/** * author: nok0 * created: 2021.04.09 23:47:15 **/ #ifdef LOCAL #define _GLIBCXX_DEBUG #endif #include using namespace std; #if __has_include() #include using namespace atcoder; #endif #pragma region Macros // rep macro #define foa(v, a) for(auto &v : a) #define REPname(a, b, c, d, e, ...) e #define REP(...) REPname(__VA_ARGS__, REP3, REP2, REP1, REP0)(__VA_ARGS__) #define REP0(x) for(int i = 0; i < (x); ++i) #define REP1(i, x) for(int i = 0; i < (x); ++i) #define REP2(i, l, r) for(int i = (l); i < (r); ++i) #define REP3(i, l, r, c) for(int i = (l); i < (r); i += (c)) #define REPSname(a, b, c, ...) c #define REPS(...) REPSname(__VA_ARGS__, REPS1, REPS0)(__VA_ARGS__) #define REPS0(x) for(int i = 1; i <= (x); ++i) #define REPS1(i, x) for(int i = 1; i <= (x); ++i) #define RREPname(a, b, c, d, e, ...) e #define RREP(...) RREPname(__VA_ARGS__, RREP3, RREP2, RREP1, RREP0)(__VA_ARGS__) #define RREP0(x) for(int i = (x)-1; i >= 0; --i) #define RREP1(i, x) for(int i = (x)-1; i >= 0; --i) #define RREP2(i, r, l) for(int i = (r)-1; i >= (l); --i) #define RREP3(i, r, l, c) for(int i = (r)-1; i >= (l); i -= (c)) #define RREPSname(a, b, c, ...) c #define RREPS(...) RREPSname(__VA_ARGS__, RREPS1, RREPS0)(__VA_ARGS__) #define RREPS0(x) for(int i = (x); i >= 1; --i) #define RREPS1(i, x) for(int i = (x); i >= 1; --i) // name macro #define pb push_back #define eb emplace_back #define SZ(x) ((int)(x).size()) #define all(x) (x).begin(), (x).end() #define rall(x) (x).rbegin(), (x).rend() #define popcnt(x) __builtin_popcountll(x) template using V = std::vector; template using VV = std::vector>; template using pqup = std::priority_queue, std::greater>; using ll = long long; using ld = long double; using int128 = __int128_t; using pii = std::pair; using pll = std::pair; // input macro template std::istream &operator>>(std::istream &is, std::pair &p) { is >> p.first >> p.second; return is; } template std::istream &operator>>(std::istream &is, std::vector &v) { for(T &i : v) is >> i; return is; } std::istream &operator>>(std::istream &is, __int128_t &a) { std::string s; is >> s; __int128_t ret = 0; for(int i = 0; i < s.length(); i++) if('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0'; a = ret * (s[0] == '-' ? -1 : 1); return is; } #if __has_include() std::istream &operator>>(std::istream &is, atcoder::modint998244353 &a) { long long v; is >> v; a = v; return is; } std::istream &operator>>(std::istream &is, atcoder::modint1000000007 &a) { long long v; is >> v; a = v; return is; } template std::istream &operator>>(std::istream &is, atcoder::static_modint &a) { long long v; is >> v; a = v; return is; } template std::istream &operator>>(std::istream &is, atcoder::dynamic_modint &a) { long long v; is >> v; a = v; return is; } #endif namespace scanner { void scan(int &a) { std::cin >> a; } void scan(long long &a) { std::cin >> a; } void scan(std::string &a) { std::cin >> a; } void scan(char &a) { std::cin >> a; } void scan(char a[]) { std::scanf("%s", a); } void scan(double &a) { std::cin >> a; } void scan(long double &a) { std::cin >> a; } template void scan(std::pair &p) { std::cin >> p; } template void scan(std::vector &a) { std::cin >> a; } void INPUT() {} template void INPUT(Head &head, Tail &... tail) { scan(head); INPUT(tail...); } } // namespace scanner #define VEC(type, name, size) \ std::vector name(size); \ scanner::INPUT(name) #define