/** * @FileName a.cpp * @Author kanpurin * @Created 2022.11.04 19:19:40 **/ #include "bits/stdc++.h" using namespace std; typedef long long ll; struct Monoid { using T = ll; T val; bool undef = true; Monoid() { *this = zero(); } Monoid(T val, bool undef = true) : val(val), undef(undef) {} static Monoid zero() { return Monoid(0); } static Monoid e() { return Monoid(1,false); } Monoid& operator+=(const Monoid &a) { if (this->undef) *this = a; else if (!a.undef) this->val += a.val; return *this; } Monoid& operator*=(int c) { return *this; } friend Monoid operator+(const Monoid& a, const Monoid& b) { return Monoid(a) += b; } friend Monoid operator*(const Monoid& a, int c) { return Monoid(a) *= c; } friend std::ostream& operator<<(std::ostream &os, const Monoid &x) { return os << x.val; } }; struct Automaton { vector> delta; vector is_accept, is_reject; int qsize; int init; int alphabet_size = 10; inline int next(int state, int c) const { return delta[state][c]; } inline bool accept(int state) const { return is_accept[state]; } inline bool reject(int state) const { return is_reject[state]; } inline int size() const {return qsize; } }; template Monoid digitDP(const string &s, const Automaton &dfa, bool eq = 1) { array,2> dp; dp[1][dfa.init] = Monoid::e(); for (int i = 0; i < s.size(); i++) { array,2> dp2; for (int tight = 0; tight <= 1; tight++) { for (auto [state,value] : dp[tight]) { if (dfa.reject(state)) continue; int lim = (tight ? s[i] - '0' : dfa.alphabet_size - 1); for (int c = 0; c <= lim; c++) { int tight_ = tight && c == lim; int state_ = dfa.next(state,c); if (dfa.reject(state_)) continue; dp2[tight_][state_] += value*c; } } } dp = move(dp2); } Monoid ans = Monoid::zero(); for (int tight = 0; tight <= eq; tight++) for (auto [state,value] : dp[tight]) if (dfa.accept(state)) ans += value; return ans; } struct PartitionRefinement { unordered_map> block; vector block_id; int t; PartitionRefinement(int k) : block_id(k) { for (int i = 0; i < k; i++) { block[0].insert(i); block_id[i] = 0; } t = 1; } vector> partition(vector &v) { unordered_map split; for (int i = 0; i < v.size(); i++) { if (block[block_id[v[i]]].size() == 1) { if (split.find(block_id[v[i]]) == split.end()) continue; for (int p : block[split[block_id[v[i]]]]) { block_id[p] = block_id[v[i]]; block[block_id[v[i]]].insert(p); } block.erase(split[block_id[v[i]]]); continue; } block[block_id[v[i]]].erase(v[i]); if (split.find(block_id[v[i]]) != split.end()) { block_id[v[i]] = split[block_id[v[i]]]; block[block_id[v[i]]].insert(v[i]); } else { split[block_id[v[i]]] = t; block_id[v[i]] = t++; block[block_id[v[i]]].insert(v[i]); } } vector> res; for (auto p : split) { res.push_back(p); } return res; } void print() { for (auto [i,st] : block) { cerr << i << " [ "; for (int v : st) { cerr << v << " "; } cerr << "]" << endl; } } }; Automaton Minimize(const Automaton& dfa) { vector>> inv_delta(dfa.size(),vector>(dfa.alphabet_size)); for (int state = 0; state < dfa.size(); state++) { for (int c = 0; c < dfa.alphabet_size; c++) { int t = dfa.delta[state][c]; inv_delta[t][c].push_back(state); } } PartitionRefinement pr(dfa.size()); vector f; for (int state = 0; state < dfa.size(); state++) { if (dfa.accept(state)) f.push_back(state); } pr.partition(f); queue> que; for (int c = 0; c < dfa.alphabet_size; c++) { que.push({c,0}); que.push({c,1}); } while(!que.empty()) { auto [c,b_id] = que.front(); que.pop(); vector v; for (int state : pr.block[b_id]) { for (int p : inv_delta[state][c]) { v.push_back(p); } } if (v.size() == 0) continue; auto par = pr.partition(v); for (auto p : par) { if (pr.block[p.first].size() > pr.block[p.second].size()) { swap(p.first,p.second); } if (pr.block[p.first].size() == 0) continue; for (int c2 = 0; c2 < dfa.alphabet_size; c2++) { que.push({c2,p.first}); } } } map mp; for (int state = 0; state < dfa.size(); state++) { int b_id = pr.block_id[state]; if (mp.find(b_id) != mp.end()) continue; mp[b_id] = mp.size(); } vector