hiho-offer102C.cpp

const int mod1 = 1000000007, mod2 = 998244353;
const int mod = mod1;
#ifdef LOCAL
//#define _GLIBCXX_DEBUG
// #pragma comment(linker, "/STACK:102400000,102400000") // 在 Windows 上有效
#endif
#include <bits/stdc++.h>
using namespace std;
// Andrew He's modular-arithmetic class
template <int MOD_> struct modnum {
    static constexpr int MOD = MOD_;
    static_assert(MOD_ > 0, "MOD must be positive");
private:
    using ll = long long;

    int v;

    static int minv(int a, int m) {
        a %= m;
        assert(a);
        return a == 1 ? 1 : int(m - ll(minv(m, a)) * ll(m) / a);
    }

public:

    modnum() : v(0) {}
    modnum(ll v_) : v(int(v_ % MOD)) { if (v < 0) v += MOD; }
    explicit operator int() const { return v; }
    friend std::ostream& operator << (std::ostream& out, const modnum& n) { return out << int(n); }
    friend std::istream& operator >> (std::istream& in, modnum& n) { ll v_; in >> v_; n = modnum(v_); return in; }

    friend bool operator == (const modnum& a, const modnum& b) { return a.v == b.v; }
    friend bool operator != (const modnum& a, const modnum& b) { return a.v != b.v; }

    modnum inv() const {
        modnum res;
        res.v = minv(v, MOD);
        return res;
    }
    modnum neg() const {
        modnum res;
        res.v = v ? MOD-v : 0;
        return res;
    }
    modnum operator- () const {
        return neg();
    }
    modnum operator+ () const {
        return modnum(*this);
    }

    modnum& operator ++ () {
        v ++;
        if (v == MOD) v = 0;
        return *this;
    }
    modnum& operator -- () {
        if (v == 0) v = MOD;
        v --;
        return *this;
    }
    modnum& operator += (const modnum& o) {
        v += o.v;
        if (v >= MOD) v -= MOD;
        return *this;
    }
    modnum& operator -= (const modnum& o) {
        v -= o.v;
        if (v < 0) v += MOD;
        return *this;
    }
    modnum& operator *= (const modnum& o) {
        v = int(ll(v) * ll(o.v) % MOD);
        return *this;
    }
    modnum& operator /= (const modnum& o) {
        return *this *= o.inv();
    }

    friend modnum operator ++ (modnum& a, int) { modnum r = a; ++a; return r; }
    friend modnum operator -- (modnum& a, int) { modnum r = a; --a; return r; }
    friend modnum operator + (const modnum& a, const modnum& b) { return modnum(a) += b; }
    friend modnum operator - (const modnum& a, const modnum& b) { return modnum(a) -= b; }
    friend modnum operator * (const modnum& a, const modnum& b) { return modnum(a) *= b; }
    friend modnum operator / (const modnum& a, const modnum& b) { return modnum(a) /= b; }
    friend modnum pow(modnum x, size_t n) {
        modnum res = 1;
        while (n) {
            if (n & 1) res *= x;
            x *= x;
            n /= 2;
        }
        return res;
    }
};

struct Modnum {
private:
    static int MOD; // declaration, not definition
    using ll = long long;
    using modnum = Modnum;
    int v;

    static int minv(int a, int m) {
        a %= m;
        assert(a);
        return a == 1 ? 1 : int(m - ll(minv(m, a)) * ll(m) / a);
    }

public:
    Modnum() : v(0) {}
    Modnum(ll v_) : v(int(v_ % MOD)) { if (v < 0) v += MOD; }
    static void set_mod(const int &m) {
        assert(m > 0);
        MOD = m;
    }
    explicit operator int() const { return v; }
    friend std::ostream& operator << (std::ostream& out, const modnum& n) { return out << int(n); }
    friend std::istream& operator >> (std::istream& in, modnum& n) { ll v_; in >> v_; n = modnum(v_); return in; }

    friend bool operator == (const modnum& a, const modnum& b) { return a.v == b.v; }
    friend bool operator != (const modnum& a, const modnum& b) { return a.v != b.v; }

    modnum inv() const {
        modnum res;
        res.v = minv(v, MOD);
        return res;
    }
    modnum neg() const {
        modnum res;
        res.v = v ? MOD-v : 0;
        return res;
    }
    modnum operator- () const {
        return neg();
    }
    modnum operator+ () const {
        return *this;
    }

