#include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { if (m < q) {m = q; return true;} else return false; } template bool chmin(T &m, const T q) { if (m > q) {m = q; return true;} else return false; } int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template pair operator+(const pair &l, const pair &r) { return make_pair(l.first + r.first, l.second + r.second); } template pair operator-(const pair &l, const pair &r) { return make_pair(l.first - r.first, l.second - r.second); } template vector sort_unique(vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template istream &operator>>(istream &is, vector &vec) { for (auto &v : vec) is >> v; return is; } template ostream &operator<<(ostream &os, const vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } #if __cplusplus >= 201703L template istream &operator>>(istream &is, tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template ostream &operator<<(ostream &os, const tuple &tpl) { std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os; } #endif template ostream &operator<<(ostream &os, const deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const pair &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; } template ostream &operator<<(ostream &os, const map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl #else #define dbg(x) (x) #endif template struct ModInt { #if __cplusplus >= 201402L #define MDCONST constexpr #else #define MDCONST #endif using lint = long long; MDCONST static int get_mod() { return mod; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set fac; int v = mod - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < mod; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((mod - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val; MDCONST ModInt() : val(0) {} MDCONST ModInt &_setval(lint v) { return val = (v >= mod ? v - mod : v), *this; } MDCONST ModInt(lint v) { _setval(v % mod + mod); } MDCONST explicit operator bool() const { return val != 0; } MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); } MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + mod); } MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); } MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); } MDCONST ModInt operator-() const { return ModInt()._setval(mod - val); } MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; } MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; } MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; } MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % mod + x.val); } friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); } friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); } friend MDCONST ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); } MDCONST bool operator==(const ModInt &x) const { return val == x.val; } MDCONST bool operator!=(const ModInt &x) const { return val != x.val; } MDCONST bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; } MDCONST ModInt pow(lint n) const { lint ans = 1, tmp = this->val; while (n) { if (n & 1) ans = ans * tmp % mod; tmp = tmp * tmp % mod, n /= 2; } return ans; } static std::vector facs, invs; MDCONST static void _precalculation(int N) { int l0 = facs.size(); if (N <= l0) return; facs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i % mod; long long facinv = ModInt(facs.back()).pow(mod - 2).val; for (int i = N - 1; i >= l0; i--) { invs[i] = facinv * facs[i - 1] % mod; facinv = facinv * i % mod; } } MDCONST lint inv() const { if (this->val < 1 << 20) { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val]; } else { return this->pow(mod - 2).val; } } MDCONST ModInt fac() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val]; } MDCONST