add min_of_mod_of_linear

number/min_of_mod_of_linear.hpp

@@ -0,0 +1,109 @@
+#pragma once
+
+#include <cassert>
+#include <vector>
+#include <utility>
+
+// min((ax + b) mod m) for 0 <= x < n
+// 0 < n, 0 < m <= 1e9
+// https://maspypy.com/library-checker-min-of-mod-of-linear
+template <class Int = int, class Long = long long>
+auto min_of_mod_of_linear(Int n, const Int &m, Int a, Int b) {
+
+    // p = (xs, ys), v = (dx, dy) として
+    // 列 (p, p + v, ..., p + (len - 1)v) が点群全体を包絡する
+    struct Result {
+        Int xs, ys;
+        Int dx, dy;
+        Int len;
+
+        Int min() const { return ys + dy * (len - 1); }
+    };
+
+    assert(0 < n);
+    assert(0 < m);
+
+    if (n > m) n = m;
+    a = (m - a % m) % m;
+    b = (b % m + m) % m;
+
+    Int x = 0, y = b;
+
+    Int p = 0, q = 1, r = 1, s = 0; // p/q <= a/m < r/s
+
+    std::vector<Result> ret{Result{x, y, 0, 0, 1}};
+
+    while (true) {
+        if (x + q > n - 1) break;
+
+        const Long aqmp = (Long)a * q - (Long)m * p;
+        if (aqmp == 0) break; // p/q == a/m
+
+        const Long mras = (Long)m * r - (Long)a * s;
+
+        assert(aqmp > 0);
+        assert(mras > 0);
+
+        if (const Long num = p + r, den = q + s; num * m <= den * a) { // num/den <= a/m 下側を更新
+
+            // q + is <= n - 1 - x
+
+            // (p + ir) / (q + is) <= a/m （下側である条件） i >= 0
+            // <=> m(p + ir) <= a(q + is) <=> (mr - as)i <= aq - mp
+
+            // a(q + is) - m(p + ir) <= Y （改善する条件） i >= 0
+            // (mr - as)i >= aq - mp - Y
+
+            const Long iub = aqmp / mras;
+
+            const Long imp_i_lb = std::max<Long>((std::max<Long>(aqmp - y, 0) + mras - 1) / mras, 0);
+
+            if (imp_i_lb <= iub) {
+                p += imp_i_lb * r;
+                q += imp_i_lb * s;
+
+                const Long c = (Long)a * q - (Long)m * p;
+                assert(0 <= c);
+                assert(c <= y);
+
+                const Long kub = y / c;
+                const Long k = std::min<Long>(kub, (n - 1 - x) / q);
+
+                if (k) ret.push_back(Result{x + q, (Int)(y - c), q, (Int)-c, (Int)k});
+
+                x += q * k;
+                y -= c * k;
+                if (k < kub) break;
+                continue;
+            } else {
+                p += iub * r;
+                q += iub * s;
+            }
+        } else { // num/den > a/m 上側を更新
+            // (r + ip)/(s + iq) > a/m <=> m(r + ip) > a(s + iq) <=> (aq - mp)i < mr - as
+            // -> i step 進んでもまだ超えない
+
+            const Long i = (mras - 1) / aqmp;
+            assert(i > 0);
+
+            // x + (s + iq) <= n - 1
+            // const Long iub = (n - 1 - q - x - s) / q;
+            // if (i > iub) break;
+
+            r += (i - 1) * p; // NOTE: overflow
+            s += (i - 1) * q;
+
+            // まだ a/m < (p + r) / (q + s)
+            assert((Long)a * (q + s) < (Long)m * (p + r));
+
+            r += p;
+            s += q;
+
+            // 逆転
+            assert((Long)a * (q + s) >= (Long)m * (p + r));
+        }
+    }
+
+    // return y;
+    return ret;
+}

number/test/min_of_mod_of_linear.test.cpp

@@ -0,0 +1,17 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/min_of_mod_of_linear"
+
+#include "../min_of_mod_of_linear.hpp"
+#include <iostream>
+using namespace std;
+
+int main() {
+    cin.tie(nullptr), ios::sync_with_stdio(false);
+
+    int T;
+    cin >> T;
+    while (T--) {
+        int n, m, a, b;
+        cin >> n >> m >> a >> b;
+        cout << min_of_mod_of_linear<int, long long>(n, m, a, b).back().min() << '\n';
+    }
+}