to_state(dfa.size()); for (int state = 0; state < dfa.size(); state++) { to_state[state] = mp[pr.block_id[state]]; } Automaton M; M.init = to_state[dfa.init]; M.alphabet_size = dfa.alphabet_size; M.qsize = mp.size(); M.delta.resize(M.qsize,vector(M.alphabet_size)); M.is_accept.resize(M.qsize); M.is_reject.resize(M.qsize); for (int state = 0; state < dfa.size(); state++) { for (int c = 0; c < dfa.alphabet_size; c++) { M.delta[to_state[state]][c] = to_state[dfa.next(state,c)]; } if (dfa.accept(state)) M.is_accept[to_state[state]] = true; if (dfa.reject(state)) M.is_reject[to_state[state]] = true; } return M; } struct KadomatsuAutomaton : public Automaton { private: void initializer() { qsize = 11*11+1; init = 0; set_delta(); set_is_accept(); set_is_reject(); } void set_delta() { delta.resize(qsize,vector(alphabet_size,0)); for (int i = -1; i < 10; i++) { for (int j = -1; j < 10; j++) { for (int k = 0; k < 10; k++) { if (i == -1 && j == -1 && k == 0) { delta[(i+1)*11+(j+1)][k] = 0; } else if (i != -1 && j == -1) { delta[(i+1)*11+(j+1)][k] = qsize-1; } else if (i == -1 && j != -1 && j != k) { delta[(i+1)*11+(j+1)][k] = (j+1)*11+(k+1); } else if (i == -1 && j != -1) { delta[(i+1)*11+(j+1)][k] = qsize-1; } else if (i == -1) { delta[(i+1)*11+(j+1)][k] = (j+1)*11+(k+1); } else if (i != j && j != k && i != k && (j==min({i,j,k})||j==max({i,j,k}))) { delta[(i+1)*11+(j+1)][k] = (j+1)*11+(k+1); } else { delta[(i+1)*11+(j+1)][k] = qsize-1; } } } } } void set_is_accept() { is_accept.resize(qsize,false); for (int i = 0; i < 10; i++) { for (int j = 0; j < 10; j++) { is_accept[(i+1)*11+(j+1)] = true; } } } void set_is_reject() { is_reject.resize(qsize,false); is_reject[qsize-1] = true; } public: KadomatsuAutomaton(int alpha_size = 10) { alphabet_size = alpha_size; initializer(); } }; struct CountNumberAutomaton : public Automaton { private: std::vector flg; int num; bool eq; void initializer() { assert(flg.size() == alphabet_size); qsize = num+3; init = num+2; set_delta(); set_is_accept(); set_is_reject(); } void set_delta() { delta.resize(qsize,std::vector(alphabet_size)); for (int state = 0; state < qsize; state++) { for (int c = 0; c < alphabet_size; c++) { if (state == init && c == 0) delta[state][c] = init; else if (state == init) delta[state][c] = flg[c]?1:0; else if (state == num+1) delta[state][c] = state; else delta[state][c] = flg[c]?state+1:state; } } } void set_is_accept() { is_accept.resize(qsize,false); if (eq) is_accept[num] = true; else { for (int state = 0; state <= num; state++) { is_accept[state] = true; } is_accept[num+2] = true; } } void set_is_reject() { is_reject.resize(qsize,false); is_reject[num+1] = true; } public: CountNumberAutomaton(std::vector flg, int num, bool eq = false, int alpha_size = 10) : flg(flg), num(num), eq(eq) { alphabet_size = alpha_size; initializer(); } }; template Automaton IntersectionAutomaton(const Automaton1 &A, const Automaton2 &B) { assert(A.alphabet_size == B.alphabet_size); Automaton M; M.alphabet_size = A.alphabet_size; vector> table(A.size(), vector(B.size(),-1)); vector x = {A.init}, y = {B.init}; table[x[0]][y[0]] = 0; M.init = 0; for (int i = 0; i < x.size(); ++i) { M.delta.push_back(vector(M.alphabet_size, -1)); M.is_accept.push_back(A.accept(x[i]) && B.accept(y[i])); M.is_reject.push_back(A.reject(x[i]) || B.reject(y[i])); for (int c = 0; c < A.alphabet_size; c++) { int u = A.next(x[i],c), v = B.next(y[i],c); if (table[u][v] == -1) { table[u][v] = x.size(); x.push_back(u); y.push_back(v); } M.delta[i][c] = table[u][v]; } } M.qsize = M.delta.size(); return M; } int main() { int t;cin >> t; auto M = KadomatsuAutomaton(); auto M2 = IntersectionAutomaton(M,CountNumberAutomaton({1,1,1,1,1,1,1,1,1,1},2)); while(t--) { ll k;cin >> k; ll l = 0, r = 37294859064823; while(r-l>1) { ll mid = (l+r)/2; string sm = to_string(mid); if (k <= digitDP(sm,M).val-digitDP(sm,M2).val) { r = mid; } else { l = mid; } } cout << r << endl; } return 0; }