#include namespace { #pragma GCC diagnostic ignored "-Wunused-function" #include #pragma GCC diagnostic warning "-Wunused-function" using namespace std; using namespace atcoder; #define rep(i,n) for(int i = 0; i < (int)(n); i++) #define rrep(i,n) for(int i = (int)(n) - 1; i >= 0; i--) #define all(x) begin(x), end(x) #define rall(x) rbegin(x), rend(x) template bool chmax(T& a, const T& b) { if (a < b) { a = b; return true; } else return false; } template bool chmin(T& a, const T& b) { if (b < a) { a = b; return true; } else return false; } using ll = long long; using P = pair; using VI = vector; using VVI = vector; using VL = vector; using VVL = vector; #pragma GCC target("popcnt") struct WaveletMatrix { int height = 0; int n = 0; using storage_type = unsigned int; static constexpr int B = numeric_limits::digits, LowBits = __lg(B); static_assert(B == 1 << LowBits); struct BVecElem { storage_type v; int acc; }; struct BitVector; vector data; vector bvecs; struct BitVector { span d; int zeros; BitVector(span d, int k, const auto& a, auto& na) : d(d) { int n = size(a); int ptr0 = 0, ptr1 = n; for (int i = 0; i < n; i += B) { int len = min(B, n - i); storage_type x = 0; for (int i_low = 0; i_low < len; i_low++) { bool bit = a[i | i_low] >> k & 1; if (!bit) { na[ptr0++] = a[i | i_low]; } else { x |= storage_type(bit) << i_low; na[--ptr1] = a[i | i_low]; } } d[i >> LowBits].v = x; } assert(ptr0 == ptr1); zeros = ptr0; reverse(na.begin() + ptr0, na.begin() + n); for (int i = 0, sz = size(d); i + 1 < sz; i++) { d[i + 1].acc = d[i].acc + popcount(d[i].v); } } int rank1(int i) { auto di = d[i >> LowBits]; int low = i & (B - 1); auto mask = (storage_type(1) << low) - 1; return di.acc + popcount(di.v & mask); } int rank(bool b, int i) { return b == 0 ? i - rank1(i) : rank1(i); } int mv(bool b, int i) { return b == 0 ? i - rank1(i) : zeros + rank1(i); } }; WaveletMatrix() = default; template WaveletMatrix(vector a) { auto mx = empty(a) ? 0 : *max_element(begin(a), end(a)); height = mx == 0 ? 0 : __lg(mx) + 1; n = size(a); int block_count = (n >> LowBits) + 1; data.resize(height * block_count); bvecs.reserve(height); vector na(n); for (int k = height - 1; k >= 0; k--) { bvecs.emplace_back(span(begin(data) + k * block_count, block_count), k, a, na); swap(a, na); } reverse(begin(bvecs), end(bvecs)); }; // returns the number of occurrence of x in a[l, r) int rank(auto x, int l, int r) { assert(0 <= l && l <= r && r <= n); if (x >> height) return 0; for (int k = height - 1; k >= 0; k--) { bool bit = x >> k & 1; l = bvecs[k].mv(bit, l); r = bvecs[k].mv(bit, r); } return r - l; } int rangefreq_below(auto vr, int l, int r) { assert(0 <= l && l <= r && r <= n); if (vr <= 0) return 0; if (vr >> height) return r - l; int res = 0; for (int k = height - 1; k >= 0; k--) { bool bit = vr >> k & 1; if (bit) { int ones_r = bvecs[k].rank1(r), ones_l = bvecs[k].rank1(l); res += (r - l) - (ones_r - ones_l); l = bvecs[k].zeros + ones_l; r = bvecs[k].zeros + ones_r; } else { l = bvecs[k].mv(bit, l); r = bvecs[k].mv(bit, r); } } return res; } // returns the number of occurrence of values in [vl, vr) in a[l, r) int rangefreq(auto vl, auto vr, int l, int r) { assert(vl <= vr); return rangefreq_below(vr, l, r) - rangefreq_below(vl, l, r); } // returns the maximum vr s.t. # of points in [l, r) * [vl, vr) <= cnt_ub template T max_upper(T vl, int l, int r, int cnt_ub, T res_for_inf) { assert(0 <= l && l <= r && r <= n); assert(cnt_ub >= 0); assert(numeric_limits::digits > height); cnt_ub += rangefreq_below(vl, l, r); if (cnt_ub >= r - l) return res_for_inf; T res = 0; for (int k = height - 1; k >= 0; k--) { int ones_l = bvecs[k].rank1(l), ones_r = bvecs[k].rank1(r); int zeros = (r - l) - (ones_r - ones_l); if (zeros <= cnt_ub) { cnt_ub -= zeros; res += T(1) << k; l = bvecs[k].zeros + ones_l; r = bvecs[k].zeros + ones_r; } else { l = l - ones_l; r = r - ones_r; } } return res; } template T min_lower(T vr, int l, int r, int cnt_ub) { assert(0 <= l && l <= r && r <= n); assert(cnt_ub >= 0); assert(numeric_limits::digits > height); cnt_ub += (r - l) - rangefreq_below(vr, l, r); if (cnt_ub >= r - l) return 0; T res = T(1) << height; for (int k = height - 1; k >= 0; k--) { int ones_l = bvecs[k].rank1(l), ones_r = bvecs[k].rank1(r); int ones = ones_r - ones_l; if (ones <= cnt_ub) { cnt_ub -= ones; res -= T(1) << k; l = l - ones_l; r = r - ones_r; } else { l = bvecs[k].zeros + ones_l; r = bvecs[k].zeros + ones_r; } } return res; } }; } int main() { ios::sync_with_stdio(false); cin.tie(0); int n, q; cin >> n >> q; VI a(n); rep(i, n) cin >> a[i]; vector> qs(q); for (auto& [t, i, x] : qs) { cin >> t; if (t == 1) cin >> i >> x, i--; else cin >> i; } constexpr int B = 224; VI evs(q, -1); constexpr int INF = 1001001001; WaveletMatrix wm; VI is_unchanged, is_changed, mn_acc; rep(iq, q) { if (iq % B == 0) { is_unchanged.clear(); is_changed.clear(); int r = min(iq + B, q); vector changes(n); for (auto [t, i, x] : span(qs).subspan(iq, r - iq)) { if (t == 1) changes[i] = true; } rep(i, n) { (changes[i] ? is_changed : is_unchanged).emplace_back(i); } VI init; for (int i : is_unchanged) init.emplace_back(a[i]); wm = WaveletMatrix(init); mn_acc = {INF}; set vals; auto get_min = [&](int x) { int res = INF; auto it = vals.lower_bound(x); if (it != vals.end()) chmin(res, x ^ *it); if (it != vals.begin()) chmin(res, x ^ *prev(it)); return res; }; for (int i : is_unchanged) { mn_acc.emplace_back(min(mn_acc.back(), get_min(a[i]))); vals.emplace(a[i]); } } auto [t, i, x] = qs[iq]; if (t == 1) { a[i] = x; } else { int r = i; int p = lower_bound(all(is_unchanged), r) - is_unchanged.begin(); int ans = mn_acc[p]; VI vs; for (int i : is_changed) { if (i >= r) break; vs.emplace_back(a[i]); int cand = wm.max_upper(a[i], 0, p, 0, INF); if (cand != INF) chmin(ans, a[i] ^ cand); cand = (int)wm.min_lower(a[i], 0, p, 0) - 1; if (cand != -1) chmin(ans, a[i] ^ cand); } sort(all(vs)); rep(i, ssize(vs) - 1) chmin(ans, vs[i] ^ vs[i+1]); cout << ans << '\n'; } } }