add monotone_minima, online_offline_dp

.verify-helper/timestamps.remote.json

@@ -44,6 +44,8 @@
 "test/yosupo-lca.2.test.cpp": "2024-05-17 22:45:13 +0900",
 "test/yosupo-line-add-get-min.test.cpp": "2023-06-19 07:11:52 +0900",
 "test/yosupo-log-of-fps.test.cpp": "2024-01-20 00:58:37 +0900",
+"test/yosupo-min-plus-convolution-convex-and-arbitary.test.cpp": "2024-04-24 16:21:33 +0900",
+"test/yosupo-min-plus-convolution-convex-and-convex.test.cpp": "2024-04-24 16:21:33 +0900",
 "test/yosupo-partition-function_1.test.cpp": "2024-01-20 00:58:37 +0900",
 "test/yosupo-partition-function_2.test.cpp": "2024-01-20 00:58:37 +0900",
 "test/yosupo-point-add-rectangle-sum.test.cpp": "2024-05-17 23:06:28 +0900",
@@ -71,6 +73,7 @@
 "test/yukicoder-186.test.cpp": "2023-09-16 00:07:15 +0900",
 "test/yukicoder-187.test.cpp": "2023-09-16 00:07:15 +0900",
 "test/yukicoder-2292.test.cpp": "2024-05-17 23:11:56 +0900",
+"test/yukicoder-705.test.cpp": "2024-04-24 16:21:33 +0900",
 "test/yukicoder-rotate-enlarge_1.test.cpp": "2023-10-12 08:50:40 +0900",
 "test/yukicoder-rotate-enlarge_2.test.cpp": "2023-10-12 08:50:40 +0900",
 "test/yukicoder-search-oji.test.cpp": "2023-10-12 08:50:40 +0900"

cpp/convex.hpp

@@ -0,0 +1,121 @@
+#pragma once
+#include <cassert>
+#include <functional>
+#include <limits>
+#include <stack>
+#include <type_traits>
+#include <optional>
+#include <vector>
+
+/**
+ * @brief monotoneな行列において、各行の最小値と、それを取る最小列番号を$O((H+W)\log H)$時間で得る
+ * @param h 行数(行番号は[0..h-1])
+ * @param w 列数(列番号は[0..w-1])
+ * @param f 行列の要素を与える関数(行番号と列番号を引数にとって値を返す)
+ * @param comp 要素の比較関数(最小値を知りたい場合はデフォルトのstd::less<T>)
+ * @return std::pair<std::vector<T>, std::vector<int>> (各行の最小値, 各行の最小値列番号)
+ */
+template <typename F, typename Comp = std::less<std::invoke_result_t<F, int, int>>>
+std::pair<std::vector<std::invoke_result_t<F, int, int>>, std::vector<int>> monotone_minima(int h, int w, const F& f, const Comp& comp = Comp()) {
+    using T = std::invoke_result_t<F, int, int>;
+    assert(h >= 0);
+    assert(w >= 0);
+    std::vector<T> res_value(h);
+    std::vector<int> res_idx(h);
+    std::stack<std::tuple<int, int, int, int>> stk; // {i, j, l, r} : [i..j]行目の答えを求める。結果は[l..r]の範囲に収まることが保証されている。
+    stk.emplace(0, h-1, 0, w-1);
+    while(!stk.empty()) {
+        auto [i, j, l, r] = stk.top(); stk.pop();
+        int m = (i+j) / 2;
+        T min_value = f(m, l);
+        int min_idx = l;
+        for(int k = l+1; k <= r; k++) {
+            T value = f(m, k);
+            if(comp(value, min_value)) {
+                min_value = value;
+                min_idx = k;
+            }
+        }
+        res_value[m] = min_value;
+        res_idx[m] = min_idx;
+        if(i <= m-1) stk.emplace(i, m-1, l, min_idx);
+        if(m+1 <= j) stk.emplace(m+1, j, min_idx, r);
+    }
+    return {res_value, res_idx};
+}
+
+/**
+ * @brief オンライン・オフライン変換による、DAGで辺のコストがmonotoneな場合の最短経路問題を$O(N \log^2 N)$時間で求める
+ * @param n ノード数-1 (ノード番号は[0..n])
+ * @param f 辺のコスト(i<jのときf(i, j)が辺のコストを返す)
+ * @param comp 比較関数
+ * @return 頂点0から頂点[0..n]へのコストを表す長さn+1のvector
+ */
+template <typename F, typename Comp = std::less<std::invoke_result_t<F, int, int>>>
+std::vector<std::invoke_result_t<F, int, int>> online_offline_dp(int n, const F& f, const Comp& comp = Comp{}) {
+    using T = std::invoke_result_t<F, int, int>;
+    std::vector<std::optional<T>> dp(n+1, std::nullopt);
+    dp[0] = 0;
+    // dp[l..r)の要素を、dp[l..r)からの遷移のみの範囲で求める
+    auto solve_subproblem = [&](auto self, int l, int r) -> void {
+        int mid = (l+r) / 2;
+        if(mid-l >= 2) self(self, l, mid);
+        auto submat = [&](int i, int j) {
+            return dp[l+j].value() + f(l+j, mid+i);
+        };
+        std::vector<T> min_val = monotone_minima(r-mid, mid-l, submat, comp).first;
+        for(int i = 0; i < r-mid; i++) {
+            if(!dp[mid+i] || comp(min_val[i], dp[mid+i].value())) {
+                dp[mid+i] = min_val[i];
+            }
+        }
+        if(r-mid >= 2) self(self, mid, r);
+    };
+    solve_subproblem(solve_subproblem, 0, n+1);
+    std::vector<T> res(n+1);
+    for(int i = 0; i <= n; i++) res[i] = dp[i].value();
+    return res;
+}
+
+/**
+ * @brief aがconvex (a_{i+1}-a_i <= a_{i+2}-a_{i+1})のとき、c_k = min_{i+j=k}(a_i + b_j)なる長さN+M-1の配列を得る O((N+m)\log(N+M))
+ * @param a vector 配列1 (convexである必要があります)
+ * @param b vector 配列2
+ * @return c vector min-plus convolutionの結果
+ */
+template <typename T, std::enable_if_t<std::is_scalar_v<T>, std::nullptr_t> = nullptr>
+std::vector<T> min_plus_convolution_convex_arbitary(const std::vector<T>& a, const std::vector<T>& b) {
+    int n = a.size(), m = b.size();
+    for(int i = 0; i < n-2; i++) assert(a[i+1]-a[i] <= a[i+2]-a[i+1]);
+    auto mat = [&](int i, int j) {
+        int k = i-j;
+        if(k < 0) return std::numeric_limits<T>::max();
+        if(k >= n) return std::numeric_limits<T>::max();
+        return a[k] + b[j];
+    };
+    return monotone_minima(n+m-1, m, mat).first;
+}
+
+/**
+ * @brief aもbもconvex (a_{i+1}-a_i <= a_{i+2}-a_{i+1})のとき、c_k = min_{i+j=k}(a_i + b_j)なる長さN+M-1の配列を得る O(N+M)
+ * @param a vector 配列1 (convexである必要があります)
+ * @param b vector 配列2
+ * @return c vector min-plus convolutionの結果
+ */
+template <typename T, std::enable_if_t<std::is_scalar_v<T>, std::nullptr_t> = nullptr>
+std::vector<T> min_plus_convolution_convex_convex(const std::vector<T>& a, const std::vector<T>& b) {
+    int n = a.size(), m = b.size();
+    for(int i = 0; i < n-2; i++) assert(a[i+1]-a[i] <= a[i+2]-a[i+1]);
+    for(int i = 0; i < m-2; i++) assert(b[i+1]-b[i] <= b[i+2]-b[i+1]);
+    int i = 0, j = 0;
+    std::vector<T> c(n+m-1);
+    c[0] = a[i] + b[j];
+    for(int k = 1; k < n+m-1; k++) {
+        if(j == m-1 || (i < n-1 && a[i+1]+b[j] < a[i]+b[j+1])) {
+            c[k] = a[++i] + b[j];
+        } else {
+            c[k] = a[i] + b[++j];
+        }
+    }
+    return c;
+}

