This documentation is automatically generated by online-judge-tools/verification-helper
#include "flow/longest_shortest_path.hpp"
#include "flow/mincostflow.hpp"
#include "graph/shortest_path/dijkstra.hpp"
/*
potential p[v]
距離を L 以上にしたい : L<=p[t]-p[s]
辺 (u,v,w)
伸ばす量 max(0,p[v]-p[u]-w)
- t から s に費用-L, 容量INF
- u から v に費用 w, 容量 1
これの循環流
*/
// https://qoj.ac/contest/1435/problem/7737
template <typename T = ll, bool DAG = false>
struct Longest_Shortest_Path {
int N, s, t;
T F, L, K;
bool solved;
vc<tuple<int, int, T, T>> dat;
vc<T> pot;
Longest_Shortest_Path(int N, int s, int t) : N(N), s(s), t(t), F(0), solved(0) {}
// 現在の長さ, 長さを+1するコスト
void add(int frm, int to, T length, T cost) {
assert(0 <= frm && frm < N && 0 <= to && to < N && !solved);
if (DAG) assert(frm < to);
dat.eb(frm, to, length, cost);
}
T init_dist() {
Graph<T, 1> G(N);
for (auto& [a, b, c, d]: dat) G.add(a, b, c);
G.build();
auto [dist, par] = dijkstra<T>(G, s);
return dist[t];
}
// 距離が L 以上になるようにせよ. return: min cost.
T solve_by_target_length(T target_length) {
L = target_length;
assert(!solved && L >= init_dist());
solved = 1;
Min_Cost_Flow<T, T, DAG> G(N, s, t);
for (auto& [a, b, length, cost]: dat) { G.add(a, b, cost, length); }
T ans = -infty<T>;
for (auto& [x, y]: G.slope()) {
if (chmax(ans, x * L - y)) F = x;
}
return K = ans;
}
// コストが K で最大距離にせよ. return: max dist.
T solve_by_cost(T K) {}
// frm, to, cost. add_edge 順.
vc<T> get_potentials() {
assert(solved);
if (len(pot)) return pot;
Min_Cost_Flow<T, T, DAG> G(N, s, t);
for (auto& [a, b, length, cost]: dat) { G.add(a, b, cost, length); }
G.flow(F);
pot = G.get_potentials();
Graph<T, 1> resG(N);
auto add = [&](int a, int b, T x) -> void {
x = x + pot[a] - pot[b];
resG.add(a, b, x);
};
for (auto& e: G.edges()) {
if (e.cap > e.flow) add(e.frm, e.to, e.cost);
if (e.flow > 0) add(e.to, e.frm, -e.cost);
}
add(s, t, L), add(t, s, -L);
resG.build();
vc<T> dist = dijkstra<ll>(resG, s).fi;
FOR(x, N) pot[x] += dist[x];
return pot;
}
// 変更後の長さ
vc<T> get_edges() {
get_potentials();
vc<T> res;
for (auto [frm, to, length, cost]: dat) { res.eb(max<T>(length, pot[to] - pot[frm])); }
return res;
}
};
#line 2 "flow/mincostflow.hpp"
// atcoder library のものを改変
namespace internal {
template <class E>
struct csr {
vector<int> start;
vector<E> elist;
explicit csr(int n, const vector<pair<int, E>>& edges) : start(n + 1), elist(edges.size()) {
for (auto e: edges) { start[e.first + 1]++; }
for (int i = 1; i <= n; i++) { start[i] += start[i - 1]; }
auto counter = start;
for (auto e: edges) { elist[counter[e.first]++] = e.second; }
}
};
template <class T>
struct simple_queue {
vector<T> payload;
int pos = 0;
void reserve(int n) { payload.reserve(n); }
int size() const { return int(payload.size()) - pos; }
bool empty() const { return pos == int(payload.size()); }
void push(const T& t) { payload.push_back(t); }
T& front() { return payload[pos]; }
void clear() {
payload.clear();
pos = 0;
}
void pop() { pos++; }
};
} // namespace internal
/*
・atcoder library をすこし改変したもの
