// いろいろ高速化 #pragma GCC target("avx2") #pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #ifndef HIDDEN_IN_VS // 折りたたみ用 // 警告の抑制 #define _CRT_SECURE_NO_WARNINGS // ライブラリの読み込み #include using namespace std; // 型名の短縮 using ll = long long; using ull = unsigned long long; // -2^63 ~ 2^63 = 9 * 10^18(int は -2^31 ~ 2^31 = 2 * 10^9) using pii = pair; using pll = pair; using pil = pair; using pli = pair; using vi = vector; using vvi = vector; using vvvi = vector; using vvvvi = vector; using vl = vector; using vvl = vector; using vvvl = vector; using vvvvl = vector; using vb = vector; using vvb = vector; using vvvb = vector; using vc = vector; using vvc = vector; using vvvc = vector; using vd = vector; using vvd = vector; using vvvd = vector; template using priority_queue_rev = priority_queue, greater>; using Graph = vvi; // 定数の定義 const double PI = acos(-1); const vi DX = { 1, 0, -1, 0 }; // 4 近傍(下,右,上,左) const vi DY = { 0, 1, 0, -1 }; int INF = 1001001001; ll INFL = 4004004003104004004LL; // (int)INFL = 1010931620; // 入出力高速化 struct fast_io { fast_io() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } fastIOtmp; // 汎用マクロの定義 #define all(a) (a).begin(), (a).end() #define sz(x) ((int)(x).size()) #define lbpos(a, x) (int)distance((a).begin(), std::lower_bound(all(a), x)) #define ubpos(a, x) (int)distance((a).begin(), std::upper_bound(all(a), x)) #define Yes(b) {cout << ((b) ? "Yes\n" : "No\n");} #define rep(i, n) for(int i = 0, i##_len = int(n); i < i##_len; ++i) // 0 から n-1 まで昇順 #define repi(i, s, t) for(int i = int(s), i##_end = int(t); i <= i##_end; ++i) // s から t まで昇順 #define repir(i, s, t) for(int i = int(s), i##_end = int(t); i >= i##_end; --i) // s から t まで降順 #define repe(v, a) for(const auto& v : (a)) // a の全要素(変更不可能) #define repea(v, a) for(auto& v : (a)) // a の全要素(変更可能) #define repb(set, d) for(int set = 0, set##_ub = 1 << int(d); set < set##_ub; ++set) // d ビット全探索(昇順) #define repis(i, set) for(int i = lsb(set), bset##i = set; i >= 0; bset##i -= 1 << i, i = lsb(bset##i)) // set の全要素(昇順) #define repp(a) sort(all(a)); for(bool a##_perm = true; a##_perm; a##_perm = next_permutation(all(a))) // a の順列全て(昇順) #define smod(n, m) ((((n) % (m)) + (m)) % (m)) // 非負mod #define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} // 重複除去 #define EXIT(a) {cout << (a) << endl; exit(0);} // 強制終了 #define inQ(x, y, u, l, d, r) ((u) <= (x) && (l) <= (y) && (x) < (d) && (y) < (r)) // 半開矩形内判定 // 汎用関数の定義 template inline ll powi(T n, int k) { ll v = 1; rep(i, k) v *= n; return v; } template inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // 最大値を更新(更新されたら true を返す) template inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // 最小値を更新(更新されたら true を返す) template inline T get(T set, int i) { return (set >> i) & T(1); } // 演算子オーバーロード template inline istream& operator>>(istream& is, pair& p) { is >> p.first >> p.second; return is; } template inline istream& operator>>(istream& is, vector& v) { repea(x, v) is >> x; return is; } template inline vector& operator--(vector& v) { repea(x, v) --x; return