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:heavy_check_mark: AOJ2334 街を駆ける道
(Test/AOJ/2334.test.cpp)

トタタ族の$i$個目の街の座標を$P_{i}$、ツテテ族の$j$個目の街の座標を$Q_{j}$とする。二点$p, q$を端点とする線分を$(p, q)$と表記する。

線分$(P_{1}, P_{2})$と線分$(Q_{1}, Q_{2})$が交わらない場合、この二本を引くことで最短の長さで目的を達成できる。

そうでないとき、$P_{1}, P_{2}$を$(Q_{1}, Q_{2})$と交わらないようにつなぐ、あるいは$Q_{1}, Q_{2}$を$(P_{1}, P_{2})$と交わらないようにつなぐ必要がある。

前者の場合、$Q_{1}, Q_{2}$をつなぐのに線分$(Q_{1}, Q_{2})$を利用すれば良い。後者の場合、$P_{1}, P_{2}$をつなぐのに線分$(P_{1}, P_{2})$を用いれば良い。

結局のところどちらか一方は丁度一本の線分で街1と街2をつなぐとして良いことがわかった。一本の線分でつなぐ方を固定するともう一方は$N$頂点$O(N^2)$辺の単一始点最短経路問題を解くことになる。辺重みがすべて正であるため、ダイクストラ法を用いれば良い。

Depends on

Code

#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/problems/2334"
#define ERROR 0.000000001

#include "../../Src/Template/IOSetting.hpp"
#include "../../Src/GeometryR2/Point.hpp"
#include "../../Src/GeometryR2/Segment.hpp"
#include "../../Src/GeometryR2/Intersect/SegmentAndSegment.hpp"

#include <iostream>
#include <queue>

using namespace zawa;
using namespace geometryR2;

const Real INF{9e18};

Real solve(int N, const std::vector<Point>& P, const Segment& S) {
    std::vector<std::vector<std::pair<int, Real>>> g(N);
    for (int i{} ; i < N ; i++) {
        for (int j{} ; j < N ; j++) {
            if (i == j) continue;
            Segment cur{P[i], P[j]};
            if (Intersect(cur, S)) continue;
            g[i].emplace_back(j, cur.length());
        }
    }
    std::vector<Real> dist(N, INF);
    using qt = std::pair<Real, int>;
    std::priority_queue<qt, std::vector<qt>, std::greater<qt>> que;
    dist[0] = 0;
    que.emplace(dist[0], 0);
    while (que.size()) {
        auto [d, v]{que.top()};
        que.pop();
        if (Smaller(dist[v], d)) continue;
        for (auto [x, w] : g[v]) {
            if (Smaller(dist[v] + w, dist[x])) {
                dist[x] = dist[v] + w;
                que.emplace(dist[x], x);
            }
        }
    }
    return dist[1] + S.length();
}

int main() {
    SetFastIO();
    int NA, NB;
    std::cin >> NA >> NB;
    std::vector<Point> A(NA), B(NB);
    for (auto& a : A) std::cin >> a;
    for (auto& b : B) std::cin >> b;
    Real ans{std::min(
            solve(NA, A, Segment{B[0], B[1]}),
            solve(NB, B, Segment{A[0], A[1]})
            )};
    SetPrecision(10);
    if (Smaller(ans, INF)) {
        std::cout << ans << '\n';
    }
    else {
        std::cout << -1 << '\n';
    }
}
#line 1 "Test/AOJ/2334.test.cpp"
#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/problems/2334"
#define ERROR 0.000000001

#line 2 "Src/Template/IOSetting.hpp"

#line 2 "Src/Template/TypeAlias.hpp"

#include <cstdint>
#include <cstddef>

namespace zawa {

using i16 = std::int16_t;
using i32 = std::int32_t;
using i64 = std::int64_t;
using i128 = __int128_t;

using u8 = std::uint8_t;
using u16 = std::uint16_t;
using u32 = std::uint32_t;
using u64 = std::uint64_t;

using usize = std::size_t;

} // namespace zawa
#line 4 "Src/Template/IOSetting.hpp"

#include <iostream>
#include <iomanip>

namespace zawa {

void SetFastIO() {
    std::cin.tie(nullptr)->sync_with_stdio(false);
}

void SetPrecision(u32 dig) {
    std::cout << std::fixed << std::setprecision(dig);
}