    modnum& operator ++ () {
        v ++;
        if (v == MOD) v = 0;
        return *this;
    }
    modnum& operator -- () {
        if (v == 0) v = MOD;
        v --;
        return *this;
    }
    modnum& operator += (const modnum& o) {
        v += o.v;
        if (v >= MOD) v -= MOD;
        return *this;
    }
    modnum& operator -= (const modnum& o) {
        v -= o.v;
        if (v < 0) v += MOD;
        return *this;
    }
    modnum& operator *= (const modnum& o) {
        v = int(ll(v) * ll(o.v) % MOD);
        return *this;
    }
    modnum& operator /= (const modnum& o) {
        return *this *= o.inv();
    }

    friend modnum operator ++ (modnum& a, int) { modnum r = a; ++a; return r; }
    friend modnum operator -- (modnum& a, int) { modnum r = a; --a; return r; }
    friend modnum operator + (const modnum& a, const modnum& b) { return modnum(a) += b; }
    friend modnum operator - (const modnum& a, const modnum& b) { return modnum(a) -= b; }
    friend modnum operator * (const modnum& a, const modnum& b) { return modnum(a) *= b; }
    friend modnum operator / (const modnum& a, const modnum& b) { return modnum(a) /= b; }
};
int Modnum::MOD = 1; // definition

#define sq(x) (x)*(x) // square
#define FAST_READ ios::sync_with_stdio(false); cin.tie(nullptr);
#ifdef LOCAL
#define see(x) cout  <<"<DBG> " << #x << ": " << (x) << endl
#endif
#ifndef LOCAL
#define see(x)
#endif
#define LSON(x) ((x) * 2)
#define RSON(x) ((x) * 2 + 1)
template<typename A, typename B>
void Min(A &a, const B &b){
    if (b < a) a = b;
}
template<typename A, typename B>
void Max(A &a, const B &b){
    if (b > a) a = b;
}
int cas;

auto kase = []() {
    cout << "Case #" << ++cas << ": ";

};
#if __cplusplus < 201402L
template<class Iterator>
std::reverse_iterator<Iterator> make_reverse_iterator(Iterator it)
{
    return std::reverse_iterator<Iterator>(it);
}
#endif

template <typename iter_t>
struct iter_pair {
    iter_t _beg, _end;
    iter_t begin(){return _beg;}
    iter_t end(){return _end;}
};

template<class cont> iter_pair<reverse_iterator<decltype(begin(declval<cont>()))>>
reverse(cont &&r) {
    return {make_reverse_iterator(end(r)), make_reverse_iterator(begin(r))};
}

template<typename T> void dprintln(const T &t) { cout << t << endl; } // for debug use
template<typename T, typename ...Args> void dprintln(const T &t, const Args &...rest) { cout << t << ' '; dprintln(rest...); }
template<typename T> void println(const T &t) { cout << t << '\n'; }


template<typename T, typename ...Args> void println(const T &t, const Args &...rest) { cout << t << ' '; println(rest...); }
template<typename T> void println(const vector<T>& vec) {
    if (!vec.empty()) {
        cout << vec[0];
        for (size_t i = 1; i < vec.size(); ++i)
            cout << ' ' << vec[i];
    }
    cout << endl;
}

template<typename T> void print(const T &t) { cout << t << ' '; }

template<typename T, typename ...Args> void print(const T &t, const Args &...rest) { cout << t << ' '; print(rest...); }