・DAG = true であれば、負辺 OK (1 回目の最短路を dp で行う)
ただし、頂点番号は toposort されていることを仮定している。
*/
template <class Cap = int, class Cost = ll, bool DAG = false>
struct Min_Cost_Flow {
public:
Min_Cost_Flow() {}
explicit Min_Cost_Flow(int n, int source, int sink) : n(n), source(source), sink(sink) {
assert(0 <= source && source < n);
assert(0 <= sink && sink < n);
assert(source != sink);
}
// frm, to, cap, cost
int add(int frm, int to, Cap cap, Cost cost) {
assert(0 <= frm && frm < n);
assert(0 <= to && to < n);
assert(0 <= cap);
assert(DAG || 0 <= cost);
if (DAG) assert(frm < to);
int m = int(_edges.size());
_edges.push_back({frm, to, cap, 0, cost});
return m;
}
void debug() {
print("flow graph");
print("frm, to, cap, cost");
for (auto&& [frm, to, cap, flow, cost]: _edges) { print(frm, to, cap, cost); }
}
struct edge {
int frm, to;
Cap cap, flow;
Cost cost;
};
edge get_edge(int i) {
int m = int(_edges.size());
assert(0 <= i && i < m);
return _edges[i];
}
vector<edge> edges() { return _edges; }
// (流量, 費用)
pair<Cap, Cost> flow() { return flow(infty<Cap>); }
// (流量, 費用)
pair<Cap, Cost> flow(Cap flow_limit) { return slope(flow_limit).back(); }
vector<pair<Cap, Cost>> slope() { return slope(infty<Cap>); }
vector<pair<Cap, Cost>> slope(Cap flow_limit) {
int m = int(_edges.size());
vector<int> edge_idx(m);
auto g = [&]() {
vector<int> degree(n), redge_idx(m);
vector<pair<int, _edge>> elist;
elist.reserve(2 * m);
for (int i = 0; i < m; i++) {
auto e = _edges[i];
edge_idx[i] = degree[e.frm]++;
redge_idx[i] = degree[e.to]++;
elist.push_back({e.frm, {e.to, -1, e.cap - e.flow, e.cost}});
elist.push_back({e.to, {e.frm, -1, e.flow, -e.cost}});
}
auto _g = internal::csr<_edge>(n, elist);
for (int i = 0; i < m; i++) {
auto e = _edges[i];
edge_idx[i] += _g.start[e.frm];
redge_idx[i] += _g.start[e.to];
_g.elist[edge_idx[i]].rev = redge_idx[i];
_g.elist[redge_idx[i]].rev = edge_idx[i];
}
return _g;
}();
auto result = slope(g, flow_limit);
for (int i = 0; i < m; i++) {
auto e = g.elist[edge_idx[i]];
_edges[i].flow = _edges[i].cap - e.cap;
}
return result;
}
// O(F(N+M)) くらい使って経路復元
vvc<int> path_decomposition() {
vvc<int> TO(n);
for (auto&& e: edges()) { FOR(e.flow) TO[e.frm].eb(e.to); }
vvc<int> res;
vc<int> vis(n);
while (!TO[source].empty()) {
vc<int> path = {source};
vis[source] = 1;
while (path.back() != sink) {
int to = POP(TO[path.back()]);
while (vis[to]) { vis[POP(path)] = 0; }
path.eb(to), vis[to] = 1;
}
for (auto&& v: path) vis[v] = 0;
res.eb(path);
}
return res;
}
vc<Cost> get_potentials() { return potential; }
private:
int n, source, sink;
vector<edge> _edges;
// inside edge
struct _edge {
int to, rev;
Cap cap;
Cost cost;
};
vc<Cost> potential;
vector<pair<Cap, Cost>> slope(internal::csr<_edge>& g, Cap flow_limit) {
if (DAG) assert(source == 0 && sink == n - 1);
vector<pair<Cost, Cost>> dual_dist(n);
vector<int> prev_e(n);
vector<bool> vis(n);
struct Q {
Cost key;
int to;
bool operator<(Q r) const { return key > r.key; }
};
vector<int> que_min;
vector<Q> que;