} // namespace zawa
#line 2 "Src/GeometryR2/Point.hpp"

#line 2 "Src/GeometryR2/Real.hpp"

#line 4 "Src/GeometryR2/Real.hpp"

#include <cmath>
#include <cassert>

namespace zawa {

namespace geometryR2 {

using Real = long double;

namespace internal {

Real EPS{1e-12};
constexpr i32 negative{-1};
constexpr i32 zero{};
constexpr i32 positive{1};

} // namespace internal

Real& Eps() {
    return internal::EPS;
}

i32 Sign(Real value) {
    if (value < -Eps()) return internal::negative;
    if (value > Eps()) return internal::positive;
    return internal::zero;
}

bool Zero(Real value) {
    return Sign(value) == internal::zero;
}

bool Positive(Real value) {
    return Sign(value) == internal::positive;
}

bool Negative(Real value) {
    return Sign(value) == internal::negative;
}

bool Equal(Real a, Real b) {
    return Zero(a - b);
}

bool Smaller(Real a, Real b) {
    return Negative(a - b);
}

bool Bigger(Real a, Real b) {
    return Positive(a - b);
}

Real Square(Real value) {
    return (Zero(value) ? value : value * value);
}

Real Sqrt(Real value) {
    assert(!Negative(value));
    return (Zero(value) ? value : sqrtl(value));
}

Real Abs(Real value) {
    return (Negative(value) ? -value : value);
}

} // namespace geometryR2
 
} // namespace zawa
#line 2 "Src/GeometryR2/Angle.hpp"

#line 4 "Src/GeometryR2/Angle.hpp"

#line 6 "Src/GeometryR2/Angle.hpp"

namespace zawa {

namespace geometryR2 {

constexpr Real PI{acosl(-1)};
constexpr Real TAU{static_cast<Real>(2) * PI};

constexpr Real ArcToRadian(Real arc) {
    return (arc * PI) / static_cast<Real>(180);
}

constexpr Real RadianToArc(Real radian) {
    return (radian * static_cast<Real>(180)) / PI;
}

} // namespace geometryR2

} // namespace zawa
#line 5 "Src/GeometryR2/Point.hpp"

#line 9 "Src/GeometryR2/Point.hpp"

namespace zawa {

namespace geometryR2 {

class Point {
private:
    Real x_{}, y_{};
public:
    /* constructor */
    Point() = default;
    Point(Real x, Real y) : x_{x}, y_{y} {}

    /* getter, setter */
    Real x() const {
        return x_;
    }
    Real& x() {
        return x_;
    }
    Real y() const {
        return y_;
    }
    Real& y() {
        return y_;
    }