// this overload is chosen when there's only one argument
template<class T> void scan(T &t) { cin >> t; }
template<class T, class ...Args> void scan(T &a, Args &...rest) { cin >> a; scan(rest...); }
template<typename T> void scan(vector<T>& vec) {for (T& x : vec) scan(x);}
using ull = unsigned long long;
using ll = long long;
using ul = unsigned long;
using vl = vector<ll>;
using vi = vector<int>;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using vb = vector<bool>;
using vpii = vector<pii>;
using ldb = long double;
template <typename int_t>
inline int_t lowbit(int_t x) {return x & -x;}
#define rng(i, a, b) for(int i = (int)(a); i < (int)(b); ++i)
#define up(i, a, b) for (int i = int(a); i <= int(b); ++i)
#define down(i, b, a) for (int i = int(b); i >= int(a); i--)
#define rep(n) for(int _ = 0, __ = (int)n; _ < __; _++)
#define stp(i, a, b, c) for (int i = (a); i < (b); i += (c))
#define FOR(x, cont) for (const auto &x: cont)
#define INC(init, x, y) for (init; x <= y; ++x)
#define For(x, cont) for (auto &x: cont)
#define all(x) begin(x), end(x)
#define pb push_back
#define mp make_pair
#define eb emplace_back
#define SZ(x) (int)(x).size()
#define UP(i, l, r) for(i = decltype(i)(l); i <= decltype(i)(r); ++i)
#define DOWN(i, r, l) for (i = decltype(i)(r); i >= decltype(i)(l); --i)
#define Dec(a, b) for (; a >= b; --a)
template <typename T, typename Comp = less<T>>
using pq = priority_queue<T, vector<T>, Comp>;
#define popcnt(x) __builtin_popcountll((x))
#define SET(arr, v) memset(arr, (v), sizeof (arr))
#define UNIQ(vec) (vec).erase(unique(all(vec)), end(vec))
#define LB(cont, x) int(lower_bound(all(cont), x) - begin(cont))
#define UB(cont, x) int(upper_bound(all(cont), x) - begin(cont))
#define AI(name, n, m) vv<int> name(n, vi(m));
#define AL(name, n, m) vv<ll> name(size_t(n), vl(size_t(m)));
#define set0(arr) memset(arr, 0, sizeof arr)
#define set1(arr) memset(arr, -1, sizeof arr)
#define AT(T, n, m, a)  vector<vector<T>> a(n, vector<T>(m))
const int dx[] = {0, 0, -1, 1}, dy[] = {-1, 1, 0, 0};
auto bet = [](const ll x, const ll y, const ll i) {
    return x <= i && i <= y;
};

template<typename T1, typename T2>
T1 ceil(T1 x, T2 y) { // y >= 1，是整数。需要注意 x + y - 1 是否会溢出
    return (x + y - 1) / y;
}


inline int h_bit(unsigned long long x) {
    return sizeof(unsigned long long) * 8 - 1 - __builtin_clzll(x);
}

int pow2(int x){ // power of 2
    return 1 << (h_bit((ull)x) + (x != lowbit(x)));
}



template <typename T>
struct bit {
    vector<T> a;
    function<T(T,T)> bin_op;
    const T init;
    explicit bit(int n, function<T(T,T)> bin_op, T init):bin_op(bin_op), init(init) {
        a.assign(n + 1, init);
    }

    T prefix(T x) {
        auto res = init;
        while (x) {
            res = bin_op(a[x], res);
            x -= x & -x;
        }
        return res;
    }
    void modify(int x, T v) {
        while (x < (int)a.size()) {
            a[x] = bin_op(a[x], v);
            x += x & -x;
        }
    }
    void clear(){
        fill(a.begin(), a.end(), init);
    }
};

template <typename T>
struct r_bit {
    vector<T> a;
    function<T(T,T)> bin_op;
    const T init;
    explicit r_bit(int n, function<T(T,T)> bin_op, T init):bin_op(move(bin_op)), init(init) {
        a.ssign(n + 1, init);
    }
    T suffix(int x) {
        T res = init;
        while (x < SZ(a)) {
            res = bin_op(res, a[x]);
            x += x & -x;
        }
        return res;
    }
    void modify(int x, T v) {
        while (x > 0) {
            a[x] = bin_op(a[x], v);
            x -= x & -x;
        }
    }
    void clear(){
        fill(a.begin(), a.end(), init);
    }
};

vi get_prime(int n) {
    vi minp((ul) n + 1), p;
    for (int i = 2; i <= n; i++) {
        if (!minp[i]) {
            minp[i] = i;
            p.pb(i);
        }
        FOR(x, p) {
            if (x <= minp[i] && x * i <= n)
                minp[x * i] = x;
            else break;
        }
    }
    return p;
}

// alias templates
template<typename T> using vv = vector<vector<T>>;
template <typename T1, typename T2 = T1> using vp = vector<pair<T1,T2>>;
template<typename T, int n> using va = vector<array<T,n>>;

#ifdef __GNUC__
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
template <typename T>
using rank_tree = __gnu_pbds::tree<
        T,
        __gnu_pbds::null_type,
        less<T>,
        __gnu_pbds::rb_tree_tag,
        __gnu_pbds::tree_order_statistics_node_update>;
#endif

