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#include "library/linear_algebra/matrix_f2.hpp"
$\mathbb{F} _ 2$ 上の行列に対する種々の演算は bitset による高速化が可能である。
#ifndef SUISEN_MATRIX_F2 #define SUISEN_MATRIX_F2 #include <cassert> #include <optional> #include <vector> #include "library/datastructure/util/dynamic_bitset.hpp" namespace suisen { struct MatrixF2 { MatrixF2() : MatrixF2(0, 0) {} MatrixF2(int n, int m, bool fill_value = false) : n(n), m(m), dat(n, DynamicBitSet(m, fill_value)) {} const DynamicBitSet& operator[](std::size_t i) const { return dat[i]; } DynamicBitSet& operator[](std::size_t i) { return dat[i]; } operator std::vector<DynamicBitSet>() const { return dat; } friend bool operator==(const MatrixF2& x, const MatrixF2& y) { return x.dat == y.dat; } friend bool operator!=(const MatrixF2& x, const MatrixF2& y) { return x.dat != y.dat; } std::pair<int, int> shape() const { return { n, m }; } int row_size() const { return n; } int col_size() const { return m; } MatrixF2 transposed() const { MatrixF2 t(m, n); for (std::size_t i = 0; i < n; ++i) for (std::size_t j = 0; j < m; ++j) t[j][i] = dat[i][j]; return t; } friend MatrixF2& operator+=(MatrixF2& x, const MatrixF2& y) { assert(x.n == y.n and x.m == y.m); for (std::size_t i = 0; i < x.n; ++i) x[i] ^= y[i]; return x; } friend MatrixF2& operator-=(MatrixF2& x, const MatrixF2& y) { return x += y; } friend MatrixF2& operator*=(MatrixF2& x, const MatrixF2& y) { return x = x * y; } friend MatrixF2& operator*=(MatrixF2& x, bool val) { if (not val) for (auto& row : x.dat) row.reset(); return x; } friend MatrixF2& operator/=(MatrixF2& x, const MatrixF2& y) { return x = x * *y.inv(); } friend MatrixF2& operator/=(MatrixF2& x, bool val) { assert(val); return x; } friend MatrixF2 operator+(MatrixF2 x, const MatrixF2& y) { x += y; return x; } friend MatrixF2 operator-(MatrixF2 x, const MatrixF2& y) { x -= y; return x; } friend MatrixF2 operator*(const MatrixF2& x, MatrixF2 y) { y = y.transposed(); assert(x.m == y.m); MatrixF2 z(x.n, y.n); for (std::size_t i = 0; i < x.n; ++i) for (std::size_t j = 0; j < y.n; ++j) { z[i][j] = (x[i] & y[j]).count() & 1; } return z; } friend MatrixF2 operator*(MatrixF2 x, bool val) { x *= val; return x; } friend MatrixF2 operator*(bool val, MatrixF2 x) { x *= val; return x; } friend MatrixF2 operator/(const MatrixF2 &x, const MatrixF2& y) { return x * *y.inv(); } friend MatrixF2 operator/(MatrixF2 x, bool val) { x /= val; return x; } DynamicBitSet operator*(const DynamicBitSet& x) const { assert(m == std::size_t(x.size())); DynamicBitSet y(n); for (std::size_t i = 0; i < n; ++i) y[i] = (dat[i] & x).count() & 1; return y; } MatrixF2 pow(long long b) const { assert(n == m); MatrixF2 p = *this, res = e1(n); for (; b; b >>= 1) { if (b & 1) res *= p; p *= p; } return res; } static MatrixF2 e0(std::size_t n) { return MatrixF2(n, n); } static MatrixF2 e1(std::size_t n) { MatrixF2 res(n, n); for (std::size_t i = 0; i < n; ++i) res[i][i] = 1; return res; } std::optional<MatrixF2> inv() const { assert(n == m); MatrixF2 A = *this, B = e1(n); for (std::size_t i = 0; i < n; ++i) { for (std::size_t j = i + 1; j < n; ++j) if (A[j][i]) { std::swap(A[i], A[j]), std::swap(B[i], B[j]); if (A[j][i]) A[j] ^= A[i], B[j] ^= B[i]; } if (not A[i][i]) return std::nullopt; } for (std::size_t i = n; i-- > 0;) { for (std::size_t j = 0; j < i; ++j) { if (A[j][i]) A[j] ^= A[i], B[j] ^= B[i]; } } return B; } bool det() const { MatrixF2 A = *this; for (std::size_t i = 0; i < n; ++i) { for (std::size_t j = i + 1; j < n; ++j) if (A[j][i]) { std::swap(A[i], A[j]); if (A[j][i]) A[j] ^= A[i]; } if (not A[i][i]) return false; } return true; } std::size_t rank() const { MatrixF2 A = *this; std::size_t r = 0; for (std::size_t j = 0; j < m; ++j) { for (std::size_t i = r + 1; i < n; ++i) if (A[i][j]) { std::swap(A[r], A[i]); if (A[i][j]) A[i] ^= A[r]; } r += A[r][j]; } return r; } private: std::size_t n, m; std::vector<DynamicBitSet> dat; }; } // namespace suisen #endif // SUISEN_MATRIX_F2
#line 1 "library/linear_algebra/matrix_f2.hpp" #include <cassert> #include <optional> #include <vector> #line 1 "library/datastructure/util/dynamic_bitset.hpp" #line 5 "library/datastructure/util/dynamic_bitset.hpp" #include <limits> #include <utility> #line 8 "library/datastructure/util/dynamic_bitset.hpp" namespace suisen { struct DynamicBitSet { private: using block = unsigned long long; static constexpr std::size_t block_size = std::numeric_limits<block>::digits; static constexpr std::size_t log_block_size = __builtin_ctz(block_size); struct bitref { block& b; std::size_t i; operator bool() const { return (b >> i) & 1; } bool test() const { return (b >> i) & 1; } void set() { b |= block(1) << i; } void reset() { b &= ~(block(1) << i); } void flip() { b ^= block(1) << i; } bitref& operator&=(bool val) { b &= block(val) << i; return *this; } bitref& operator|=(bool val) { b |= block(val) << i; return *this; } bitref& operator^=(bool val) { b ^= block(val) << i; return *this; } bitref& operator =(bool val) { val ? set() : reset(); return *this; } bitref& operator =(const bitref& v) { return (*this) = bool(v); } }; std::size_t n; std::vector<block> blocks; public: DynamicBitSet(std::size_t n = 0, bool fill_value = false) : n(n), blocks((n + block_size - 1) >> log_block_size, fill_value ? ~block(0) : 0) {} bool empty() const { return n == 0; } int size() const { return n; } void resize(std::size_t new_size, bool fill_value = false) { std::size_t new_block_num = (new_size + block_size - 1) >> log_block_size; if (new_block_num < block_num()) { n = new_size; return blocks.resize(new_block_num); } blocks.resize(new_block_num); std::size_t old_size = std::exchange(n, new_size); if (old_size <= new_size) range_update(old_size, new_size, fill_value); } void push_back(bool val) { if (n & (block_size - 1)) { (*this)[n] = val; } else { blocks.push_back(val); } ++n; } void pop_back() { if ((n & (block_size - 1)) == 1) blocks.pop_back(); --n; } friend bool operator==(const DynamicBitSet& x, const DynamicBitSet& y) { if (x.n != y.n) return false; if (x.empty()) return true; for (std::size_t i = 0; i < x.block_num() - 1; ++i) { if (x.blocks[i] != y.blocks[i]) return false; } const std::size_t num = x.n - ((x.block_num() - 1) << log_block_size); return get_lower_bits(x.blocks.back(), num) == get_lower_bits(y.blocks.back(), num); } friend bool operator!