test/yosupo-min-plus-convolution-convex-and-arbitary.test.cpp

@@ -0,0 +1,14 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/min_plus_convolution_convex_arbitrary"
+#include "../cpp/convex.hpp"
+#include <iostream>
+
+int main() {
+    int n, m; std::cin >> n >> m;
+    std::vector<int> a(n), b(m);
+    for(int i = 0; i < n; i++) std::cin >> a[i];
+    for(int i = 0; i < m; i++) std::cin >> b[i];
+    std::vector<int> c = min_plus_convolution_convex_arbitary(a, b);
+    for(int i = 0; i < n+m-1; i++) {
+        std::cout << c[i] << (i==n+m-2 ? '\n' : ' ');
+    }
+}

test/yosupo-min-plus-convolution-convex-and-convex.test.cpp

@@ -0,0 +1,14 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/min_plus_convolution_convex_convex"
+#include "../cpp/convex.hpp"
+#include <iostream>
+
+int main() {
+    int n, m; std::cin >> n >> m;
+    std::vector<int> a(n), b(m);
+    for(int i = 0; i < n; i++) std::cin >> a[i];
+    for(int i = 0; i < m; i++) std::cin >> b[i];
+    std::vector<int> c = min_plus_convolution_convex_convex(a, b);
+    for(int i = 0; i < n+m-1; i++) {
+        std::cout << c[i] << (i==n+m-2 ? '\n' : ' ');
+    }
+}

test/yukicoder-705.test.cpp

@@ -0,0 +1,18 @@
+#define PROBLEM "https://yukicoder.me/problems/no/705"
+#include "../cpp/convex.hpp"
+#include <iostream>
+
+int main() {
+    using ll = long long;
+    int n; std::cin >> n;
+    std::vector<ll> a(n), x(n), y(n);
+    for(int i = 0; i < n; i++) std::cin >> a[i];
+    for(int i = 0; i < n; i++) std::cin >> x[i];
+    for(int i = 0; i < n; i++) std::cin >> y[i];
+    auto f = [&](int i, int j) {
+        ll dx = std::abs(x[i] - a[j-1]);
+        ll dy = std::abs(y[i]);
+        return dx*dx*dx + dy*dy*dy;
+    };
+    std::cout << online_offline_dp(n, f)[n] << std::endl;
+}