VVEC(type, name, h, w) \ std::vector> name(h, std::vector(w)); \ scanner::INPUT(name) #define INT(...) \ int __VA_ARGS__; \ scanner::INPUT(__VA_ARGS__) #define LL(...) \ long long __VA_ARGS__; \ scanner::INPUT(__VA_ARGS__) #define STR(...) \ std::string __VA_ARGS__; \ scanner::INPUT(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ scanner::INPUT(__VA_ARGS__) #define DOUBLE(...) \ double __VA_ARGS__; \ scanner::INPUT(__VA_ARGS__) #define LD(...) \ long double __VA_ARGS__; \ scanner::INPUT(__VA_ARGS__) // output-macro template std::ostream &operator<<(std::ostream &os, const std::pair &p) { os << p.first << " " << p.second; return os; } template std::ostream &operator<<(std::ostream &os, const std::vector &a) { for(int i = 0; i < int(a.size()); ++i) { if(i) os << " "; os << a[i]; } return os; } std::ostream &operator<<(std::ostream &dest, __int128_t &value) { std::ostream::sentry s(dest); if(s) { __uint128_t tmp = value < 0 ? -value : value; char buffer[128]; char *d = std::end(buffer); do { --d; *d = "0123456789"[tmp % 10]; tmp /= 10; } while(tmp != 0); if(value < 0) { --d; *d = '-'; } int len = std::end(buffer) - d; if(dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); } } return dest; } #if __has_include() std::ostream &operator<<(std::ostream &os, const atcoder::modint998244353 &a) { return os << a.val(); } std::ostream &operator<<(std::ostream &os, const atcoder::modint1000000007 &a) { return os << a.val(); } template std::ostream &operator<<(std::ostream &os, const atcoder::static_modint &a) { return os << a.val(); } template std::ostream &operator<<(std::ostream &os, const atcoder::dynamic_modint &a) { return os << a.val(); } #endif template void print(const T a) { std::cout << a << '\n'; } template void print(Head H, Tail... T) { std::cout << H << ' '; print(T...); } template void printel(const T a) { std::cout << a << '\n'; } template void printel(const std::vector &a) { for(const auto &v : a) std::cout << v << '\n'; } template void printel(Head H, Tail... T) { std::cout << H << '\n'; printel(T...); } void Yes(const bool b = true) { std::cout << (b ? "Yes\n" : "No\n"); } void No() { std::cout << "No\n"; } void YES(const bool b = true) { std::cout << (b ? "YES\n" : "NO\n"); } void NO() { std::cout << "NO\n"; } void err(const bool b = true) { if(b) { std::cout << "-1\n", exit(0); } } //debug macro namespace debugger { template void view(const std::vector &a) { std::cerr << "{ "; for(const auto &v : a) { std::cerr << v << ", "; } std::cerr << "\b\b }"; } template void view(const std::vector> &a) { std::cerr << "{\n"; for(const auto &v : a) { std::cerr << "\t"; view(v); std::cerr << "\n"; } std::cerr << "}"; } template void view(const std::vector> &a) { std::cerr << "{\n"; for(const auto &p : a) std::cerr << "\t(" << p.first << ", " << p.second << ")\n"; std::cerr << "}"; } template void view(const std::map &m) { std::cerr << "{\n"; for(const auto &p : m) std::cerr << "\t[" << p.first << "] : " << p.second << "\n"; std::cerr << "}"; } template void view(const std::pair &p) { std::cerr << "(" << p.first << ", " << p.second << ")"; } template void view(const std::set &s) { std::cerr << "{ "; for(auto &v : s) { view(v); std::cerr << ", "; } std::cerr << "\b\b }"; } template void view(const T &e) { std::cerr << e; } } // namespace debugger #ifdef LOCAL void debug_out() {} template void debug_out(Head