node.
+```
+
+## Main Features
+- Supports custom comparison functions for building Cartesian trees based on different criteria.
+- Uses monostack algorithm, time complexity $O(n)$
+- Does not modify the original data (pass by value)
+
+## Important Functions and Time Complexity
+
+| Function    | Description                                                           | Time Complexity |
+| ----------- | --------------------------------------------------------------------- | --------------- |
+| `cartesian` | Build a cartesian tree based on the given array and comparison method | $O(n)$          |
+

src/alfred/data_structure/cartesian.hpp

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+#ifndef AFDS_CARTESIAN
+#define AFDS_CARTESIAN
+
+#include <pair>
+#include <vector>
+
+// Returns pair of (ls, rs).
+template <class T, class Comp = std::less<T>>
+std::vector<std::pair<int, int>> cartesian(std::vector<T> a, Comp comp) {
+    std::vector<int> stk; // stack of right chain.
+    std::vector<std::pair<int, int>> ans(a.size());
+    for (size_t i = 0; i < a.size(); i++) {
+        int lst = -1;
+        while (!stk.empty() && comp(a[i], a[stk.back()])) {
+            lst = stk.back(), stk.pop_back();
+        }
+        if (!stk.empty()) ans[stk.back()].second = i;
+        if (lst != -1) ans[i].first = lst;
+        stk.push_back(i);
+    }
+    return ans;
+}
+
+#endif