=(const DynamicBitSet& x, const DynamicBitSet& y) { return not (x == y); } friend bool operator<(const DynamicBitSet& x, const DynamicBitSet& y) { assert(x.n == y.n); if (x.empty()) return false; std::size_t num = x.n - ((x.block_num() - 1) << log_block_size); block tx = get_lower_bits(x.blocks.back(), num); block ty = get_lower_bits(y.blocks.back(), num); if (tx != ty) return tx < ty; for (std::size_t i = x.block_num() - 1; i-- > 0;) { if (x.blocks[i] != y.blocks[i]) return x.blocks[i] < y.blocks[i]; } return false; } friend bool operator<=(const DynamicBitSet& x, const DynamicBitSet& y) { assert(x.n == y.n); if (x.empty()) return true; std::size_t num = x.n - ((x.block_num() - 1) << log_block_size); block tx = get_lower_bits(x.blocks.back(), num); block ty = get_lower_bits(y.blocks.back(), num); if (tx != ty) return tx < ty; for (std::size_t i = x.block_num() - 1; i-- > 0;) { if (x.blocks[i] != y.blocks[i]) return x.blocks[i] < y.blocks[i]; } return true; } friend bool operator>(const DynamicBitSet& x, const DynamicBitSet& y) { return not (x <= y); } friend bool operator>=(const DynamicBitSet& x, const DynamicBitSet& y) { return not (x < y); } operator bool() const { return any(); } friend DynamicBitSet& operator&=(DynamicBitSet& x, const DynamicBitSet& y) { assert(x.n == y.n); for (std::size_t i = 0; i < y.block_num(); ++i) x.blocks[i] &= y.blocks[i]; return x; } friend DynamicBitSet& operator|=(DynamicBitSet& x, const DynamicBitSet& y) { assert(x.n == y.n); for (std::size_t i = 0; i < y.block_num(); ++i) x.blocks[i] |= y.blocks[i]; return x; } friend DynamicBitSet& operator^=(DynamicBitSet& x, const DynamicBitSet& y) { assert(x.n == y.n); for (std::size_t i = 0; i < y.block_num(); ++i) x.blocks[i] ^= y.blocks[i]; return x; } friend DynamicBitSet operator&(DynamicBitSet x, const DynamicBitSet& y) { x &= y; return x; } friend DynamicBitSet operator|(DynamicBitSet x, const DynamicBitSet& y) { x |= y; return x; } friend DynamicBitSet operator^(DynamicBitSet x, const DynamicBitSet& y) { x ^= y; return x; } friend DynamicBitSet& operator<<=(DynamicBitSet &x, std::size_t shamt) { return x = x << shamt; } friend DynamicBitSet& operator>>=(DynamicBitSet &x, std::size_t shamt) { return x = x >> shamt; } friend DynamicBitSet operator<<(const DynamicBitSet &x, std::size_t shamt) { if (shamt >= x.n) return DynamicBitSet(x.size()); DynamicBitSet res(x.size()); std::size_t block_shamt = shamt >> log_block_size; std::size_t bit_shamt = shamt & (block_size - 1); for (std::size_t i = 0; i + block_shamt < res.block_num(); ++i) { if (bit_shamt == 0) { res.blocks[i + block_shamt] = x.blocks[i]; } else { res.blocks[i + block_shamt] |= x.blocks[i] << bit_shamt; if (i + block_shamt + 1 != res.block_num()) { res.blocks[i + block_shamt + 1] |= x.blocks[i] >> (block_size - bit_shamt); } } } return res; } friend DynamicBitSet operator>>(const DynamicBitSet& x, std::size_t shamt) { if (shamt >= x.n) return DynamicBitSet(x.size()); DynamicBitSet res(x.size()); std::size_t block_shamt = shamt >> log_block_size; std::size_t bit_shamt = shamt & (block_size - 1); for (std::size_t i = 0; i + block_shamt < x.block_num(); ++i) { if (bit_shamt == 0) { res.blocks[i] = x.blocks[i + block_shamt]; } else { res.blocks[i] |= x.blocks[i + block_shamt] >> bit_shamt; if (i + block_shamt + 1 != x.block_num()) { res.blocks[i] |= x.blocks[i + block_shamt + 1] << (block_size - bit_shamt); } } } res.range_reset(x.n - shamt, x.n); return res; } DynamicBitSet operator~() const { DynamicBitSet neg(n); for (std::size_t i = 0; i < block_num(); ++i) neg.blocks[i] = ~blocks[i]; return neg; } bool operator[](std::size_t i) const { return (blocks[block_index(i)] >> bit_index(i)) & 1; } bitref operator[](std::size_t i) { return { blocks[block_index(i)], bit_index(i) }; } void range_set(std::size_t l, std::size_t r) { assert(l <= r and r <= n); if (l == r) return; std::size_t lb = block_index(l), rb = block_index(r - 1); std::size_t li = bit_index(l), ri = bit_index(r); if (ri == 0) ri = block_size; if (lb == rb) { blocks[lb] |= mask_range_bits(~block(0), li, ri); return; } blocks[lb] |= mask_upper_bits(~block(0), block_size - li); blocks[rb] |= mask_lower_bits(~block(0), ri); for (std::size_t i = lb + 1; i < rb; ++i) blocks[i] = ~block(0); } void range_reset(std::size_t l, std::size_t r) { assert(l <= r and r <= n); if (l == r) return; std::size_t lb = block_index(l), rb = block_index(r - 1); std::size_t li = bit_index(l), ri = bit_index(r); if (ri == 0) ri = block_size; if (lb == rb) { blocks[lb] &= ~mask_range_bits(~block(0), li, ri); return; } blocks[lb] &= ~mask_upper_bits(~block(0), block_size - li); blocks[rb] &= ~mask_lower_bits(~block(0), ri); for (std::size_t i = lb + 1; i < rb; ++i) blocks[i] = block(0); } void range_flip(std::size_t l, std::size_t r) { assert(l <= r and r <= n); if (l == r) return; std::size_t lb = block_index(l), rb = block_index(r - 1); std::size_t li = bit_index(l), ri = bit_index(r); if (ri == 0) ri = block_size; if (lb == rb) { blocks[lb] ^= mask_range_bits(~block(0), li, ri); return; } blocks[lb] ^= mask_upper_bits(~block(0), block_size - li); blocks[rb] ^= mask_lower_bits(~block(0), ri); for (std::size_t i = lb + 1; i < rb; ++i) blocks[i] ^= ~block(0); } void range_update(std::size_t l, std::size_t r, bool val) { val ? range_set(l, r) : range_reset(l, r); } int range_count(std::size_t l, std::size_t r) const { assert(l <= r and r <= n); if (l == r) return 0; std::size_t lb = block_index(l), rb = block_index(r - 1); std::size_t li = bit_index(l), ri = bit_index(r); if (ri == 0) ri = block_size; if (lb == rb) { return __builtin_popcountll(blocks[lb] & mask_range_bits(~block(0), li, ri)); } int res = 0; res += __builtin_popcountll(blocks[lb] & mask_upper_bits(~block(0), block_size - li)); res += __builtin_popcountll(blocks[rb] & mask_lower_bits(~block(0), ri)); for (std::size_t i = lb + 1; i < rb; ++i) res += __builtin_popcountll(blocks[i]); return res; } void set() { for (block& b : blocks) b = ~block(0); } void reset() { for (block& b : blocks) b = 0; } bool all() const { if (empty()) return true; for (std::size_t i = 0; i < block_num() - 1; ++i) { if (blocks[i] != ~block(0)) return false; } const std::size_t num = n - ((block_num() - 1) << log_block_size); assert(num); const block upper = ((block(1) << (block_size - num)) - 1) << num; return (upper | blocks.back()) == ~block(0); } bool none() const { if (empty()) return true; for (std::size_t i = 0; i < block_num() - 1; ++i) { if (blocks[i] != 0) return false; } const std::size_t num = n - ((block_num() - 