auto dual_ref = [&]() {
for (int i = 0; i < n; i++) { dual_dist[i].second = infty<Cost>; }
fill(vis.begin(), vis.end(), false);
que_min.clear();
que.clear();
// que[0..heap_r) was heapified
size_t heap_r = 0;
dual_dist[source].second = 0;
que_min.push_back(source);
while (!que_min.empty() || !que.empty()) {
int v;
if (!que_min.empty()) {
v = que_min.back();
que_min.pop_back();
} else {
while (heap_r < que.size()) {
heap_r++;
push_heap(que.begin(), que.begin() + heap_r);
}
v = que.front().to;
pop_heap(que.begin(), que.end());
que.pop_back();
heap_r--;
}
if (vis[v]) continue;
vis[v] = true;
if (v == sink) break;
Cost dual_v = dual_dist[v].first, dist_v = dual_dist[v].second;
for (int i = g.start[v]; i < g.start[v + 1]; i++) {
auto e = g.elist[i];
if (!e.cap) continue;
Cost cost = e.cost - dual_dist[e.to].first + dual_v;
if (dual_dist[e.to].second > dist_v + cost) {
Cost dist_to = dist_v + cost;
dual_dist[e.to].second = dist_to;
prev_e[e.to] = e.rev;
if (dist_to == dist_v) {
que_min.push_back(e.to);
} else {
que.push_back(Q{dist_to, e.to});
}
}
}
}
if (!vis[sink]) { return false; }
for (int v = 0; v < n; v++) {
if (!vis[v]) continue;
dual_dist[v].first -= dual_dist[sink].second - dual_dist[v].second;
}
return true;
};
auto dual_ref_dag = [&]() {
FOR(i, n) dual_dist[i].se = infty<Cost>;
dual_dist[source].se = 0;
fill(vis.begin(), vis.end(), false);
vis[0] = true;
FOR(v, n) {
if (!vis[v]) continue;
Cost dual_v = dual_dist[v].fi, dist_v = dual_dist[v].se;
for (int i = g.start[v]; i < g.start[v + 1]; i++) {
auto e = g.elist[i];
if (!e.cap) continue;
Cost cost = e.cost - dual_dist[e.to].fi + dual_v;
if (dual_dist[e.to].se > dist_v + cost) {
vis[e.to] = true;
Cost dist_to = dist_v + cost;
dual_dist[e.to].second = dist_to;
prev_e[e.to] = e.rev;
}
}
}
if (!vis[sink]) { return false; }
for (int v = 0; v < n; v++) {
if (!vis[v]) continue;
dual_dist[v].fi -= dual_dist[sink].se - dual_dist[v].se;
}
return true;
};
Cap flow = 0;
Cost cost = 0, prev_cost_per_flow = -1;
vector<pair<Cap, Cost>> result = {{Cap(0), Cost(0)}};
while (flow < flow_limit) {
if (DAG && flow == 0) {
if (!dual_ref_dag()) break;
} else {
if (!dual_ref()) break;
}
Cap c = flow_limit - flow;
for (int v = sink; v != source; v = g.elist[prev_e[v]].to) { c = min(c, g.elist[g.elist[prev_e[v]].rev].cap); }
for (int v = sink; v != source; v = g.elist[prev_e[v]].to) {
auto& e = g.elist[prev_e[v]];
e.cap += c;
g.elist[e.rev].cap -= c;
}
Cost d = -dual_dist[source].first;
flow += c;
cost += c * d;
if (prev_cost_per_flow == d) { result.pop_back(); }
result.push_back({flow, cost});
prev_cost_per_flow = d;
}
dual_ref();
potential.resize(n);
FOR(v, n) potential[v] = dual_dist[v].fi;
return result;
}
};
#line 2 "ds/hashmap.hpp"
// u64 -> Val
template <typename Val>
struct HashMap {
// n は入れたいものの個数で ok
HashMap(u32 n = 0) { build(n); }
void build(u32 n) {
u32 k = 8;
while (k < n * 2) k *= 2;
cap = k / 2, mask = k - 1;
key.resize(k), val.resize(k), used.assign(k, 0);
}
// size を保ったまま. size=0 にするときは build すること.