v; } template inline vector& operator++(vector& v) { repea(x, v) ++x; return v; } #endif // 折りたたみ用 #if __has_include() #include using namespace atcoder; #ifdef _MSC_VER #include "localACL.hpp" #endif //using mint = modint1000000007; using mint = modint998244353; //using mint = modint; // mint::set_mod(m); namespace atcoder { inline istream& operator>>(istream& is, mint& x) { ll x_; is >> x_; x = x_; return is; } inline ostream& operator<<(ostream& os, const mint& x) { os << x.val(); return os; } } using vm = vector; using vvm = vector; using vvvm = vector; using vvvvm = vector; #endif #ifdef _MSC_VER // 手元環境(Visual Studio) #include "local.hpp" #else // 提出用(gcc) inline int popcount(int n) { return __builtin_popcount(n); } inline int popcount(ll n) { return __builtin_popcountll(n); } inline int lsb(int n) { return n != 0 ? __builtin_ctz(n) : -1; } inline int lsb(ll n) { return n != 0 ? __builtin_ctzll(n) : -1; } inline int msb(int n) { return n != 0 ? (31 - __builtin_clz(n)) : -1; } inline int msb(ll n) { return n != 0 ? (63 - __builtin_clzll(n)) : -1; } #define dump(...) #define dumpel(v) #define dump_list(v) #define dump_mat(v) #define input_from_file(f) #define output_to_file(f) #define Assert(b) { if (!(b)) while (1) cout << "OLE"; } #endif //【アダマール変換】: O(2^n n) /* * a[0..2^n) をアダマール変換したものに上書きする. */ template void hadamard(vector& a) { // verify : https://judge.yosupo.jp/problem/bitwise_xor_convolution // 具体例: // A[0] = a[0] + a[1] + a[2] + a[3] + a[4] + a[5] + a[6] + a[7] + ... // A[1] = a[0] - a[1] + a[2] - a[3] + a[4] - a[5] + a[6] - a[7] + ... // A[2] = a[0] + a[1] - a[2] - a[3] + a[4] + a[5] - a[6] - a[7] + ... // A[3] = a[0] - a[1] - a[2] + a[3] + a[4] - a[5] - a[6] + a[7] + ... // A[4] = a[0] + a[1] + a[2] + a[3] - a[4] - a[5] - a[6] - a[7] + ... // A[5] = a[0] - a[1] + a[2] - a[3] - a[4] + a[5] - a[6] + a[7] + ... // A[6] = a[0] + a[1] - a[2] - a[3] - a[4] - a[5] + a[6] + a[7] + ... // A[7] = a[0] - a[1] - a[2] + a[3] - a[4] + a[5] + a[6] - a[7] + ... int n = msb(sz(a)); rep(i, n) repb(set, n) { if (!(set & (1 << i))) { T x = a[set]; T y = a[set | (1 << i)]; a[set] = x + y; a[set + (1 << i)] = x - y; } } } //【逆アダマール変換】: O(2^n n) /* * A[0..2^n) を逆アダマール変換したものに上書きする. * * 制約:A の要素は 2 で割れる. */ template void hadamard_inv(vector& A) { // verify : https://judge.yosupo.jp/problem/bitwise_xor_convolution int n = msb(sz(A)); rep(i, n) repb(set, n) { if (!(set & (1 << i))) { T x = A[set]; T y = A[set | (1 << i)]; A[set] = (x + y) / 2; A[set + (1 << i)] = (x - y) / 2; } } } //【XOR 畳込み】O(2^n n) /* * 与えられた a[0..2^n), b[0..2^n) に対して * c[set] = Σ(set1 XOR set2 = set) a[set1] b[set2] * なる c[0..2^n) を返す. * * 利用:【アダマール変換】,【逆アダマール変換】 */ template vector xor_convolution(vector a, vector b) { // 参考 : https://kazuma8128.hatenablog.com/entry/2018/05/31/144519 // verify : https://judge.yosupo.jp/problem/bitwise_xor_convolution int n = msb(sz(a)); hadamard(a); hadamard(b); repb(set, n) a[set] *= b[set]; hadamard_inv(a); return a; } bool naive(int n, int k, vi