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#include "knapsack/subset_sum.hpp"
#include "ds/my_bitset.hpp" #include "enumerate/bits.hpp" // O(N MAX(vals)) template <typename T> vc<int> subset_sum_solution_1(vc<T>& vals, int target) { int n = len(vals); if (n == 0) return {}; int mx = MAX(vals); int b = 0, sb = 0; while (b < n && sb + vals[b] <= target) { sb += vals[b++]; } if (b == n && sb != target) return {}; int off = target - mx + 1; vc<int> dp(2 * mx, -1); vv(int, PAR, n, 2 * mx, -1); dp[sb - off] = b; FOR3(i, b, n) { auto newdp = dp; auto& par = PAR[i]; int a = vals[i]; FOR(j, mx) { if (chmax(newdp[j + a], dp[j])) { par[j + a] = -2; } } FOR3_R(j, mx, 2 * mx) { FOR3_R(k, max(dp[j], 0), newdp[j]) { if (chmax(newdp[j - vals[k]], k)) par[j - vals[k]] = k; } } swap(dp, newdp); } if (dp[mx - 1] == -1) return {}; vc<bool> use(n); int i = n - 1, j = mx - 1; while (i >= b) { int p = PAR[i][j]; if (p == -2) { use[i] = !use[i]; j -= vals[i--]; } elif (p == -1) { --i; } else { use[p] = !use[p]; j += vals[p]; } } while (i >= 0) { use[i] = !use[i]; --i; } vc<int> I; FOR(i, n) if (use[i]) I.eb(i); ll sm = 0; for (auto&& i: I) sm += vals[i]; assert(sm == target); return I; } // O(N target / w) template <typename T> vc<int> subset_sum_solution_2(vc<T>& vals, int target) { int n = len(vals); auto I = argsort(vals); My_Bitset dp(1, 1); vc<int> last(target + 1, -1); FOR(k, n) { int v = vals[I[k]]; if (v > target) continue; My_Bitset newdp = dp; int new_size = len(dp) + v; newdp.resize(new_size); newdp.or_to_range(v, new_size, dp); if (len(newdp) > target + 1) newdp.resize(target + 1); // update したところをメモ FOR(i, len(newdp.dat)) { u64 upd = (i < len(dp.dat) ? dp.dat[i] : u64(0)) ^ newdp.dat[i]; enumerate_bits_64(upd, [&](int p) -> void { last[(i << 6) | p] = I[k]; }); } swap(dp, newdp); } if (target >= len(dp) || !dp[target]) return {}; vc<int> ANS; while (target > 0) { int i = last[target]; ANS.eb(i); target -= vals[i]; } return ANS; } // O(sum^{1.5} / w) // sum=10^6 で 150ms:https://codeforces.com/contest/755/problem/F template <typename T> vc<int> subset_sum_solution_3(vc<T>& vals, int target) { int SM = SUM<int>(vals); int N = len(vals); vvc<int> IDS(SM + 1); FOR(i, N) IDS[vals[i]].eb(i); vc<pair<int, int>> par(N, {-1, -1}); vc<int> grp_vals; vc<int> raw_idx; FOR(x, 1, SM + 1) { auto& I = IDS[x]; while (len(I) >= 3) { int a = POP(I), b = POP(I); int c = len(par); par.eb(a, b); IDS[2 * x].eb(c); } for (auto& i: I) { grp_vals.eb(x); raw_idx.eb(i); } } auto I = subset_sum_solution_2<int>(grp_vals, target); vc<int> ANS; for (auto& i: I) { vc<int> st = {raw_idx[i]}; while (len(st)) { auto c = POP(st); if (c < N) { ANS.eb(c); continue; } auto [a, b] = par[c]; st.eb(a), st.eb(b); } } return ANS; } template <typename T> vc<int> subset_sum_solution_4(vc<T>& vals, T target) { if (target <= 0) return {}; int N = len(vals); int M = N / 2; auto calc = [&](int L, int R) -> vc<T> { int n = R - L; vc<T> dp = {0}; FOR(i, n) { T a = vals[L + i]; vc<T> dp1(len(dp)); FOR(k, len(dp)) dp1[k] = dp[k] + a; vc<T> newdp; merge(all(dp), all(dp1), back_inserter(newdp)); swap(dp, newdp); } return dp; }; auto restore = [&](int L, int R, T v) -> vc<int> { int n = R - L; vc<T> dp(1 << n); FOR(i, n) FOR(s, 1 << i) dp[s | 1 << i] = dp[s] + vals[L + i]; FOR(s, 1 << n) { if (dp[s] == v) { vc<int> I; FOR(i, n) if (s >> i & 1) I.eb(L + i); return I; } } assert(0); return {}; }; auto dp1 = calc(0, M); auto dp2 = calc(M, N); int t = 0; FOR_R(s, len(dp1)) { while (t + 1 < len(dp2) && dp1[s] + dp2[t + 1] <= target) { ++t; } if (dp1[s] + dp2[t] == target) { vc<int> I1 = restore(0, M, dp1[s]); vc<int> I2 = restore(M, N, dp2[t]); concat(I1, I2); return I1; } } return {}; } template <typename T> vc<int> subset_sum(vc<T>& vals, T target) { if (target <= 0) return {}; int n = len(vals); if (n == 0) return {}; int mx = MAX(vals); // しきい値の調整をしていない // solution 1: O(N mx)) // solution 2: O(N target / w) // solution 3: O(sum^1.5 / w) // solution 4: O(2^(N/2)) double x1 = double(len(vals)) * mx; double x2 = double(len(vals)) * target / 500.0; double x3 = pow(SUM<double>(vals), 1.5) / 500.0; double x4 = pow(2.0, 0.5 * len(vals)); double mi = min({x1, x2, x3, x4}); if (x1 == mi) return subset_sum_solution_1(vals, target); if (x2 == mi) return subset_sum_solution_2(vals, target); if (x3 == mi) return subset_sum_solution_3(vals, target); return subset_sum_solution_4(vals, target); }
#line 2 "ds/my_bitset.hpp" // https://codeforces.com/contest/914/problem/F // https://yukicoder.me/problems/no/142 // わずかに普通の bitset より遅いときもあるようだが, // 固定長にしたくないときや slice 操作が必要なときに使う struct My_Bitset { using T = My_Bitset; int N; vc<u64> dat; // x で埋める My_Bitset(int N = 0, int x = 0) : N(N) { assert(x == 0 || x == 1); u64 v = (x == 0 ? 0 : -1); dat.assign((N + 63) >> 6, v); if (N) dat.back() >>= (64 * len(dat) - N); } int size() { return N; } void resize(int size) { dat.resize((size + 63) >> 6); int remainingBits = size & 63; if (remainingBits != 0) { u64 mask = (u64(1) << remainingBits) - 1; dat.back() &= mask; } N = size; } static T from_string(string &S) { int N = len(S); T ANS(N); FOR(i, N) ANS[i] = (S[i] == '1'); return ANS; } // thanks to chatgpt! class Proxy { public: Proxy(vc<u64> &d, int i) : dat(d), index(i) {} operator bool() const { return (dat[index >> 6] >> (index & 63)) & 1; } Proxy &operator=(u64 value) { dat[index >> 6] &= ~(u64(1) << (index & 63)); dat[index >> 6] |= (value & 1) << (index & 63); return *this; } void flip() { dat[index >> 6] ^= (u64(1) << (index & 63)); // XOR to flip the bit } private: vc<u64> &dat; int index; }; Proxy operator[](int i) { return Proxy(dat, i); } bool operator==(const T &p) { assert(N == p.N); FOR(i, len(dat)) if (dat[i] != p.dat[i]) return false; return true; } T &operator&=(const T &p) { assert(N == p.N); FOR(i, len(dat)) dat[i] &= p.dat[i]; return *this; } T &operator|=(const T &p) { assert(N == p.N); FOR(i, len(dat)) dat[i] |= p.dat[i]; return *this; } T &operator^=(const T &p) { assert(N == p.N); FOR(i, len(dat)) dat[i] ^= p.dat[i]; return *this; } T operator&(const T &p) const { return T(*this) &= p; } T operator|(const T &p) const { return T(*this) |= p; } T operator^(const T &p) const { return T(*this) ^= p; } T operator~() const { T p = (*this); p.flip_range(0, N); return p; } int count() { int ans = 0; for (u64 val: dat) ans += popcnt(val); return ans; } int dot(T &p) { assert(N == p.N); int ans = 0; FOR(i, len(dat)) ans += popcnt(dat[i] & p.dat[i]); return ans; } int dot_mod_2(T &p) { assert(N == p.N); int ans = 0; FOR(i, len(dat)) ans ^= popcnt_mod_2(dat[i] & p.dat[i]); return ans; } int next(int i) { chmax(i, 0); if (i >= N) return