void clear() {
used.assign(len(used), 0);
cap = (mask + 1) / 2;
}
int size() { return len(used) / 2 - cap; }
int index(const u64& k) {
int i = 0;
for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
return i;
}
Val& operator[](const u64& k) {
if (cap == 0) extend();
int i = index(k);
if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
return val[i];
}
Val get(const u64& k, Val default_value) {
int i = index(k);
return (used[i] ? val[i] : default_value);
}
bool count(const u64& k) {
int i = index(k);
return used[i] && key[i] == k;
}
// f(key, val)
template <typename F>
void enumerate_all(F f) {
FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
}
private:
u32 cap, mask;
vc<u64> key;
vc<Val> val;
vc<bool> used;
u64 hash(u64 x) {
static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
x += FIXED_RANDOM;
x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
return (x ^ (x >> 31)) & mask;
}
void extend() {
vc<pair<u64, Val>> dat;
dat.reserve(len(used) / 2 - cap);
FOR(i, len(used)) {
if (used[i]) dat.eb(key[i], val[i]);
}
build(2 * len(dat));
for (auto& [a, b]: dat) (*this)[a] = b;
}
};
#line 3 "graph/base.hpp"
template <typename T>
struct Edge {
int frm, to;
T cost;
int id;
};
template <typename T = int, bool directed = false>
struct Graph {
static constexpr bool is_directed = directed;
int N, M;
using cost_type = T;
using edge_type = Edge<T>;
vector<edge_type> edges;
vector<int> indptr;
vector<edge_type> csr_edges;
vc<int> vc_deg, vc_indeg, vc_outdeg;
bool prepared;
class OutgoingEdges {
public:
OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}
const edge_type* begin() const {
if (l == r) { return 0; }
return &G->csr_edges[l];
}
const edge_type* end() const {
if (l == r) { return 0; }
return &G->csr_edges[r];
}
private:
const Graph* G;
int l, r;
};
bool is_prepared() { return prepared; }
Graph() : N(0), M(0), prepared(0) {}
Graph(int N) : N(N), M(0), prepared(0) {}
void build(int n) {
N = n, M = 0;
prepared = 0;
edges.clear();
indptr.clear();
csr_edges.clear();
vc_deg.clear();
vc_indeg.clear();
vc_outdeg.clear();
}
void add(int frm, int to, T cost = 1, int i = -1) {
assert(!prepared);
assert(0 <= frm && 0 <= to && to < N);
if (i == -1) i = M;
auto e = edge_type({frm, to, cost, i});
edges.eb(e);
++M;
}
#ifdef FASTIO
// wt, off
void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }
void read_graph(int M, bool wt = false, int off = 1) {
for (int m = 0; m < M; ++m) {
INT(a, b);
a -= off, b -= off;
if (!wt) {
add(a, b);
} else {
T c;
read(c);
add(a, b, c);
}
}
build();
}
#endif
void build() {
assert(!prepared);
prepared = true;
indptr.assign(N + 1, 0);
for (auto&& e: edges) {
indptr[e.frm + 1]++;
if (!directed) indptr[e.to + 1]++;
}
for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
auto counter = indptr;
csr_edges.resize(indptr.back() + 1);
for (auto&& e: edges) {
csr_edges[counter[e.frm]++] = e;
if (!directed) csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
}
}
OutgoingEdges operator[](int v) const {
assert(prepared);
return {this, indptr[v], indptr[v + 1]};
}
vc<int> deg_array() {
