Problem:
Score: 600 points

Problem Statement

M-kun is a brilliant air traffic controller.

On the display of his radar, there are N airplanes numbered 1, 2, ..., N, all flying at the same altitude.

Each of the airplanes flies at a constant speed of 0.1 per second in a constant direction. The current coordinates of the airplane numbered i are (X_i, Y_i), and the direction of the airplane is as follows:

if U_i is U, it flies in the positive y direction;

if U_i is R, it flies in the positive x direction;

if U_i is D, it flies in the negative y direction;

if U_i is L, it flies in the negative x direction.

To help M-kun in his work, determine whether there is a pair of airplanes that will collide with each other if they keep flying as they are now.

If there is such a pair, find the number of seconds after which the first collision will happen.

We assume that the airplanes are negligibly small so that two airplanes only collide when they reach the same coordinates simultaneously.

Constraints

1 \leq N \leq 200000

0 \leq X_i, Y_i \leq 200000

U_i is U, R, D, or L.

The current positions of the N airplanes, (X_i, Y_i), are all distinct.

N, X_i, and Y_i are integers.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1 U_1
X_2 Y_2 U_2
X_3 Y_3 U_3
:
X_N Y_N U_N

Output

If there is a pair of airplanes that will collide with each other if they keep flying as they are now, print an integer representing the number of seconds after which the first collision will happen.

If there is no such pair, print SAFE.

Sample Input 1

2
11 1 U
11 47 D

Sample Output 1

230

If the airplanes keep flying as they are now, two airplanes numbered 1 and 2 will reach the coordinates (11, 24) simultaneously and collide.

Sample Input 2

4
20 30 U
30 20 R
20 10 D
10 20 L

Sample Output 2

SAFE

No pair of airplanes will collide.

Sample Input 3

8
168 224 U
130 175 R
111 198 D
121 188 L
201 116 U
112 121 R
145 239 D
185 107 L

Sample Output 3

100

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#define x first
#define y second
#define int int long long // 2e9 & 9e18
using namespace std; const int inf = numeric_limits<int>::max();
struct __io{__io(){ios_base::Init i; ios_base::sync_with_stdio(0); cin.tie(0);}} __io; // fast I/O
#ifdef LOCAL // setting up print debugging (yes lol)
template<class K, class V>ostream& operator<<(ostream&s,const pair<K,V>&p){s<<'<'<<p.x<<", "<<p.y<<'>';return s;}
template<class T, class=typename T::value_type, class=typename enable_if<!is_same<T,string>::value>::type>
ostream& operator<<(ostream&s,const T&v){s<<'[';for(auto&x:v){s<<x<<", ";}if(!v.empty()){s<<"\b\b";}s<<']';return s;}
void __prnt(){cerr<<endl;} template<class T, class...Ts>void __prnt(T&&a,Ts&&...etc){cerr<<a<<' ';__prnt(etc...);}
#define print(...) __prnt(__VA_ARGS__)
#else
#define print(...)
#endif

struct plane {
    int x;
    int y;
    char d;
};
ostream& operator<<(ostream&s,const plane &p){
    s<<'<'<<p.x<<", "<<p.y<< ", " << p.d << '>';
    return s;
}

int reference(int n, vector<plane> &v);

signed main() {
    int n;
    cin >> n;
    vector<plane> v, u, d, l, r;
    // map<int, plane> mu, md, ml, mr;
    for (int i = 0; i < n; ++i) {
        plane p;
        cin >> p.x >> p.y >> p.d;
        switch (p.d) {
            case 'U': u.push_back(p); /*mu[] = p;*/ break;
            case 'D': d.push_back(p); /*md[] = p;*/ break;
            case 'L': l.push_back(p); /*ml[] = p;*/ break;
            case 'R': r.push_back(p); /*mr[] = p;*/ break;
        }
        v.push_back(p);
    }
    print(u);
    print(d);
    print(l);
    print(r);