    /* operator */
    Point& operator+=(const Point& rhs) {
        x_ += rhs.x();
        y_ += rhs.y();
        return *this;
    }
    friend Point operator+(const Point& lhs, const Point& rhs) {
        return Point{lhs} += rhs;
    }
    Point operator+() const {
        return *this;
    }
    Point& operator-=(const Point& rhs) {
        x_ -= rhs.x();
        y_ -= rhs.y();
        return *this;
    }
    friend Point operator-(const Point& lhs, const Point& rhs) {
        return Point{lhs} -= rhs;
    }
    Point operator-() const {
        return Point{} - *this;
    }
    Point& operator*=(Real k) {
        x_ *= k;
        y_ *= k;
        return *this;
    }
    friend Point operator*(Real k, const Point& p) {
        return Point{p} *= k;
    }
    friend Point operator*(const Point& p, Real k) {
        return Point{p} *= k;
    }
    Point& operator/=(Real k) {
        assert(!Zero(k));
        x_ /= k;
        y_ /= k;
        return *this;
    }
    friend Point operator/(Real k, const Point& p) {
        return Point{p} /= k;
    }
    friend Point operator/(const Point& p, Real k) {
        return Point{p} /= k;
    }
    friend bool operator==(const Point& lhs, const Point& rhs) {
        return Equal(lhs.x(), rhs.x()) and Equal(lhs.y(), rhs.y());
    }
    friend bool operator!=(const Point& lhs, const Point& rhs) {
        return !Equal(lhs.x(), rhs.x()) or !Equal(lhs.y(), rhs.y());
    }
    friend bool operator<(const Point& lhs, const Point& rhs) {
        return Smaller(lhs.x(), rhs.x()) or 
            (Equal(lhs.x(), rhs.x()) and Smaller(lhs.y(), rhs.y()));
    }
    friend bool operator<=(const Point& lhs, const Point& rhs) {
        return Smaller(lhs.x(), rhs.x()) or 
            (Equal(lhs.x(), rhs.x()) and (Smaller(lhs.y(), rhs.y()) or Equal(lhs.y(), rhs.y())));
    }
    friend bool operator>(const Point& lhs, const Point& rhs) {
        return Bigger(lhs.x(), rhs.x()) or
            (Equal(lhs.x(), rhs.x()) and Bigger(lhs.y(), rhs.y()));
    }
    friend bool operator>=(const Point& lhs, const Point& rhs) {
        return Bigger(lhs.x(), rhs.x()) or
            (Equal(lhs.x(), rhs.x()) and (Bigger(lhs.y(), rhs.y()) or Equal(lhs.y(), rhs.y())));
    }
    friend std::istream& operator>>(std::istream& is, Point& p) {
        is >> p.x_ >> p.y_;
        return is;
    }
    friend std::ostream& operator<<(std::ostream& os, const Point& p) {
        os << '(' << p.x_ << ',' << p.y_ << ')';
        return os;
    }
    
    /* member function */
    Real normSquare() const {
        return Square(x_) + Square(y_);
    }
    Real norm() const {
        return Sqrt(normSquare());
    }
    void normalize() {
        assert((*this) != Point{});
        (*this) /= norm(); 
    }
    Point normalized() const {
        Point res{*this};
        res.normalize();
        return res;
    }
    Point rotated(Real radian) const {
        return Point{
            x_ * cosl(radian) - y_ * sinl(radian),
            x_ * sinl(radian) + y_ * cosl(radian)
        };
    }
    void rotate(Real radian) {
        *this = rotated(radian); 
    }
    Point rotatedByArc(Real arc) const {
        return rotated(ArcToRadian(arc));
    }
    void rotateByArc(Real arc) {
        *this = rotatedByArc(arc);
    }
    Real argument() const {
        return (Negative(y_) ? TAU : static_cast<Real>(0)) + atan2l(y_, x_);
    }
    Real argumentByArc() const {
        return RadianToArc(argument());
    }

    /* friend function */
    friend Real Dot(const Point& lhs, const Point& rhs) {
        return lhs.x() * rhs.x() + lhs.y() * rhs.y();
    }
    friend Real Cross(const Point& lhs, const Point& rhs) {
        return lhs.x() * rhs.y() - lhs.y() * rhs.x();
    }
    friend Real Argument(const Point& lhs, const Point& rhs) {
        return rhs.argument() - lhs.argument();
    }
    friend bool ArgComp(const Point& lhs, const Point& rhs) {
        return Smaller(lhs.argument(), rhs.argument());
    }
};

using Vector = Point;

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Segment.hpp"

#line 2 "Src/GeometryR2/Relation.hpp"

#line 5 "Src/GeometryR2/Relation.hpp"

namespace zawa {

namespace geometryR2 {

enum RELATION {
    // p0 -> p1 -> p2の順で直線上に並んでいる
    ONLINE_FRONT = -2,
    // (p1 - p0) -> (p2 - p0)が時計回りになっている
    CLOCKWISE,
    // p0 -> p2 -> p1の順で直線上に並んでいる
    ON_SEGMENT,
    // (p1 - p0) -> (p2 - p0)が反時計回りになっている
    COUNTER_CLOCKWISE,
    // p2 -> p0 -> p1、またはp1 -> p0 -> p2の順で直線上に並んでいる
    ONLINE_BACK
};

RELATION Relation(const Point& p0, const Point& p1, const Point& p2) {
    Point a{p1 - p0}, b{p2 - p0};
    if (Positive(Cross(a, b))) return COUNTER_CLOCKWISE;
    if (Negative(Cross(a, b))) return CLOCKWISE;
    if (Negative(Dot(a, b))) return ONLINE_BACK;
    if (Smaller(a.normSquare(), b.normSquare())) return ONLINE_FRONT;
    return ON_SEGMENT;
};