a) { rep(i, n) { repi(na, max(1, a[i] - k), a[i] + k) { vi a2(a); a2[i] = na; vi acc(n + 1); rep(i, n) acc[i + 1] = acc[i] ^ a2[i]; uniq(acc); if (sz(acc) == n + 1) return true; } } return false; } bool TLE(int n, int k, vi a) { int A = 1 << (msb(*max_element(all(a))) + 1); vi acc(n + 1); rep(i, n) acc[i + 1] = acc[i] ^ a[i]; dump("acc:", acc); vi imos(n + 1); int v = 0; vi l_min(A, -1); repi(r, 0, n) { int l = l_min[acc[r]]; if (l != -1) { imos[l]++; imos[r]--; v++; } l_min[acc[r]] = r; } if (v == 0) return true; rep(i, n) imos[i + 1] += imos[i]; dump("imos:", imos); int L = INF, R = -INF; rep(i, n) { if (imos[i] == v) { chmin(L, i); chmax(R, i); } } dump("L, R:", L, R); if (L == INF) return false; vl cnt_l(A), cnt_r(A); repi(i, 0, L - 1) cnt_l[acc[i]]++; repi(i, L, n) cnt_r[acc[i]]++; dump("cnt_l:", cnt_l); dump("cnt_r:", cnt_r); repi(i, L, R) { cnt_l[acc[i]]++; cnt_r[acc[i]]--; vl c2 = xor_convolution(cnt_l, cnt_r); dump("c2:", c2); bool ok = false; repi(na, max(1, a[i] - k), a[i] + k) { if ((a[i] ^ na) >= A || c2[a[i] ^ na] == 0) { ok = true; break; } } if (ok) return true; } return false; } bool solve(int n, int k, vi a) { int A = 1 << (msb(*max_element(all(a))) + 1); vi acc(n + 1); rep(i, n) acc[i + 1] = acc[i] ^ a[i]; dump("acc:", acc); vi imos(n + 1); int v = 0; vi l_min(A, -1); repi(r, 0, n) { int l = l_min[acc[r]]; if (l != -1) { imos[l]++; imos[r]--; v++; } l_min[acc[r]] = r; } if (v == 0) return true; rep(i, n) imos[i + 1] += imos[i]; dump("imos:", imos); int L = INF, R = -INF; rep(i, n) { if (imos[i] == v) { chmin(L, i); chmax(R, i); } } dump("L, R:", L, R); if (L == INF) return false; vl cnt_l(A), cnt_r(A); repi(i, 0, L - 1) cnt_l[acc[i]]++; repi(i, L, n) cnt_r[acc[i]]++; dump("cnt_l:", cnt_l); dump("cnt_r:", cnt_r); vl c2 = xor_convolution(cnt_l, cnt_r); dump("c2:", c2); repi(i, L, R) { repi(j, 0, i - 1) c2[acc[j] ^ acc[i]]--; repi(j, i + 1, n) c2[acc[j] ^ acc[i]]++; bool ok = false; repi(na, max(1, a[i] - k), a[i] + k) { if ((a[i] ^ na) >= A || c2[a[i] ^ na] == 0) { ok = true; break; } } if (ok) return true; } return false; } void bug_find() { #ifdef _MSC_VER // 合わない入力例を見つける. mute_dump = true; mt19937_64 mt; mt.seed((int)time(NULL)); uniform_int_distribution rnd(0LL, 1LL << 60); rep(hoge, 10000) { int n = rnd(mt) % 10 + 1; int k = rnd(mt) % 5 + 1; vi a(n); rep(i, n) a[i] = rnd(mt) % 15 + 1; auto res_naive = naive(n, k, a); auto res_solve = solve(n, k, a); if (res_naive != res_solve) { cout << "----------error!----------" << endl; cout << "input:" << endl; cout << n << " " << k << endl; cout << a << endl; cout << "results:" << endl; cout << res_naive << endl; cout << res_solve << endl; cout << "--------------------------" << endl; } } mute_dump = false; exit(0); #endif } /* ----------error!---------- input: 3 1 2 2 1 results: 1 0 -------------------------- */ int main() { input_from_file("input.txt"); // output_to_file("output.txt"); bug_find(); int n, k; cin >> n >> k; vi a(n); cin >> a; dump(naive(n, k, a)); dump("-----"); Yes(solve(n, k, a)); }