    int t = inf;
    {
        map<int, plane> ma;
        for (plane &p : u) {
            ma[p.x + p.y] = p;
        }
        for (plane &p : r) {
            auto it = ma.find(p.x + p.y);
            if (it != ma.end()) {
                int w = it->y.x - p.x;
                if (w > 0) {
                    t = min(t, w*10);
                }
            }
        }
    }
    {
        map<int, plane> ma;
        for (plane &p : d) {
            ma[p.x + p.y] = p;
        }
        for (plane &p : l) {
            auto it = ma.find(p.x + p.y);
            if (it != ma.end()) {
                int w = p.x - it->y.x;
                if (w > 0) {
                    t = min(t, w*10);
                }
            }
        }
    }
    /*{
        map<int, plane> ma;
        for (plane &p : u) {
            ma[p.x - p.y] = p;
        }
        for (plane &p : l) {
            auto it = ma.find(p.x - p.y);
            if (it != ma.end()) {
                int w = p.x - it->y.x;
                if (w > 0) {
                    t = min(t, w*10);
                }
            }
        }
    }
    {
        map<int, plane> ma;
        for (plane &p : d) {
            ma[p.x - p.y] = p;
        }
        for (plane &p : r) {
            auto it = ma.find(p.x - p.y);
            if (it != ma.end()) {
                int w = it->y.x - p.x;
                if (w > 0) {
                    t = min(t, w*10);
                }
            }
        }
    }*/
    t = min(t, reference(n, v));
    print("cross-direction:", t);

    {
        map<int, vector<plane>> m;
        for (auto &p : d) m[p.x].push_back(p);
        for (auto &p : u) m[p.x].push_back(p);
        for (auto &[_, v] : m) {
            sort(v.begin(), v.end(), [](plane &a, plane &b){return a.y < b.y;});
            print("vert", v);
            for (int i = 1; i < (int)v.size(); ++i) {
                if (v[i-1].d == 'U' && v[i].d == 'D') {
                    t = min(t, (v[i].y - v[i-1].y) * 5);
                }
            }
        }
    }
    {
        map<int, vector<plane>> m;
        for (auto &p : l) m[p.y].push_back(p);
        for (auto &p : r) m[p.y].push_back(p);
        for (auto &[_, v] : m) {
            sort(v.begin(), v.end(), [](plane &a, plane &b){return a.x < b.x;});
            print("hor", v);
            for (int i = 1; i < (int)v.size(); ++i) {
                if (v[i-1].d == 'R' && v[i].d == 'L') {
                    t = min(t, (v[i].x - v[i-1].x) * 5);
                }
            }
        }
    }


    if (t < inf) {
        cout << t << endl;
    } else {
        cout << "SAFE" << endl;
    }
}


///////////////////////////////// DEBUG ///////////////////////////
// https://atcoder.jp/contests/m-solutions2020/submissions/15455643