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Distance/PointAndPoint.hpp"

#line 4 "Src/GeometryR2/Distance/PointAndPoint.hpp"

namespace zawa {

namespace geometryR2 {

Real Distance(const Point& p0, const Point& p1) {
    return Point{p1 - p0}.norm();
}

Real DistanceSquare(const Point& p0, const Point& p1) {
    return Point{p1 - p0}.normSquare();
}

} // namespace geometryR2

} // namespace zawa
#line 6 "Src/GeometryR2/Segment.hpp"

#include <algorithm>
#line 9 "Src/GeometryR2/Segment.hpp"

namespace zawa {

namespace geometryR2 {

class Segment {
private:
    Point p0_{}, p1_{};
public:
    /* constructor */
    Segment() = default;
    Segment(const Point& p0, const Point& p1) : p0_{p0}, p1_{p1} {}
    Segment(Real x0, Real y0, Real x1, Real y1) : p0_{x0, y0}, p1_{x1, y1} {}

    /* getter setter */
    const Point& p0() const {
        return p0_;
    }
    Point& p0() {
        return p0_;
    }
    const Point& p1() const {
        return p1_;
    }
    Point& p1() {
        return p1_;
    }

    /* member function */
    bool valid() const {
        return p0_ != p1_;
    }
    bool straddle(const Segment& s) const {
        return Relation(p0_, p1_, s.p0()) * Relation(p0_, p1_, s.p1()) <= 0;
    }
    Real length() const {
        assert(valid());
        return Distance(p0_, p1_);
    }
    Point midpoint() const {
        assert(valid());
        return p0_ + Vector{p1_ - p0_} / static_cast<Real>(2);
    }
};

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Intersect/SegmentAndSegment.hpp"

#line 4 "Src/GeometryR2/Intersect/SegmentAndSegment.hpp"

#line 6 "Src/GeometryR2/Intersect/SegmentAndSegment.hpp"

namespace zawa {

namespace geometryR2 {

bool Intersect(const Segment& s0, const Segment& s1) {
    assert(s0.valid());
    assert(s1.valid());
    return s0.straddle(s1) and s1.straddle(s0);
}

} // namespace geometryR2

} // namespace zawa
#line 8 "Test/AOJ/2334.test.cpp"

#line 10 "Test/AOJ/2334.test.cpp"
#include <queue>

using namespace zawa;
using namespace geometryR2;

const Real INF{9e18};

Real solve(int N, const std::vector<Point>& P, const Segment& S) {
    std::vector<std::vector<std::pair<int, Real>>> g(N);
    for (int i{} ; i < N ; i++) {
        for (int j{} ; j < N ; j++) {
            if (i == j) continue;
            Segment cur{P[i], P[j]};
            if (Intersect(cur, S)) continue;
            g[i].emplace_back(j, cur.length());
        }
    }
    std::vector<Real> dist(N, INF);
    using qt = std::pair<Real, int>;
    std::priority_queue<qt, std::vector<qt>, std::greater<qt>> que;
    dist[0] = 0;
    que.emplace(dist[0], 0);
    while (que.size()) {
        auto [d, v]{que.top()};
        que.pop();
        if (Smaller(dist[v], d)) continue;
        for (auto [x, w] : g[v]) {
            if (Smaller(dist[v] + w, dist[x])) {
                dist[x] = dist[v] + w;
                que.emplace(dist[x], x);
            }
        }
    }
    return dist[1] + S.length();
}

int main() {
    SetFastIO();
    int NA, NB;
    std::cin >> NA >> NB;
    std::vector<Point> A(NA), B(NB);
    for (auto& a : A) std::cin >> a;
    for (auto& b : B) std::cin >> b;
    Real ans{std::min(
            solve(NA, A, Segment{B[0], B[1]}),
            solve(NB, B, Segment{A[0], A[1]})
            )};
    SetPrecision(10);
    if (Smaller(ans, INF)) {
        std::cout << ans << '\n';
    }
    else {
        std::cout << -1 << '\n';
    }
}
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