if (vc_deg.empty()) calc_deg();
return vc_deg;
}
pair<vc<int>, vc<int>> deg_array_inout() {
if (vc_indeg.empty()) calc_deg_inout();
return {vc_indeg, vc_outdeg};
}
int deg(int v) {
if (vc_deg.empty()) calc_deg();
return vc_deg[v];
}
int in_deg(int v) {
if (vc_indeg.empty()) calc_deg_inout();
return vc_indeg[v];
}
int out_deg(int v) {
if (vc_outdeg.empty()) calc_deg_inout();
return vc_outdeg[v];
}
#ifdef FASTIO
void debug() {
#ifdef LOCAL
print("Graph");
if (!prepared) {
print("frm to cost id");
for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
} else {
print("indptr", indptr);
print("frm to cost id");
FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
}
#endif
}
#endif
vc<int> new_idx;
vc<bool> used_e;
// G における頂点 V[i] が、新しいグラフで i になるようにする
// {G, es}
// sum(deg(v)) の計算量になっていて、
// 新しいグラフの n+m より大きい可能性があるので注意
Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
if (len(new_idx) != N) new_idx.assign(N, -1);
int n = len(V);
FOR(i, n) new_idx[V[i]] = i;
Graph<T, directed> G(n);
vc<int> history;
FOR(i, n) {
for (auto&& e: (*this)[V[i]]) {
if (len(used_e) <= e.id) used_e.resize(e.id + 1);
if (used_e[e.id]) continue;
int a = e.frm, b = e.to;
if (new_idx[a] != -1 && new_idx[b] != -1) {
history.eb(e.id);
used_e[e.id] = 1;
int eid = (keep_eid ? e.id : -1);
G.add(new_idx[a], new_idx[b], e.cost, eid);
}
}
}
FOR(i, n) new_idx[V[i]] = -1;
for (auto&& eid: history) used_e[eid] = 0;
G.build();
return G;
}
Graph<T, true> to_directed_tree(int root = -1) {
if (root == -1) root = 0;
assert(!is_directed && prepared && M == N - 1);
Graph<T, true> G1(N);
vc<int> par(N, -1);
auto dfs = [&](auto& dfs, int v) -> void {
for (auto& e: (*this)[v]) {
if (e.to == par[v]) continue;
par[e.to] = v, dfs(dfs, e.to);
}
};
dfs(dfs, root);
for (auto& e: edges) {
int a = e.frm, b = e.to;
if (par[a] == b) swap(a, b);
assert(par[b] == a);
G1.add(a, b, e.cost);
}
G1.build();
return G1;
}
HashMap<int> MP_FOR_EID;
int get_eid(u64 a, u64 b) {
if (len(MP_FOR_EID) == 0) {
MP_FOR_EID.build(N - 1);
for (auto& e: edges) {
u64 a = e.frm, b = e.to;
u64 k = to_eid_key(a, b);
MP_FOR_EID[k] = e.id;
}
}
return MP_FOR_EID.get(to_eid_key(a, b), -1);
}
u64 to_eid_key(u64 a, u64 b) {
if (!directed && a > b) swap(a, b);
return N * a + b;
}
private:
void calc_deg() {
assert(vc_deg.empty());
vc_deg.resize(N);
for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
}
void calc_deg_inout() {
assert(vc_indeg.empty());
vc_indeg.resize(N);
vc_outdeg.resize(N);
for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
}
};
#line 3 "graph/shortest_path/dijkstra.hpp"
template <typename T, typename GT>
pair<vc<T>, vc<int>> dijkstra_dense(GT& G, int s) {
const int N = G.N;
vc<T> dist(N, infty<T>);
vc<int> par(N, -1);
vc<bool> done(N);
dist[s] = 0;
while (1) {
int v = -1;
T mi = infty<T>;
FOR(i, N) {
if (!done[i] && chmin(mi, dist[i])) v = i;
}
if (v == -1) break;
done[v] = 1;
for (auto&& e: G[v]) {
if (chmin(dist[e.to], dist[v] + e.cost)) par[e.to] = v;
}
}
return {dist, par};
}
template <typename T, typename GT, bool DENSE = false>
pair<vc<T>, vc<int>> dijkstra(GT& G, int v) {