ModInt doublefac() const { lint k = (this->val + 1) / 2; return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } MDCONST ModInt nCr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() / ((*this - r).fac() * r.fac()); } ModInt sqrt() const { if (val == 0) return 0; if (mod == 2) return val; if (pow((mod - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((mod - 1) / 2) == 1) b += 1; int e = 0, m = mod - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val, mod - x.val)); } }; template std::vector ModInt::facs = {1}; template std::vector ModInt::invs = {0}; using mint = ModInt<1000000007>; // Directed graph library to find strongly connected components (強連結成分分解) // 0-indexed directed graph // Complexity: O(V + E) struct DirectedGraphSCC { int V; // # of Vertices std::vector> to, from; std::vector used; // Only true/false std::vector vs; std::vector cmp; int scc_num = -1; DirectedGraphSCC(int V = 0) : V(V), to(V), from(V), cmp(V) {} void _dfs(int v) { used[v] = true; for (auto t : to[v]) if (!used[t]) _dfs(t); vs.push_back(v); } void _rdfs(int v, int k) { used[v] = true; cmp[v] = k; for (auto t : from[v]) if (!used[t]) _rdfs(t, k); } void add_edge(int from_, int to_) { assert(from_ >= 0 and from_ < V and to_ >= 0 and to_ < V); to[from_].push_back(to_); from[to_].push_back(from_); } // Detect strongly connected components and return # of them. // Also, assign each vertex `v` the scc id `cmp[v]` (0-indexed) int FindStronglyConnectedComponents() { used.assign(V, false); vs.clear(); for (int v = 0; v < V; v++) if (!used[v]) _dfs(v); used.assign(V, false); scc_num = 0; for (int i = (int)vs.size() - 1; i >= 0; i--) if (!used[vs[i]]) _rdfs(vs[i], scc_num++); return scc_num; } // Find and output the vertices that form a closed cycle. // output: {v_1, ..., v_C}, where C is the length of cycle, // {} if there's NO cycle (graph is DAG) int _c, _init; std::vector _ret_cycle; bool _dfs_detectcycle(int now, bool b0) { if (now == _init and b0) return true; for (auto nxt : to[now]) if (cmp[nxt] == _c and !used[nxt]) { _ret_cycle.emplace_back(nxt), used[nxt] = 1; if (_dfs_detectcycle(nxt, true)) return true; _ret_cycle.pop_back(); } return false; } std::vector DetectCycle() { int ns = FindStronglyConnectedComponents(); if (ns == V) return {}; std::vector cnt(ns); for (auto x : cmp) cnt[x]++; _c = std::find_if(cnt.begin(), cnt.end(), [](int x) { return x > 1; }) - cnt.begin(); _init = std::find(cmp.begin(), cmp.end(), _c) - cmp.begin(); used.assign(V, false); _ret_cycle.clear(); _dfs_detectcycle(_init, false); return _ret_cycle; } // After calling `FindStronglyConnectedComponents()`, generate a new graph by uniting all vertices // belonging to the same component(The resultant graph is DAG). DirectedGraphSCC GenerateTopologicalGraph() { DirectedGraphSCC newgraph(scc_num); for (int s = 0; s < V; s++) for (auto t : to[s]) { if (cmp[s] != cmp[t]) newgraph.add_edge(cmp[s], cmp[t]); } return newgraph; } }; // 2-SAT solver: Find a solution for `(Ai v Aj) ^ (Ak v Al) ^ ... = true` // - `nb_sat_vars`: Number of variables // - Considering a graph with `2 * nb_sat_vars` vertices // - Vertices [0, nb_sat_vars) means `Ai` // - vertices [nb_sat_vars, 2 * nb_sat_vars) means `not Ai` struct SATSolver : DirectedGraphSCC { int nb_sat_vars; std::vector solution; SATSolver(int nb_variables = 0) : DirectedGraphSCC(nb_variables * 2), nb_sat_vars(nb_variables), solution(nb_sat_vars) {} void add_x_or_y_constraint(bool is_x_true, int x, bool is_y_true, int y) { assert(x >= 0 and x < nb_sat_vars); assert(y >= 0 and y < nb_sat_vars); if (!is_x_true) x += nb_sat_vars; if (!is_y_true) y += nb_sat_vars; add_edge((x + nb_sat_vars) % (nb_sat_vars * 2), y); add_edge((y + nb_sat_vars) % (nb_sat_vars * 2), x); } // Solve the 2-SAT problem. If no solution exists, return `false`. // Otherwise, dump one solution to `solution` and return `true`. bool run() { FindStronglyConnectedComponents(); for (int i = 0; i < nb_sat_vars; i++) { if (cmp[i] == cmp[i + nb_sat_vars]) return false; solution[i] = cmp[i] > cmp[i + nb_sat_vars]; } return true; } }; template struct ShortestPath { int V, E; int INVALID = -1; std::vector>> to; ShortestPath() = default; ShortestPath(int V) : V(V), E(0), to(V) {} void add_edge(int s, int t, T len) { assert(0 <= s and s < V); assert(0 <= t and t < V); to[s].emplace_back(t, len); E++; } std::vector dist; std::vector prev; // Dijkstra algorithm // Complexity: O(E log E) void Dijkstra(int s) { assert(0 <= s and s < V); dist.assign(V, std::numeric_limits::max()); dist[s] = 0; prev.assign(V, INVALID); using P = std::pair; std::priority_queue, std::greater

> pq; pq.emplace(0, s); while (!pq.empty()) { T d; int v; std::tie(d, v) = pq.top(); pq.pop(); if (dist[v] < d) continue; for (auto nx : to[v]) { T dnx = d + nx.second; if (dist[nx.first] > dnx) { dist[nx.first] = dnx, prev[nx.first] = v; pq.emplace(dnx, nx.first); } } } } // Bellman-Ford algorithm // Complexity: O(VE) bool BellmanFord(int s, int nb_loop) { assert(0 <= s and s < V); dist.assign(V, std::numeric_limits::max()); dist[s] = 0; prev.assign(V, INVALID); for (int l = 0; l < nb_loop; l++) { bool upd = false; for (int v = 0; v < V; v++) { if (dist[v] == std::numeric_limits::max()) continue; for (auto nx : to[v]) { T dnx = dist[v] + nx.second; if (dist[nx.first] > dnx) { dist[nx.first] = dnx, prev[nx.first] = v; upd = true; } } } if (!upd) return true; } return false; } void ZeroOneBFS(int s) { assert(0 <= s and s < V); dist.assign(V, std::numeric_limits::max()); dist[s] = 0; prev.assign(V, INVALID); std::deque que; que.push_back(s); while (!que.empty()) { int v = que.front(); que.pop_front(); for (auto nx : to[v]) { T dnx = dist[v] + nx.second; if (dist[nx.first] > dnx) { dist[nx.first] = dnx, prev[nx.first] = v; if (nx.second) { que.push_back(nx.first); } else { que.push_front(nx.first); } } } } } // Warshall-Floyd algorithm // Complexity: O(E + V^3) std::vector> dist2d; void WarshallFloyd() { dist2d.assign(V, std::vector(V, std::numeric_limits::max())); for (int i = 0; i < V; i++) { dist2d[i][i] = 0; for (auto p : to[i]) dist2d[i][p.first] = min(dist2d[i][p.first], p.second); } for (int k = 0; k < V; k++) { for (int i = 0; i < V; i++) { if (dist2d[i][k] = std::numeric_limits::max()) continue; for (int j = 0; j < V; j++) { if (dist2d[k][j] = std::numeric_limits::max()) continue; dist2d[i][j] = min(dist2d[i][j], dist2d[i][k] + dist2d[k][j]); } } } } }; int main() { int N, M; cin >> N >> M; N++; vector>>> to(N); DirectedGraphSCC graph(N); ShortestPath g(N), ginv(N); while (M--) { lint u, v, l, a; cin >> u >> v >> l >> a; graph.add_edge(u, v); to[u].emplace_back(v, make_pair(l, a)); g.add_edge(u, v, 0); ginv.add_edge(v, u, 0); } graph.FindStronglyConnectedComponents(); vector ord(N); REP(i, N) ord[i] = make_pair(graph.cmp[i], i); sort(ord.begin(), ord.end()); vector dpcnt(N); vector dptot(N); dpcnt[0] = 1; vector arrive(N); arrive[0] = 1; for (auto [_, i] : ord) if (arrive[i]) { for (auto [j, la] : to[i]) { arrive[j] = true; dptot[j] += dptot[i] * la.second + dpcnt[i] * la.first * la.second; dpcnt[j] += dpcnt[i] * la.second; } } dbg(dptot); dbg(dpcnt); dbg(ord); g.ZeroOneBFS(0); ginv.ZeroOneBFS(N - 1); vector cmpsz(N); REP(i, N) cmpsz[graph.cmp[i]]++; REP(i, N) if (!g.dist[i] and !ginv.dist[i] and cmpsz[graph.cmp[i]] > 1) { puts("INF"); return 0; } cout << dptot.back() << '\n'; }