N; int k = i >> 6; { u64 x = dat[k]; int s = i & 63; x = (x >> s) << s; if (x) return (k << 6) | lowbit(x); } FOR(idx, k + 1, len(dat)) { if (dat[idx] == 0) continue; return (idx << 6) | lowbit(dat[idx]); } return N; } int prev(int i) { chmin(i, N - 1); if (i <= -1) return -1; int k = i >> 6; if ((i & 63) < 63) { u64 x = dat[k]; x &= (u64(1) << ((i & 63) + 1)) - 1; if (x) return (k << 6) | topbit(x); --k; } FOR_R(idx, k + 1) { if (dat[idx] == 0) continue; return (idx << 6) | topbit(dat[idx]); } return -1; } My_Bitset range(int L, int R) { assert(L <= R); My_Bitset p(R - L); int rm = (R - L) & 63; FOR(rm) { p[R - L - 1] = bool((*this)[R - 1]); --R; } int n = (R - L) >> 6; int hi = L & 63; int lo = 64 - hi; int s = L >> 6; if (hi == 0) { FOR(i, n) { p.dat[i] ^= dat[s + i]; } } else { FOR(i, n) { p.dat[i] ^= (dat[s + i] >> hi) ^ (dat[s + i + 1] << lo); } } return p; } My_Bitset slice(int L, int R) { return range(L, R); } int count_range(int L, int R) { assert(L <= R); int cnt = 0; while ((L < R) && (L & 63)) cnt += (*this)[L++]; while ((L < R) && (R & 63)) cnt += (*this)[--R]; int l = L >> 6, r = R >> 6; FOR(i, l, r) cnt += popcnt(dat[i]); return cnt; } // [L,R) に p を代入 void assign_to_range(int L, int R, My_Bitset &p) { assert(p.N == R - L); int a = 0, b = p.N; while (L < R && (L & 63)) { (*this)[L++] = bool(p[a++]); } while (L < R && (R & 63)) { (*this)[--R] = bool(p[--b]); } // p[a:b] を [L:R] に int l = L >> 6, r = R >> 6; int s = a >> 6, t = b >> t; int n = r - l; if (!(a & 63)) { FOR(i, n) dat[l + i] = p.dat[s + i]; } else { int hi = a & 63; int lo = 64 - hi; FOR(i, n) dat[l + i] = (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo); } } // [L,R) に p を xor void xor_to_range(int L, int R, My_Bitset &p) { assert(p.N == R - L); int a = 0, b = p.N; while (L < R && (L & 63)) { dat[L >> 6] ^= u64(p[a]) << (L & 63); ++a, ++L; } while (L < R && (R & 63)) { --b, --R; dat[R >> 6] ^= u64(p[b]) << (R & 63); } // p[a:b] を [L:R] に int l = L >> 6, r = R >> 6; int s = a >> 6, t = b >> t; int n = r - l; if (!(a & 63)) { FOR(i, n) dat[l + i] ^= p.dat[s + i]; } else { int hi = a & 63; int lo = 64 - hi; FOR(i, n) dat[l + i] ^= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo); } } // 行列基本変形で使うやつ // p は [i:N) にしかないとして p を xor する void xor_suffix(int i, My_Bitset &p) { assert(N == p.N && 0 <= i && i < N); FOR(k, i / 64, len(dat)) { dat[k] ^= p.dat[k]; } } // [L,R) に p を and void and_to_range(int L, int R, My_Bitset &p) { assert(p.N == R - L); int a = 0, b = p.N; while (L < R && (L & 63)) { if (!p[a]) (*this)[L] = 0; a++, L++; } while (L < R && (R & 63)) { --b, --R; if (!p[b]) (*this)[R] = 0; } // p[a:b] を [L:R] に int l = L >> 6, r = R >> 6; int s = a >> 6, t = b >> t; int n = r - l; if (!(a & 63)) { FOR(i, n) dat[l + i] &= p.dat[s + i]; } else { int hi = a & 63; int lo = 64 - hi; FOR(i, n) dat[l + i] &= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo); } } // [L,R) に p を or void or_to_range(int L, int R, My_Bitset &p) { assert(p.N == R - L); int a = 0, b = p.N; while (L < R && (L & 63)) { dat[L >> 6] |= u64(p[a]) << (L & 63); ++a, ++L; } while (L < R && (R & 63)) { --b, --R; dat[R >> 6] |= u64(p[b]) << (R & 63); } // p[a:b] を [L:R] に int l = L >> 6, r = R >> 6; int s = a >> 6, t = b >> t; int n = r - l; if (!