#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #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 bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } const std::vector> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}; int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); } template std::pair operator+(const std::pair &l, const std::pair &r) { return std::make_pair(l.first + r.first, l.second + r.second); } template std::pair operator-(const std::pair &l, const std::pair &r) { return std::make_pair(l.first - r.first, l.second - r.second); } template std::vector sort_unique(std::vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template IStream &operator>>(IStream &is, std::vector &vec) { for (auto &v : vec) is >> v; return is; } template OStream &operator<<(OStream &os, const std::vector &vec); template OStream &operator<<(OStream &os, const std::array &arr); template OStream &operator<<(OStream &os, const std::unordered_set &vec); template OStream &operator<<(OStream &os, const pair &pa); template OStream &operator<<(OStream &os, const std::deque &vec); template OStream &operator<<(OStream &os, const std::set &vec); template OStream &operator<<(OStream &os, const std::multiset &vec); template OStream &operator<<(OStream &os, const std::unordered_multiset &vec); template OStream &operator<<(OStream &os, const std::pair &pa); template OStream &operator<<(OStream &os, const std::map &mp); template OStream &operator<<(OStream &os, const std::unordered_map &mp); template OStream &operator<<(OStream &os, const std::tuple &tpl); template OStream &operator<<(OStream &os, const std::vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } template std::istream &operator>>(std::istream &is, std::tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template OStream &operator<<(OStream &os, const std::tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } template OStream &operator<<(OStream &os, const std::unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::pair &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; } template OStream &operator<<(OStream &os, const std::map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::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) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl #define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr) #else #define dbg(x) ((void)0) #define dbgif(cond, x) ((void)0) #endif #include using S = pint; // min, max constexpr int inf = 1 << 28; S op(S l, S r) { return {min(l.first, r.first), max(l.second, r.second)}; } S e() { return {inf, -inf}; } int main() { int N; cin >> N; vector F(N); cin >> F; map> v2is; REP(i, N) v2is[F.at(i)].push_back(i); // vector nexts(N, N); // { // map mp; // IREP(i, N) { // if (mp.count(F.at(i))) nexts.at(i) = mp.at(F.at(i)); // mp[F.at(i)] = i; // } // } // dbg(nexts); // vector prevs(N, -1); // { // map mp; // REP(i, N) { // if (mp.count(F.at(i))) prevs.at(i) = mp.at(F.at(i)); // mp[F.at(i)] = i; // } // } // dbg(prevs); // vector vs; // REP(i, N) { // vs.emplace_back(prevs.at(i), nexts.at(i)); // } // dbg(vs); int steps = 0; // set init; auto build_se = [&](const map &v2cnt) { set cnt2v; for (auto [v, cnt] : v2cnt) { cnt2v.emplace(cnt, v); } return cnt2v; }; auto rec = [&](auto &&self, int l, int r, map &v2cnt, set &cnt2v) -> void { if (cnt2v.empty()) return; auto [cnt, v] = *cnt2v.begin(); if (cnt > 1) return; ++steps; const int idx = arglb(v2is.at(v), l); const int mid = v2is.at(v).at(idx); auto rm = [&](int val) { int c = v2cnt[val]; cnt2v.erase(pint(c, val)); if (c > 1) { v2cnt[val]--; cnt2v.emplace(c - 1, val); } else { v2cnt.erase(val); } }; rm(F.at(mid)); map v2cnt2; if (mid - l < r - mid) { FOR(i, l, mid) { rm(F.at(i)); v2cnt2[F.at(i)]++; } auto cnt2v2 = build_se(v2cnt2); self(self, l, mid, v2cnt2, cnt2v2); self(self, mid + 1, r, v2cnt, cnt2v); } else { FOR(i, mid + 1, r) { rm(F.at(i)); v2cnt2[F.at(i)]++; } auto cnt2v2 = build_se(v2cnt2); self(self, mid + 1, r, v2cnt2, cnt2v2); self(self, l, mid, v2cnt, cnt2v); } }; map v2cnt; for (int f : F) v2cnt[f]++; set cnt2v = build_se(v2cnt); rec(rec, 0, N, v2cnt, cnt2v); cout << (steps % 2 ? "Alice" : "Bob") << '\n'; }