if (DENSE) return dijkstra_dense<T>(G, v);
auto N = G.N;
vector<T> dist(N, infty<T>);
vector<int> par(N, -1);
using P = pair<T, int>;
priority_queue<P, vector<P>, greater<P>> que;
dist[v] = 0;
que.emplace(0, v);
while (!que.empty()) {
auto [dv, v] = que.top();
que.pop();
if (dv > dist[v]) continue;
for (auto&& e: G[v]) {
if (chmin(dist[e.to], dist[e.frm] + e.cost)) {
par[e.to] = e.frm;
que.emplace(dist[e.to], e.to);
}
}
}
return {dist, par};
}
// 多点スタート。[dist, par, root]
template <typename T, typename GT>
tuple<vc<T>, vc<int>, vc<int>> dijkstra(GT& G, vc<int> vs) {
assert(G.is_prepared());
int N = G.N;
vc<T> dist(N, infty<T>);
vc<int> par(N, -1);
vc<int> root(N, -1);
using P = pair<T, int>;
priority_queue<P, vector<P>, greater<P>> que;
for (auto&& v: vs) {
dist[v] = 0;
root[v] = v;
que.emplace(T(0), v);
}
while (!que.empty()) {
auto [dv, v] = que.top();
que.pop();
if (dv > dist[v]) continue;
for (auto&& e: G[v]) {
if (chmin(dist[e.to], dist[e.frm] + e.cost)) {
root[e.to] = root[e.frm];
par[e.to] = e.frm;
que.push(mp(dist[e.to], e.to));
}
}
}
return {dist, par, root};
}
#line 3 "flow/longest_shortest_path.hpp"
/*
potential p[v]
距離を L 以上にしたい : L<=p[t]-p[s]
辺 (u,v,w)
伸ばす量 max(0,p[v]-p[u]-w)
- t から s に費用-L, 容量INF
- u から v に費用 w, 容量 1
これの循環流
*/
// https://qoj.ac/contest/1435/problem/7737
template <typename T = ll, bool DAG = false>
struct Longest_Shortest_Path {
int N, s, t;
T F, L, K;
bool solved;
vc<tuple<int, int, T, T>> dat;
vc<T> pot;
Longest_Shortest_Path(int N, int s, int t) : N(N), s(s), t(t), F(0), solved(0) {}
// 現在の長さ, 長さを+1するコスト
void add(int frm, int to, T length, T cost) {
assert(0 <= frm && frm < N && 0 <= to && to < N && !solved);
if (DAG) assert(frm < to);
dat.eb(frm, to, length, cost);
}
T init_dist() {
Graph<T, 1> G(N);
for (auto& [a, b, c, d]: dat) G.add(a, b, c);
G.build();
auto [dist, par] = dijkstra<T>(G, s);
return dist[t];
}
// 距離が L 以上になるようにせよ. return: min cost.
T solve_by_target_length(T target_length) {
L = target_length;
assert(!solved && L >= init_dist());
solved = 1;
Min_Cost_Flow<T, T, DAG> G(N, s, t);
for (auto& [a, b, length, cost]: dat) { G.add(a, b, cost, length); }
T ans = -infty<T>;
for (auto& [x, y]: G.slope()) {
if (chmax(ans, x * L - y)) F = x;
}
return K = ans;
}
// コストが K で最大距離にせよ. return: max dist.
T solve_by_cost(T K) {}
// frm, to, cost. add_edge 順.
vc<T> get_potentials() {
assert(solved);
if (len(pot)) return pot;
Min_Cost_Flow<T, T, DAG> G(N, s, t);
for (auto& [a, b, length, cost]: dat) { G.add(a, b, cost, length); }
G.flow(F);
pot = G.get_potentials();
Graph<T, 1> resG(N);
auto add = [&](int a, int b, T x) -> void {
x = x + pot[a] - pot[b];
resG.add(a, b, x);
};
for (auto& e: G.edges()) {
if (e.cap > e.flow) add(e.frm, e.to, e.cost);
if (e.flow > 0) add(e.to, e.frm, -e.cost);
}
add(s, t, L), add(t, s, -L);
resG.build();
vc<T> dist = dijkstra<ll>(resG, s).fi;
FOR(x, N) pot[x] += dist[x];
return pot;
}
// 変更後の長さ
vc<T> get_edges() {
get_potentials();
vc<T> res;
for (auto [frm, to, length, cost]: dat) { res.eb(max<T>(length, pot[to] - pot[frm])); }
return res;
}
};