H, Tail... T) { debugger::view(H); std::cerr << ", "; debug_out(T...); } #define debug(...) \ do { \ std::cerr << __LINE__ << " [" << #__VA_ARGS__ << "] : ["; \ debug_out(__VA_ARGS__); \ std::cerr << "\b\b]\n"; \ } while(false) #else #define debug(...) (void(0)) #endif // vector macro template int lb(const std::vector &a, const T x) { return std::distance((a).begin(), std::lower_bound((a).begin(), (a).end(), (x))); } template int ub(const std::vector &a, const T x) { return std::distance((a).begin(), std::upper_bound((a).begin(), (a).end(), (x))); } template void UNIQUE(std::vector &a) { std::sort(a.begin(), a.end()); a.erase(std::unique(a.begin(), a.end()), a.end()); } template std::vector press(std::vector &a) { auto res = a; UNIQUE(res); for(auto &v : a) v = lb(res, v); return res; } #define SORTname(a, b, c, ...) c #define SORT(...) SORTname(__VA_ARGS__, SORT1, SORT0, ...)(__VA_ARGS__) #define SORT0(a) std::sort((a).begin(), (a).end()) #define SORT1(a, c) std::sort((a).begin(), (a).end(), [](const auto x, const auto y) { return x c y; }) template void ADD(std::vector &a, const T x) { for(auto &v : a) v += x; } template void SUB(std::vector &a, const T x = 1) { for(auto &v : a) v -= x; } template void MUL(std::vector &a, const T x) { for(auto &v : a) v *= x; } template void DIV(std::vector &a, const T x) { for(auto &v : a) v /= x; } // math macro template inline bool chmin(T &a, const U &b) { return a > b ? a = b, true : false; } template inline bool chmax(T &a, const U &b) { return a < b ? a = b, true : false; } template T divup(T x, T y) { return (x + y - 1) / y; } template T POW(T a, long long n) { T ret = 1; while(n) { if(n & 1) ret *= a; a *= a; n >>= 1; } return ret; } // modpow long long POW(long long a, long long n, const int mod) { long long ret = 1; while(n) { if(n & 1) (ret *= a) %= mod; (a *= a) %= mod; n >>= 1; } return ret; } // others struct fast_io { fast_io() { ios::sync_with_stdio(false); cin.tie(nullptr); cout << fixed << setprecision(15); } } fast_io_; const int inf = 1e9; const ll INF = 1e18; #pragma endregion #pragma region Graph Graph #include #include #include #include #include #include struct Edge { int to; long long cost; Edge() = default; Edge(int to_, long long cost_) : to(to_), cost(cost_) {} bool operator<(const Edge &a) const { return cost < a.cost; } bool operator>(const Edge &a) const { return cost > a.cost; } friend std::ostream &operator<<(std::ostream &s, Edge &a) { s << "to: " << a.to << ", cost: " << a.cost; return s; } }; class Graph { std::vector> edges; public: inline const std::vector &operator[](int k) const { return edges[k]; } inline std::vector &operator[](int k) { return edges[k]; } int size() const { return edges.size(); } void resize(const int n) { edges.resize(n); } Graph() = default; Graph(int n) : edges(n) {} Graph(int n, int e, bool weight = 0, bool directed = 0, int idx = 1) : edges(n) { input(e, weight, directed, idx); } const long long INF = 3e18; void input(int e = -1, bool weight = 0, bool directed = false, int idx = 1) { if(e == -1) e = size() - 1; while(e--) { int u, v; long long cost = 1; std::cin >> u >> v; if(weight) std::cin >> cost; u -= idx, v -= idx; edges[u].emplace_back(v, cost); if(!directed) edges[v].emplace_back(u, cost); } } void add_edge(int u, int v, long long cost = 1, bool directed = false, int idx = 0) { u -= idx, v -= idx; edges[u].emplace_back(v, cost); if(!directed) edges[v].emplace_back(u, cost); } // Ο(V+E) std::vector bfs(int s) { std::vector dist(size(), INF); std::queue que; dist[s] = 0; que.push(s); while(!que.empty()) { int v = que.front(); que.pop(); for(auto &e : edges[v]) { if(dist[e.to] != INF) continue; dist[e.to] = dist[v] + e.cost; que.push(e.to); } } return dist; } // Ο(V+E) // constraint: cost of each edge is zero or one std::vector zero_one_bfs(int s) { std::vector dist(size(), INF); std::deque deq; dist[s] = 0; deq.push_back(s); while(!deq.empty()) { int v = deq.front(); deq.pop_front(); for(auto &e : edges[v]) { assert(0LL <= e.cost and e.cost < 2LL); if(e.cost and dist[e.to] > dist[v] + 1) { dist[e.to] = dist[v] + 1; deq.push_back(e.to); } else if(!e.cost and dist[e.to] > dist[v]) { dist[e.to] = dist[v]; deq.push_front(e.to); } } } return dist; } // Ο((E+V)logV) // cannot reach: INF std::vector dijkstra(int s) { // verified std::vector dist(size(), INF); const auto compare = [](const std::pair &a, const std::pair &b) { return a.first > b.first; }; std::priority_queue, std::vector>, decltype(compare)> que{compare}; dist[s] = 0; que.emplace(0, s); while(!que.empty()) { std::pair p = que.top(); que.pop(); int v = p.second; if(dist[v] < p.first) continue; for(auto &e : edges[v]) { if(dist[e.to] > dist[v] + e.cost) { dist[e.to] = dist[v] + e.cost; que.emplace(dist[e.to], e.to); } } } return dist; } // Ο(VE) // cannot reach: INF // negative cycle: -INF std::vector bellman_ford(int s) { // verified int n = size(); std::vector res(n, INF); res[s] = 0; for(int loop = 0; loop < n - 1; loop++) { for(int v = 0; v < n; v++) { if(res[v] == INF) continue; for(auto &e : edges[v]) { res[e.to] = std::min(res[e.to], res[v] + e.cost); } } } std::queue que; std::vector chk(n); for(int v = 0; v < n; v++) { if(res[v] == INF) continue; for(auto &e : edges[v]) { if(res[e.to] > res[v] + e.cost and !chk[e.to]) { que.push(e.to); chk[e.to] = 1; } } } while(!que.empty()) { int now = que.front(); que.pop(); for(auto &e : edges[now]) { if(!chk[e.to]) { chk[e.to] = 1; que.push(e.to); } } } for(int i = 0; i < n; i++) if(chk[i]) res[i] = -INF; return res; } // Ο(V^3) std::vector> warshall_floyd() { // verified int n = size(); std::vector> dist(n, std::vector(n, INF)); for(int i = 0; i < n; i++) dist[i][i] = 0; for(int i = 0; i < n; i++) for(auto &e : edges[i]) dist[i][e.to] = std::min(dist[i][e.to], e.cost); for(int k = 0; k < n; k++) for(int i = 0; i < n; i++) { if(dist[i][k] == INF) continue; for(int j = 0; j < n; j++) { if(dist[k][j] == INF) continue; dist[i][j] = std::min(dist[i][j], dist[i][k] + dist[k][j]); } } return dist; } // Ο(V) (using DFS) // if a directed cycle exists, return {} std::vector topological_sort() { // verified std::vector res; int n = size(); std::vector used(n, 0); bool not_DAG = false; auto dfs = [&](auto self, int k) -> void { if(not_DAG) return; if(used[k]) { if(used[k] == 1) not_DAG = true; return; } used[k] = 1; for(auto &e : edges[k]) self(self, e.to); used[k] = 2; res.push_back(k); }; for(int i = 0; i < n; i++) dfs(dfs, i); if(not_DAG) return std::vector{}; std::reverse(res.begin(), res.end()); return res; } bool is_DAG() { return !topological_sort().empty(); } // verified // Ο(V) // array of the distance from each vertex to the most distant vertex std::vector height() { // verified auto vec1 = bfs(0); int v1 = -1, v2 = -1; long long dia = -1; for(int i = 0; i < int(size()); i++) if(dia < vec1[i]) dia = vec1[i], v1 = i; vec1 = bfs(v1); dia = -1; for(int i = 0; i < int(size()); i++) if(dia < vec1[i]) dia = vec1[i], v2 = i; auto vec2 = bfs(v2); for(int i = 0; i < int(size()); i++) { if(vec1[i] < vec2[i]) vec1[i] = vec2[i]; } return vec1; } // O(V+E) // vector<(int)(0 or 1)> // if it is not bipartite, return {} std::vector bipartite_grouping() { std::vector colors(size(), -1); auto dfs = [&](auto self, int now, int col) -> bool { colors[now] = col; for(auto &e : edges[now]) { if(col == colors[e.to]) return false; if(colors[e.to] == -1 and !self(self, e.to, !col)) return false; } return true; }; for(int i = 0; i < int(size()); i++) if(!colors[i] and !dfs(dfs, i, 0)) return std::vector{}; return colors; } bool is_bipartite() { return !bipartite_grouping().empty(); } // Ο(V+E) // ((v1, v2), diameter) std::pair, long long> diameter() { // verified auto vec = bfs(0); int v1 = -1, v2 = -1; long long dia = -1; for(int i = 0; i < int(size()); i++) if(dia < vec[i]) dia = vec[i], v1 = i; vec = bfs(v1); dia = -1; for(int i = 0; i < int(size()); i++) if(dia < vec[i]) dia = vec[i], v2 = i; std::pair, long long> res = {{v1, v2}, dia}; return res; } // Ο(ElogV) long long prim() { // verified long long res = 0; std::priority_queue, std::greater> que; for(auto &e : edges[0]) que.push(e); std::vector chk(size()); chk[0] = 1; int cnt = 1; while(cnt < size()) { auto e = que.top(); que.pop(); if(chk[e.to]) continue; cnt++; res += e.cost; chk[e.to] = 1; for(auto &e2 : edges[e.to]) que.push(e2); } return res; } // Ο(ElogE) long long kruskal() { // verified std::vector> Edges; for(int i = 0; i < int(size()); i++) for(auto &e : edges[i]) Edges.emplace_back(i, e.to, e.cost); std::sort(Edges.begin(), Edges.end(), [](const std::tuple &a, const std::tuple &b) { return std::get<2>(a) < std::get<2>(b); }); std::vector uf_data(size(), -1); auto root = [&uf_data](auto self, int x) -> int { if(uf_data[x] < 0) return x; return uf_data[x] = self(self, uf_data[x]); }; auto unite = [&uf_data, &root](int u, int v) -> bool { u = root(root, u), v = root(root, v); if(u == v) return false; if(uf_data[u] > uf_data[v]) std::swap(u, v); uf_data[u] += uf_data[v]; uf_data[v] = u; return true; }; long long ret = 0; for(auto &e : Edges) if(unite(std::get<0>(e), std::get<1>(e))) ret += std::get<2>(e); return ret; } // O(V) std::vector centroid() { int n = size(); std::vector centroid, sz(n); auto dfs = [&](auto self, int now, int per) -> void { sz[now] = 1; bool is_centroid = true; for(auto &e : edges[now]) { if(e.to != per) { self(self, e.to, now); sz[now] += sz[e.to]; if(sz[e.to] > n / 2) is_centroid = false; } } if(n - sz[now] > n / 2) is_centroid = false; if(is_centroid) centroid.push_back(now); }; dfs(dfs, 0, -1); return centroid; } // Ο(V+E) // directed graph from root to leaf Graph root_to_leaf(int root = 0) { Graph res(size()); std::vector chk(size(), 0); chk[root] = 1; auto dfs = [&](auto self, int now) -> void { for(auto &e : edges[now]) { if(chk[e.to] == 1) continue; chk[e.to] = 1; res.add_edge(now, e.to, e.cost, 1, 0); self(self, e.to); } }; dfs(dfs, root); return res; } // Ο(V+E) // directed graph from leaf to root Graph leaf_to_root(int root = 0) { Graph res(size()); std::vector chk(size(), 0); chk[root] = 1; auto dfs = [&](auto self, int now) -> void { for(auto &e : edges[now]) { if(chk[e.to] == 1) continue; chk[e.to] = 