1) << log_block_size); return get_lower_bits(blocks.back(), num) == 0; } bool any() const { return not none(); } int count() const { if (empty()) return 0; int res = 0; for (std::size_t i = 0; i < block_num() - 1; ++i) { res += __builtin_popcountll(blocks[i]); } const std::size_t num = n - ((block_num() - 1) << log_block_size); return res + __builtin_popcountll(get_lower_bits(blocks.back(), num)); } // Returns the position of first set bit. If there is no such positions, then returns size(). int find_first() const { if (empty()) return size(); for (std::size_t i = 0; i < block_num(); ++i) { if (blocks[i] != 0) return std::min(n, __builtin_ctzll(blocks[i]) | (i << log_block_size)); } return n; } // Returns the position of first set bit after the given position (exclusive). If there is no such positions, then returns size(). int find_next(std::size_t pos) const { std::size_t i = block_index(++pos); if (i >= blocks.size()) return n; block upper = mask_upper_bits(blocks[i], block_size - bit_index(pos)); if (upper != 0) return std::min(n, __builtin_ctzll(upper) | (i << log_block_size)); while (++i < block_num()) { if (blocks[i] != 0) return std::min(n, __builtin_ctzll(blocks[i]) | (i << log_block_size)); } return n; } bool has_intersection(const DynamicBitSet& y) const { if (n > y.n) return y.has_intersection(*this); if (empty()) return false; for (std::size_t i = 0; i < block_num() - 1; ++i) { if (blocks[i] & y.blocks[i]) return true; } const std::size_t num = n - ((block_num() - 1) << log_block_size); return get_lower_bits(blocks.back(), num) & y.blocks[block_num() - 1]; } bool is_disjoint(const DynamicBitSet& y) const { return not has_intersection(y); } private: static constexpr std::size_t block_index(std::size_t i) { return i >> log_block_size; } static constexpr std::size_t bit_index(std::size_t i) { return i & (block_size - 1); } static constexpr block get_lower_bits(block b, std::size_t num) { return num ? (b << (block_size - num) >> (block_size - num)) : block(0); } static constexpr block get_upper_bits(block b, std::size_t num) { return num ? (b >> (block_size - num)) : block(0); } static constexpr block get_range_bits(block b, std::size_t l, std::size_t r) { return l < r ? b << (block_size - r) >> (block_size - r + l) : block(0); } static constexpr block mask_lower_bits(block b, std::size_t num) { return get_lower_bits(b, num); } static constexpr block mask_upper_bits(block b, std::size_t num) { return num ? (b >> (block_size - num) << (block_size - num)) : block(0); } static constexpr block mask_range_bits(block b, std::size_t l, std::size_t r) { return l < r ? b << (block_size - r) >> (block_size - r + l) << l : block(0); } std::size_t block_num() const { return blocks.size(); } }; } // namespace suisen #line 9 "library/linear_algebra/matrix_f2.hpp" namespace suisen { struct MatrixF2 { MatrixF2() : MatrixF2(0, 0) {} MatrixF2(int n, int m, bool fill_value = false) : n(n), m(m), dat(n, DynamicBitSet(m, fill_value)) {} const DynamicBitSet& operator[](std::size_t i) const { return dat[i]; } DynamicBitSet& operator[](std::size_t i) { return dat[i]; } operator std::vector<DynamicBitSet>() const { return dat; } friend bool operator==(const MatrixF2& x, const MatrixF2& y) { return x.dat == y.dat; } friend bool operator!