//union-find 并查集
struct UF {
    vi par, sz;
    int n_tree;
    explicit UF(int n) { //0-indexed
        par.assign(n, 0);
        sz.assign(n, 1);
        rng (i, 0, n) par[i] = i;
        n_tree = n;
    }
    int n_cluster() const {
        return n_tree;
    }
    int size(int x) {
        return sz[root(x)];
    }
    int root(int x) {
        return x == par[x] ? x : par[x] = root(par[x]);
    }
    bool unite(int x, int y) {
        int rx = root(x), ry = root(y);
        if(rx != ry) {
            par[rx] = ry;
            --n_tree;
            sz[ry] += sz[rx];
            return true;
        }
        return false;
    }
};



using idx_t = long;
template <typename  T, typename Compare = std::less<T>>
struct ST{
    size_t n{}; // 0-indexed
    vv<T> a;
    ST()=default;
    template <typename ptr_t>
    ST(ptr_t beg, ptr_t end):n(end-beg){
        a.resize((size_t)h_bit(n)+1); // 注意：不能写成 h_bit(n)
        a[0].assign(beg, end);
        rng (i, 0, SZ(a)-1) {
            a[i+1].resize(n);
            rng(j, 0, n - (1 << i)) {
                a[i+1][j] = min(a[i][j], a[i][j+(1<<i)], Compare());
            }
            rng(j, n-(1 << i), n) {
                a[i+1][j] = a[i][j];
            }
        }
    }

    T query(idx_t l, idx_t r) { // l <=  r
        int i = h_bit(r - l + 1ul);
        return min(a[i][l], a[i][r+1-(1<<i)], Compare());
    }
};

vi get_popcnt(int n) {
    vi res((ul)n + 1);
    rng (i, 0, n) {
        if (i * 2 <= n) res[i * 2] = res[i];
        if ((i & 1) == 0) res[i + 1] = res[i] + 1;
    }
    return res;
}

vi get_mu(int n) {
    assert(n > 0);
    vi mu(n + 1);
    vi min_p(n + 1);
    vi prime;
    mu[1] = 1;
    rng (i, 2, n + 1) {
        if (!min_p[i]) {
            prime.pb(i);
            min_p[i] = i;
            mu[i] = -1;
        }
        FOR (p, prime) {
            if (p > min_p[i]) {
                break;
            }
            int t = p * i;
            if (t > n) break;
            min_p[t] = p;
            mu[t] = p == min_p[i] ? 0 : -mu[i];
        }
    }
    return mu;
}

template <typename num>
num fp (num x, ll n) { //fast power: hat off to quailty
    if(n < 0) {
        x = fp(x, mod - 2);
        n = -n;
    }
    num ans = 1;
    while(n){
        if(n&1) ans *= x;
        n /= 2;
        x *= x;
    }
    return ans;
}
template <typename num>
void bit_reverse_swap(vector<num> &a) {
    int n = SZ(a);
    for (int i = 1, j = n >> 1, k; i < n - 1; i++) {
        if (i < j) swap(a[i], a[j]);
        //tricky
        for (k = n >> 1; j >= k; j -= k, k >>= 1);    //inspect the highest "1"
        j += k;
    }
}
template <typename num>
void FFT(vector<num> &a, int type) {
    bit_reverse_swap(a);
    int n = SZ(a);
    for (int i = 2; i <= n; i *= 2) {
        const auto wi = fp(3, type * (mod - 1) / i); // i次单位根
        for (int j = 0; j < n; j += i) {
            num w(1);
            for (int k = j, h = i >> 1; k < j + h; k++) {
                auto t = w * a[k + h], u = a[k];
                a[k] = u + t;
                a[k + h] = u - t;
                w *= wi;
            }
        }
    }
    const auto inv = num(n).inv();
    if (type == -1) for (auto &x : a) x *= inv;
}
template <typename num>
void fp(vector<num> &a, const int n) {
    a.resize((ul)pow2((SZ(a)-1) * n + 1));
    FFT(a, 1);
    for(auto &x: a) x = fp(x, n);
    FFT(a, -1);
}
// DEBUG code by tourist
string to_string(const string& s) {
    return '"' + s + '"';
}

string to_string(const char* s) {
    return to_string((string) s);
}

string to_string(bool b) {
    return (b ? "true" : "false");
}

template <typename A, typename B>
string to_string(pair<A, B> p) {
    return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}
template <typename A>
string to_string(A v) {
    bool first = true;
    string res = "{";
    for (const auto &x : v) {
        if (!first) {
            res += ", ";
        }
        first = false;
        res += to_string(x);
    }
    res += "}";
    return res;
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
    cerr << " " << to_string(H);
    debug_out(T...);
}

