smawk

algorithm/smawk.hpp

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+#pragma once
+#include <vector>
+#include <numeric>
+namespace po167{
+// return argmax
+// 
+// https://noshi91.github.io/Library/algorithm/smawk.cpp
+template<class S> std::vector<int> smawk(int N, int M, S select){
+    auto f = [&](auto self, std::vector<int> p,std::vector<int> q) -> std::vector<int> {
+        // reduce
+        std::vector<int> n_q;
+        n_q.reserve(p.size());
+        int k = -1;
+        for(auto x:q){
+            while(!n_q.empty()){
+                if(!select(p[k], n_q.back(), x)) break;
+                n_q.pop_back();
+                k--;
+            }
+            if (k + 1 != (int)p.size()){
+                n_q.push_back(x);
+                k++;
+            }
+        }
+        std::swap(n_q, q);
+
+        int n = p.size();
+        int m = q.size();
+
+        // recursive
+        if (n == 1){
+            return q;
+        }
+        std::vector<int> n_p;
+        n_p.reserve(n / 2);
+
+        for (int i = 1; i < n; i += 2){
+            n_p.push_back(p[i]);
+        }
+
+        auto fr = self(self, n_p, q);
+        // interpolate
+        std::vector<int> ans(n);
+        int l = 0, r = 0;
+
+        for (int i = 0; i < n; i++){
+            if(i & 1){
+                ans[i] = fr[i / 2];
+                l = r;
+            } else {
+                if (i + 1 != n) {
+                    while (q[r] != fr[i / 2]) r++;
+                } else {
+                    r = m - 1;
+                }
+                ans[i] = q[l];
+                for (int j = l + 1; j <= r; j++){
+                    if(select(p[i], ans[i], q[j])){
+                        ans[i] = q[j];
+                    }
+                }
+            }
+        }
+        return ans;
+    };
+    std::vector<int> P(N), Q(M);
+    std::iota(P.begin(), P.end(), 0);
+    std::iota(Q.begin(), Q.end(), 0);
+    return f(f, P, Q);
+}
+}

algorithm/sum_max_convolution.hpp

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+#pragma once
+#include <vector>
+#include <algorithm>
+#include "smawk.hpp"
+
+namespace po167{
+// X is convex upward
+template<class T>
+std::vector<T> sum_max_convolution(std::vector<T> X, std::vector<T> Y){
+    int n = X.size();
+    int m = Y.size();
+    std::vector<T> ans(n + m - 1);
+    auto get = [&](int i, int j) -> T {
+        return X[i - j] + Y[j];
+    };
+    auto select = [&](int i, int j1, int j2) -> bool {
+        if (i < j2) return false;
+        if (i - j1 >= n) return true;
+        return get(i, j1) <= get(i, j2);
+    };
+    const int D = 10;
+    if (std::min(n, m) <= D) {
+        for (int i = 0; i < n + m - 1; i++){
+            int ind = std::max(0, i - n);
+            for (int j = ind + 1; j < m && j <= i; j++){
+                if (select(i, ind, j)) ind = j;
+            }
+            ans[i] = get(i, ind);
+        }
+        return ans;
+    }
+    std::vector<int> res = smawk(n + m - 1, m, select);
+    for (int i = 0; i < n + m - 1; i++){
+        ans[i] = get(i, res[i]);
+    }
+    return ans;
+}
+}

test/algorithm/sum_min_conv.test.cpp

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+#define PROBLEM https://judge.yosupo.jp/problem/min_plus_convolution_convex_arbitrary
+
+#include "../../algorithm/sum_max_convolution.hpp"
+
+#include <iostream>
+#include <vector>
+
+int main(){
+    std::ios::sync_with_stdio(false);
+    std::cin.tie(nullptr);
+    int N, M;
+    std::cin >> N >> M;
+    std::vector<int> A(N), B(M);
+    for (auto &a : A) std::cin >> a, a *= -1;
+    for (auto &b : B) std::cin >> b, b *= -1;
+    auto c = po167::sum_max_convolution(A, B);
+    for (int i = 0; i < N + M - 1; i++){
+        if (i) std::cout << " ";
+        std::cout << -c[i];
+    }
+    std::cout << "\n";
+}