=(const MatrixF2& x, const MatrixF2& y) { return x.dat != y.dat; } std::pair<int, int> shape() const { return { n, m }; } int row_size() const { return n; } int col_size() const { return m; } MatrixF2 transposed() const { MatrixF2 t(m, n); for (std::size_t i = 0; i < n; ++i) for (std::size_t j = 0; j < m; ++j) t[j][i] = dat[i][j]; return t; } friend MatrixF2& operator+=(MatrixF2& x, const MatrixF2& y) { assert(x.n == y.n and x.m == y.m); for (std::size_t i = 0; i < x.n; ++i) x[i] ^= y[i]; return x; } friend MatrixF2& operator-=(MatrixF2& x, const MatrixF2& y) { return x += y; } friend MatrixF2& operator*=(MatrixF2& x, const MatrixF2& y) { return x = x * y; } friend MatrixF2& operator*=(MatrixF2& x, bool val) { if (not val) for (auto& row : x.dat) row.reset(); return x; } friend MatrixF2& operator/=(MatrixF2& x, const MatrixF2& y) { return x = x * *y.inv(); } friend MatrixF2& operator/=(MatrixF2& x, bool val) { assert(val); return x; } friend MatrixF2 operator+(MatrixF2 x, const MatrixF2& y) { x += y; return x; } friend MatrixF2 operator-(MatrixF2 x, const MatrixF2& y) { x -= y; return x; } friend MatrixF2 operator*(const MatrixF2& x, MatrixF2 y) { y = y.transposed(); assert(x.m == y.m); MatrixF2 z(x.n, y.n); for (std::size_t i = 0; i < x.n; ++i) for (std::size_t j = 0; j < y.n; ++j) { z[i][j] = (x[i] & y[j]).count() & 1; } return z; } friend MatrixF2 operator*(MatrixF2 x, bool val) { x *= val; return x; } friend MatrixF2 operator*(bool val, MatrixF2 x) { x *= val; return x; } friend MatrixF2 operator/(const MatrixF2 &x, const MatrixF2& y) { return x * *y.inv(); } friend MatrixF2 operator/(MatrixF2 x, bool val) { x /= val; return x; } DynamicBitSet operator*(const DynamicBitSet& x) const { assert(m == std::size_t(x.size())); DynamicBitSet y(n); for (std::size_t i = 0; i < n; ++i) y[i] = (dat[i] & x).count() & 1; return y; } MatrixF2 pow(long long b) const { assert(n == m); MatrixF2 p = *this, res = e1(n); for (; b; b >>= 1) { if (b & 1) res *= p; p *= p; } return res; } static MatrixF2 e0(std::size_t n) { return MatrixF2(n, n); } static MatrixF2 e1(std::size_t n) { MatrixF2 res(n, n); for (std::size_t i = 0; i < n; ++i) res[i][i] = 1; return res; } std::optional<MatrixF2> inv() const { assert(n == m); MatrixF2 A = *this, B = e1(n); for (std::size_t i = 0; i < n; ++i) { for (std::size_t j = i + 1; j < n; ++j) if (A[j][i]) { std::swap(A[i], A[j]), std::swap(B[i], B[j]); if (A[j][i]) A[j] ^= A[i], B[j] ^= B[i]; } if (not A[i][i]) return std::nullopt; } for (std::size_t i = n; i-- > 0;) { for (std::size_t j = 0; j < i; ++j) { if (A[j][i]) A[j] ^= A[i], B[j] ^= B[i]; } } return B; } bool det() const { MatrixF2 A = *this; for (std::size_t i = 0; i < n; ++i) { for (std::size_t j = i + 1; j < n; ++j) if (A[j][i]) { std::swap(A[i], A[j]); if (A[j][i]) A[j] ^= A[i]; } if (not A[i][i]) return false; } return true; } std::size_t rank() const { MatrixF2 A = *this; std::size_t r = 0; for (std::size_t j = 0; j < m; ++j) { for (std::size_t i = r + 1; i < n; ++i) if (A[i][j]) { std::swap(A[r], A[i]); if (A[i][j]) A[i] ^= A[r]; } r += A[r][j]; } return r; } private: std::size_t n, m; std::vector<DynamicBitSet> dat; }; } // namespace suisen