using lint = long long;
template<class T = lint, class R = lint>
struct Point2D {
    T x, y;
    constexpr Point2D(T x = 0, T y = 0) noexcept : x(x), y(y) {};
    constexpr Point2D(pair<T, T> p) noexcept : x(p.first), y(p.second) {};
    constexpr bool operator==(const Point2D& rhs) const noexcept { return x == rhs.x && y == rhs.y; }
    constexpr bool operator!=(const Point2D& rhs) const noexcept { return !((*this) == rhs); }
    constexpr bool operator< (const Point2D& rhs) const noexcept { return x < rhs.x || (x == rhs.x && y < rhs.y); }
    constexpr bool operator> (const Point2D& rhs) const noexcept { return rhs < (*this); }
    constexpr bool operator<=(const Point2D& rhs) const noexcept { return !((*this) > rhs); }
    constexpr bool operator>=(const Point2D& rhs) const noexcept { return !((*this) < rhs); }
    constexpr Point2D  operator+ (const Point2D& rhs) const noexcept { return { x + rhs.x, y + rhs.y }; }
    constexpr Point2D  operator- (const Point2D& rhs) const noexcept { return { x - rhs.x, y - rhs.y }; }
    constexpr Point2D  operator* (const T& lmap) const noexcept { return { lmap * x, lmap * y }; }
    constexpr Point2D  operator/ (const T& lmap) const noexcept { return { x / lmap, y / lmap }; }
    constexpr Point2D& operator+=(const Point2D& rhs) noexcept { return ((*this) = (*this) + rhs); }
    constexpr Point2D& operator-=(const Point2D& rhs) noexcept { return ((*this) = (*this) - rhs); }
    constexpr Point2D& operator*=(const T& lmap) noexcept { return ((*this) = (*this) * lmap); }
    constexpr Point2D& operator/=(const T& lmap) noexcept { return ((*this) = (*this) / lmap); }
    constexpr Point2D  operator- () const noexcept { return { -x, -y }; }
    constexpr T operator* (const Point2D& rhs) const noexcept { return x * rhs.x + y * rhs.y; }
    constexpr Point2D nor() const noexcept { return { y, -x }; }
    constexpr long double hypot() const noexcept { return ::hypotl(x, y); }
    friend istream& operator>>(istream& is, Point2D& p) { return is >> p.x >> p.y; }
    friend ostream& operator<<(ostream& os, const Point2D& p) { return os << p.x << " " << p.y << "\n"; }
    R& operator[](vector<vector<R>>& v) { return v[x][y]; }
};
 
using pnt = Point2D<>;
 
template<class T>
inline bool chmin(T& a, const T b) { return a > b && (a = b, true); }
 
int reference(int n, vector<plane> &v) {
 
    int N = n;
 
    vector<pnt> U, R, L, D;
    for (int ir = 0; ir < N; ir++) {
        pnt xy(v[ir].x, v[ir].y);
        char d = v[ir].d;
        if (d == 'R') R.emplace_back(xy);
        if (d == 'L') L.emplace_back(xy);
        if (d == 'U') U.emplace_back(xy);
        if (d == 'D') D.emplace_back(xy);
    }
    sort(R.begin(), R.end(), [](auto l, auto r) {return l.x < r.x; });
    sort(L.begin(), L.end(), [](auto l, auto r) {return l.x < r.x; });
    sort(U.begin(), U.end(), [](auto l, auto r) {return l.x < r.x; });
    sort(D.begin(), D.end(), [](auto l, auto r) {return l.x < r.x; });
 
    lint ans = inf;
    auto update = [&ans](auto &l, auto &r, pnt grad, int speed) {
 
        map<lint, vector<lint>> lmap, rmap;
        for (const auto& xy : l) lmap[xy * grad].emplace_back(xy.x);
        for (const auto& xy : r) rmap[xy * grad].emplace_back(xy.x);
 
        for (const auto& [key, vecl] : lmap) {
            auto it = rmap.find(key);
            if (it != rmap.end()) {
 
                auto vecr = it->second;
 
                auto ir = vecr.begin();
                for (const auto& xl : vecl) {
                    while (ir != vecr.end() && *ir < xl) ir++;
                    if (ir == vecr.end()) break;
                    chmin(ans, (*ir - xl) * speed);
                }
            }
        }
    
    };
 
    // update(R, U, pnt(1, 1), 10);
    // update(D, L, pnt(1, 1), 10);
    update(R, D, pnt(1, -1), 10);
    update(U, L, pnt(1, -1), 10);
 
    for (auto& e : U) swap(e.x, e.y);
    for (auto& e : D) swap(e.x, e.y);
    sort(U.begin(), U.end(), [](auto l, auto r) {return l.x < r.x; });
    sort(D.begin(), D.end(), [](auto l, auto r) {return l.x < r.x; });
 
    return ans;
}