1; res.add_edge(e.to, now, e.cost, 1, 0); self(self, e.to); } }; dfs(dfs, root); return res; } // long long Chu_Liu_Edmonds(int root = 0) {} }; struct tree_doubling { private: std::vector> parent; std::vector depth; std::vector dist; int max_jump = 1; void build() { for(int i = 0; i < max_jump - 1; i++) { for(int v = 0; v < (int)dist.size(); v++) { if(parent[i][v] == -1) parent[i + 1][v] = -1; else parent[i + 1][v] = parent[i][parent[i][v]]; } } } public: tree_doubling() = default; tree_doubling(const Graph &g, const int root = 0) : dist(g.size()), depth(g.size()) { int n = g.size(); while((1 << max_jump) < n) max_jump++; parent.assign(max_jump, std::vector(n, -1)); auto dfs = [&](auto self, int now, int per, int d, long long cost) -> void { parent[0][now] = per; depth[now] = d; dist[now] = cost; for(auto &e : g[now]) if(e.to != per) self(self, e.to, now, d + 1, cost + e.cost); }; dfs(dfs, root, -1, 0, 0LL); build(); } int lowest_common_ancestor(int u, int v) { if(depth[u] < depth[v]) std::swap(u, v); int k = parent.size(); for(int i = 0; i < k; i++) if((depth[u] - depth[v]) >> i & 1) u = parent[i][u]; if(u == v) return u; for(int i = k - 1; i >= 0; i--) if(parent[i][u] != parent[i][v]) u = parent[i][u], v = parent[i][v]; return parent[0][u]; } long long length_of_path(const int u, const int v) { return dist[u] + dist[v] - dist[lowest_common_ancestor(u, v)] * 2; } int level_ancestor(int v, int level) { assert(level >= 0); for(int jump = 0; jump < max_jump and level; jump++) { if(level & 1) v = parent[jump][v]; level >>= 1; } return v; } }; struct strongly_connected_components { private: enum { CHECKED = -1, UNCHECKED = -2 }; const Graph &graph_given; Graph graph_reversed; std::vector order, group_number; /* at the beginning of the building, 'group_number' is used as 'checked' */ void dfs(int now) { if(group_number[now] != UNCHECKED) return; group_number[now] = CHECKED; for(auto &e : graph_given[now]) dfs(e.to); order.push_back(now); } void rdfs(int now, int group_count) { if(group_number[now] != UNCHECKED) return; group_number[now] = group_count; for(auto &e : graph_reversed[now]) rdfs(e.to, group_count); } void build(bool create_compressed_graph) { for(int i = 0; i < (int)graph_given.size(); i++) dfs(i); reverse(order.begin(), order.end()); group_number.assign(graph_given.size(), UNCHECKED); int group = 0; for(auto &i : order) if(group_number[i] == UNCHECKED) rdfs(i, group), group++; graph_compressed.resize(group); groups.resize(group); for(int i = 0; i < (int)graph_given.size(); i++) groups[group_number[i]].push_back(i); if(create_compressed_graph) { std::vector edges(group, -1); for(int i = 0; i < group; i++) for(auto &vertex : groups[i]) for(auto &e : graph_given[vertex]) if(group_number[e.to] != i and edges[group_number[e.to]] != i) { edges[group_number[e.to]] = i; graph_compressed[i].emplace_back(group_number[e.to], 1); } } return; } public: std::vector> groups; Graph graph_compressed; strongly_connected_components(const Graph &g_, bool create_compressed_graph = false) : graph_given(g_), graph_reversed(g_.size()), group_number(g_.size(), UNCHECKED) { for(size_t i = 0; i < g_.size(); i++) for(auto &e : graph_given[i]) graph_reversed[e.to].emplace_back(i, 1); build(create_compressed_graph); } const int &operator[](const int k) { return group_number[k]; } }; struct low_link { private: const Graph &graph_given; int order_next; void build() { int