(a & 63)) { FOR(i, n) dat[l + i] |= p.dat[s + i]; } else { int hi = a & 63; int lo = 64 - hi; FOR(i, n) dat[l + i] |= (p.dat[s + i] >> hi) | (p.dat[1 + s + i] << lo); } } // 行列基本変形で使うやつ // p は [i:N) にしかないとして p を or する void or_suffix(int i, My_Bitset &p) { assert(N == p.N && 0 <= i && i < N); FOR(k, i / 64, len(dat)) { dat[k] |= p.dat[k]; } } // [L,R) を 1 に変更 void set_range(int L, int R) { while (L < R && (L & 63)) { set(L++); } while (L < R && (R & 63)) { set(--R); } FOR(i, L >> 6, R >> 6) dat[i] = u64(-1); } // [L,R) を 1 に変更 void reset_range(int L, int R) { while (L < R && (L & 63)) { reset(L++); } while (L < R && (R & 63)) { reset(--R); } FOR(i, L >> 6, R >> 6) dat[i] = u64(0); } // [L,R) を flip void flip_range(int L, int R) { while (L < R && (L & 63)) { flip(L++); } while (L < R && (R & 63)) { flip(--R); } FOR(i, L >> 6, R >> 6) dat[i] ^= u64(-1); } // bitset に仕様を合わせる void set(int i) { (*this)[i] = 1; } void reset(int i) { (*this)[i] = 0; } void flip(int i) { (*this)[i].flip(); } void set() { fill(all(dat), u64(-1)); resize(N); } void reset() { fill(all(dat), 0); } void flip() { FOR(i, len(dat) - 1) { dat[i] = u64(-1) ^ dat[i]; } int i = len(dat) - 1; FOR(k, 64) { if (64 * i + k >= size()) break; flip(64 * i + k); } } bool any() { FOR(i, len(dat)) { if (dat[i]) return true; } return false; } bool ALL() { dat.resize((N + 63) >> 6); int r = N & 63; if (r != 0) { u64 mask = (u64(1) << r) - 1; if (dat.back() != mask) return 0; } for (int i = 0; i < N / 64; ++i) if (dat[i] != u64(-1)) return false; return true; } int _Find_first() { return next(0); } int _Find_next(int p) { return next(p + 1); } static string TO_STR[256]; string to_string() const { if (TO_STR[0].empty()) precompute(); string S; for (auto &x: dat) { FOR(i, 8) S += TO_STR[(x >> (8 * i) & 255)]; } S.resize(N); return S; } static void precompute() { FOR(s, 256) { string x; FOR(i, 8) x += '0' + (s >> i & 1); TO_STR[s] = x; } } }; string My_Bitset::TO_STR[256]; #line 1 "enumerate/bits.hpp" template <typename F> void enumerate_bits_32(u32 s, F f) { while (s) { int i = __builtin_ctz(s); f(i); s ^= 1 << i; } } template <typename F> void enumerate_bits_64(u64 s, F f) { while (s) { int i = __builtin_ctzll(s); f(i); s ^= u64(1) << i; } } template <typename BS, typename F> void enumerate_bits_bitset(BS& b, int L, int R, F f) { int p = (b[L] ? L : b._Find_next(L)); while (p < R) { f(p); p = b._Find_next(p); } } #line 3 "knapsack/subset_sum.hpp" // O(N MAX(vals)) template <typename T> vc<int> subset_sum_solution_1(vc<T>& vals, int target) { int n = len(vals); if (n == 0) return {}; int mx = MAX(vals); int b = 0, sb = 0; while (b < n && sb + vals[b] <= target) { sb += vals[b++]; } if (b == n && sb != target) return {}; int off = target - mx + 1; vc<int> dp(2 * mx, -1); vv(int, PAR, n, 2 * mx, -1); dp[sb - off] = b; FOR3(i, b, n) { auto newdp = dp; auto& par = PAR[i]; int a = vals[i]; FOR(j, mx) { if (chmax(newdp[j + a], dp[j])) { par[j + a] = -2; } } FOR3_R(j, mx, 2 * mx) { FOR3_R(k, max(dp[j], 0), newdp[j]) { if (chmax(newdp[j - vals[k]], k)) par[j - vals[k]] = k; } } swap(dp, newdp); } if (dp[mx - 1] == -1) return {}; vc<bool> use(n); int i = n - 1, j = mx - 1; while (i >= b) { int p = PAR[i][j]; if (p == -2) { use[i] = !use[i]; j -= vals[i--]; } elif (p == -1) { --i; } else { use[p] = !use[p]; j += vals[p]; } } while (i >= 0) { use[i] = !use[i]; --i; } vc<int> I; FOR(i, n) if (use[i]) I.eb(i); ll sm = 0; for (auto&& i: I) sm += vals[i]; assert(sm == target); return I; } // O(N target / w) template <typename T> vc<int> subset_sum_solution_2(vc<T>& vals, int target) { int n = len(vals); auto I = argsort(vals); My_Bitset dp(1, 1); vc<int> last(target + 1, -1); FOR(k, n) { int v = vals[I[k]]; if (v > target) continue; My_Bitset newdp = dp; int new_size = len(dp) + v; newdp.resize(new_size); newdp.or_to_range(v, new_size, dp); if (len(newdp) > target + 1) newdp.resize(target + 1); // update したところをメモ FOR(i, len(newdp.dat)) { u64 upd = (i < len(dp.dat) ? dp.dat[i] : u64(0)) ^ newdp.dat[i]; enumerate_bits_64(upd, [&](int p) -> void { last[(i << 6) | p] = I[k]; }); } swap(dp, newdp); } if (target >= len(dp) || !dp[target]) return {}; vc<int> ANS; while (target > 0) { int i = last[target]; ANS.eb(i); target -= vals[i]; } return ANS; } // O(sum^{1.5} / w) // sum=10^6 で 150ms:https://codeforces.com/contest/755/problem/F template <typename T> vc<int> subset_sum_solution_3(vc<T>& vals, int target) { int SM = SUM<int>(vals); int N = len(vals); vvc<int> IDS(SM + 1); FOR(i, N) IDS[vals[i]].eb(i); vc<pair<int, int>> par(N, {-1, -1}); vc<int> grp_vals; vc<int> raw_idx; FOR(x, 1, SM + 1) { auto& I = IDS[x]; while (len(I) >= 3) { int a = POP(I), b = POP(I); int c = len(par); par.eb(a, b); IDS[2 * x].eb(c); } for (auto& i: I) { grp_vals.eb(x); raw_idx.eb(i); } } auto I = subset_sum_solution_2<int>(grp_vals, target); vc<int> ANS; for (auto& i: I) { vc<int> st = {raw_idx[i]}; while (len(st)) { auto c = POP(st); if (c < N) { ANS.eb(c); continue; } auto [a, b] = par[c]; st.eb(a), st.eb(b); } } return ANS; } template <typename T> vc<int> subset_sum_solution_4(vc<T>& vals, T target) { if (target <= 0) return {}; int N = len(vals); int M = N / 2; auto calc = [&](int L, int R) -> vc<T> { int n = R - L; vc<T> dp = {0}; FOR(i, n) { T a = vals[L + i]; vc<T> dp1(len(dp)); FOR(k, len(dp)) dp1[k] = dp[k] + a; vc<T> newdp; merge(all(dp), all(dp1), back_inserter(newdp)); swap(dp, newdp); } return dp; }; auto restore = [&](int L, int R, T v) -> vc<int> { int n = R - L; vc<T> dp(1 << n); FOR(i, n) FOR(s, 1 << i) dp[s | 1 << i] = dp[s] + vals[L + i]; FOR(s, 1 << n) { if (dp[s] == v) { vc<int> I; FOR(i, n) if (s >> i & 1) I.eb(L + i); return I; } } assert(0); return {}; }; auto dp1 = calc(0, M); auto dp2 = calc(M, N); int t = 0; FOR_R(s, len(dp1)) { while (t + 1 < len(dp2) && dp1[s] + dp2[t + 1] <= target) { ++t; } if (dp1[s] + dp2[t] == target) { vc<int> I1 = restore(0, M, dp1[s]); vc<int> I2 = restore(M, N, dp2[t]); concat(I1, I2); return I1; } } return {}; } template <typename T> vc<int> subset_sum(vc<T>& vals, T target) { if (target <= 0) return {}; int n = len(vals); if (n == 0) return {}; int mx = MAX(vals); // しきい値の調整をしていない // solution 1: O(N mx)) // solution 2: O(N target / w) // solution 3: O(sum^1.5 / w) // solution 4: O(2^(N/2)) double x1 = double(len(vals)) * mx; double x2 = double(len(vals)) * target / 500.0; double x3 = pow(SUM<double>(vals), 1.5) / 500.0; double x4 = pow(2.0, 0.5 * len(vals)); double mi = min({x1, x2, x3, x4}); if (x1 == mi) return subset_sum_solution_1(vals, target); if (x2 == mi) return subset_sum_solution_2(vals, target); if (x3 == mi) return subset_sum_solution_3(vals, target); return subset_sum_solution_4(vals, target); }