#ifdef LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif
// end DEBUG

// minimal geometry classes
struct point {
    int x, y;

    void read() {
        scan(x, y);
    }

    ll cross(const point &b) const {
        return 1LL * x * b.y - 1LL * y * b.x;
    }

    point operator-(const point &b) const {
        return {x - b.x, y - b.y};
    }

    bool para(const point &b) const { // parallel
        return 1LL * x * b.y == 1LL * y * b.x;
    }

    bool operator<(const point &b) const {
        return cross(b) < 0;
    }
};

struct segment {
    point p, dir;

    bool on_same_line(const segment &other) const {
        return dir.para(other.dir) && (p - other.p).para(dir);
    }

    bool para(const segment &b) const {
        return dir.para(b.dir);
    }

    bool operator==(const segment &b) const {
        return this->on_same_line(b);
    }
};
struct fraction {
    int n, d;
    bool operator<(const fraction& that) const {
        return 1LL * n * that.d < 1LL * that.n * d;
    }
    bool operator==(const fraction& that) const {
        return n == that.n && d == that.d;
    }
    bool operator>=(const fraction& that) const {
        return !(*this < that);
    }
    bool operator>(const fraction& that) const {
        return 1LL * n * that.d > 1LL * that.n * d;
    }
    fraction inverse() const {
        return n < 0 ? fraction{-d, -n} : fraction{d, n};
    }
    fraction(int n, int d) {
        if (d < 0) n = -n, d = -d;
        int g = __gcd(abs(n), d);
        n /= g, d /= g;
        this->n = n, this->d = d;
    }
    fraction(int x): n(x), d(1) {}
    fraction operator+(const fraction& that) const {
        return fraction{n * that.d + that.n * d, d * that.d};
    }
    fraction operator-(const fraction& that) const {
        return fraction{n * that.d - that.n * d, d * that.d};
    }
    fraction floor() const {
        if (d == 0) return *this;
        return (n < 0 ? n - (d - 1) : n) / d;
    }
    friend ostream& operator<<(ostream& out, const fraction& f) {
        return out << "(" << f.n <<',' << f.d << ')';
    }
    pii to_pair() const {
        return {n, d};
    }
};
using num = modnum<mod>;
struct comb {
    vector<num> f;
    explicit comb(int n){
        f.resize(n + 1);
        f[0] = 1;
        up (i, 1, n) f[i] = f[i - 1] * i;
    }
    num get(int x, int y) const {
        assert(x <= SZ(f) - 1);
        assert(x >= 0 && y >= 0);
        if (x < y) return 0;
        return f[x] / f[y] / f[x - y];
    }
};

//////////////////^"^////////////////////////////////////

int main() {
    FAST_READ
    cout << fixed << setprecision(12);
#ifdef LOCAL
    ifstream in("main.in");
    cin.rdbuf(in.rdbuf());
//    ofstream out("main.out");
//    cout.rdbuf(out.rdbuf());
#endif

int n, m; scan(n, m);
vv<vi> g(n);
rng (i, 0, m) {
    int u, v, c;
    scan(u, v, c);
    --u, --v;
    g[u].pb({v, c, i});
    g[v].pb({u, c, i});
}

auto dijkstra = [&](int s, vi& d) {
    fill(all(d), INT_MAX);
    d[s] = 0;
    pq<pii, greater<>> que;
    que.emplace(0, s);

    while (!que.empty()) {
        auto top = que.top();
        que.pop();
        int u = top.second;
        if (d[u] != top.first) continue;
        FOR (e, g[u]) {
            int v = e[0], l = e[1];
            if (d[v] > d[u] + l) {
                d[v] = d[u] + l;
                que.emplace(d[v], v);
            }
        }
    }
};

vi d1(n), d2(n);

dijkstra(0, d1);
dijkstra(n - 1, d2);

vb flag(m);

int sp = d1[n - 1];
rng (i, 0, n) {
    FOR (e, g[i]) {
        flag[e[2]] = d1[i] + d2[e[0]] + e[1]  == sp || d1[e[0]] + d2[i] + e[1] == sp;
    }
}

rng (i, 0, m) println(flag[i] ? "YES" : "NO");







#ifdef LOCAL
    cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC << " s.\n";
//    fflush(stdout);
#endif
    return 0;
}