n = graph_given.size(); order.resize(n, -1); low.resize(n); order_next = 0; for(int i = 0; i < n; i++) if(order[i] == -1) dfs(i); } void dfs(int now, int par = -1) { low[now] = order[now] = order_next++; bool is_articulation = false; int cnt = 0, cnt_par = 0; for(const auto &ed : graph_given[now]) { const int &nxt = ed.to; if(order[nxt] == -1) { cnt++; dfs(nxt, now); if(order[now] < low[nxt]) bridge.push_back(std::minmax(now, nxt)); if(order[now] <= low[nxt]) is_articulation = true; low[now] = std::min(low[now], low[nxt]); } else if(nxt != par or cnt_par++ == 1) { low[now] = std::min(low[now], order[nxt]); } } if(par == -1 and cnt < 2) is_articulation = false; if(is_articulation) articulation.push_back(now); return; } public: std::vector order, low, articulation; std::vector> bridge; low_link() = default; low_link(const Graph &g_) : graph_given(g_) { build(); } }; struct two_edge_connected_components { private: const Graph &graph_given; int group_next; low_link li; std::vector group_number; void build(bool create_compressed_graph) { int n = graph_given.size(); group_number.resize(n, -1); group_next = 0; for(int i = 0; i < n; i++) if(group_number[i] == -1) dfs(i); groups.resize(group_next); for(int i = 0; i < graph_given.size(); i++) groups[group_number[i]].push_back(i); if(create_compressed_graph) { graph_compressed.resize(group_next); for(const auto &[u, v] : li.bridge) { int x = group_number[u], y = group_number[v]; graph_compressed.add_edge(x, y); } } } void dfs(int now, int par = -1) { if(par != -1 and li.order[par] >= li.low[now]) group_number[now] = group_number[par]; else group_number[now] = group_next++; for(const auto &e : graph_given[now]) if(group_number[e.to] == -1) dfs(e.to, now); } public: Graph graph_compressed; std::vector> groups; two_edge_connected_components(const Graph &g_, bool create_compressed_graph = false) : graph_given(g_), li(g_) { build(create_compressed_graph); } const int &operator[](const int k) { return group_number[k]; } }; struct heavy_light_decomposition { public: std::vector sz, in, out, head, rev, par; private: Graph &g; void dfs_sz(int v, int p = -1) { par[v] = p; if(!g[v].empty() and g[v].front().to == p) std::swap(g[v].front(), g[v].back()); for(auto &e : g[v]) { if(e.to == p) continue; dfs_sz(e.to, v); sz[v] += sz[e.to]; if(sz[g[v].front().to] < sz[e.to]) std::swap(g[v].front(), e); } } void dfs_hld(int v, int &t, int p = -1) { in[v] = t++; rev[in[v]] = v; for(auto &e : g[v]) { if(e.to == p) continue; head[e.to] = (g[v].front().to == e.to ? head[v] : e.to); dfs_hld(e.to, t, v); } out[v] = t; } void build(int root = 0) { dfs_sz(root); int t = 0; head[root] = root; dfs_hld(root, t); } public: heavy_light_decomposition(Graph &g_, int root = 0) : g(g_) { int n = g.size(); sz.resize(n, 1); in.resize(n); out.resize(n); head.resize(n); rev.resize(n); par.resize(n); build(root); } int level_ancestor(int v, int level) { while(true) { int u = head[v]; if(in[v] - level >= in[u]) return rev[in[v] - level]; level -= in[v] - in[u] + 1; v = par[u]; } } int lowest_common_ancestor(int u, int v) { for(;; v = par[head[v]]) { if(in[u] > in[v]) std::swap(u, v); if(head[u] == head[v]) return u; } } // u, v: vertex, unit: unit, q: query on a path, f: binary operation ((T, T) -> T) template T query(int u, int v, const T &unit, const Q &q, const F &f, bool edge = false) { T l = unit, r = unit; for(;; v = par[head[v]]) { if(in[u] > in[v]) std::swap(u, v), std::swap(l, r); if(head[u] == head[v]) break; l = f(q(in[head[v]], in[v] + 1), l); } return f(f(q(in[u] + edge, in[v] + 1), l), r); } // u, v: vertex, q: update query template void add(int u, int v, const Q &q, bool edge = false) { for(;; v = par[head[v]]) { if(in[u] > in[v]) std::swap(u, v); if(head[u] == head[v]) break; q(in[head[v]], in[v] + 1); } q(in[u] + edge, in[v] + 1); } std::pair subtree(int v, bool edge = false) { return std::pair(in[v] + edge, out[v]); } }; #pragma endregion using mint = modint998244353; mint cycle_res, noncycle_res; int cycle_vertex_cnt; void main_() { INT(n, k); ll u = k; k = mint(1 - k).inv().val(); Graph G(n, n); vector is_cycle(n); { //1 途中 2 終わった vector seen(n), finished(n); int pos = -1; stack hist; auto dfs = [&](auto self, int now, int per) -> void { seen[now] = 1; hist.push(now); for(auto &[to, cost] : G[now]) { if(to == per) continue; if(finished[to]) continue; if(seen[to] and !finished[to]) { pos = to; return; } self(self, to, now); if(pos != -1) return; } hist.pop(); finished[now] = 1; }; dfs(dfs, 0, -1); while(!hist.empty()) { int t = hist.top(); is_cycle[t] = 1; hist.pop(); if(t == pos) break; } } REP(i, n) { cycle_vertex_cnt += is_cycle[i]; } vector tree_size(n); { auto dfs = [&](auto self, int now, int p) -> void { for(auto &[to, cost] : G[now]) if(to != p and !is_cycle[to]) { self(self, to, now); tree_size[now] += tree_size[to]; } tree_size[now]++; }; REP(i, n) { if(is_cycle[i]) dfs(dfs, i, -1); } } debug(is_cycle); debug(tree_size); cycle_res = mint(k).pow(n - cycle_vertex_cnt) * cycle_vertex_cnt; { auto dfs = [&](auto self, int now, int per) -> void { for(auto &[to, cost] : G[now]) { if(to == per or is_cycle[to]) continue; cycle_res += (mint(k).pow(n - cycle_vertex_cnt - tree_size[to])) * (mint(k).pow(tree_size[to]) - 1); self(self, to, now); } }; REP(i, n) { dfs(dfs, i, -1); } } cycle_res *= mint(k).pow(cycle_vertex_cnt); debug(cycle_res); vector g; { queue> que; for(int i = 0; i < n; i++) { if(is_cycle[i]) { vector fp(2); fp[0] = 1; fp[1] = POW(k, tree_size[i], 998244353) - 1; que.push(fp); } } while(que.size() > 1) { auto p = que.front(); que.pop(); auto q = que.front(); que.pop(); p = convolution(p, q); que.push(p); } for(auto &v : que.front()) g.push_back(v); } map> mf; { set is_seen; REP(i, n) { if(is_cycle[i]) { if(!is_seen.count(tree_size[i])) { is_seen.insert(tree_size[i]); auto div = [&](mint c) { int m = g.size(); vector r = g; REP(i, m - 1) { r[i + 1] -= r[i] * c; } return r; }; mf[tree_size[i]] = div(POW(k, tree_size[i], 998244353) - 1); } } } } REP(i, cycle_vertex_cnt + 1) { noncycle_res += g[i] * i; } REPS(i, cycle_vertex_cnt) { noncycle_res += g[i] * (i - 1) * cycle_vertex_cnt; } map cycle_cnt; REP(i, n) { auto dfs = [&](auto self, int now, int per, int r) -> void { for(auto [to, cost] : G[now]) { if(to == per or is_cycle[to]) continue; noncycle_res -= mint(k).pow(tree_size[to]) - 1; cycle_cnt[tree_size[r]] += (mint(k).pow(tree_size[to]) - 1) * mint(k).pow(tree_size[r] - tree_size[to]); self(self, to, now, r); } }; if(is_cycle[i]) { dfs(dfs, i, -1, i); } } for(auto [key, f] : mf) { REP(j, f.size()) { noncycle_res += cycle_cnt[key] * f[j] * (j + 1); } } cout << ((cycle_res)*mint(1 - u).pow(n) + (noncycle_res) * (mint(1 - u).pow(n))).val() << endl; } int main() { int t = 1; //cin >> t; while(t--) main_(); return 0; }