#include #include #include #include #include #include #include #include #include #include #include int KthBit(const std::integral auto x, const int k) { return x >> k & 1; } void Solve() { static constexpr int kMaxNM = 50000, kDigits = 30; int n, m; std::cin >> n >> m; assert(1 <= n && 1 <= m && n * m <= kMaxNM); std::vector rc(n + m); for (int i = 0; i < n + m; ++i) { std::cin >> rc[i]; assert(0 <= rc[i] && rc[i] < (1 << kDigits)); } std::vector a(n, std::vector(m, 0)); for (int i = 0; i < n; ++i) { for (int j = 0; j < m; ++j) { std::cin >> a[i][j]; assert(0 <= a[i][j] && a[i][j] < (1 << kDigits)); } } std::vector group(n + m, 0); // ビットごと 2 グループに分ける for (int bit = 0; bit < kDigits; ++bit) { const int criteria = KthBit(a.front().front(), bit); for (int i = 1; i < n; ++i) { const int is_diff = (KthBit(a[i].front(), bit) != criteria ? 1 : 0); for (int j = 1; j < m; ++j) { if (KthBit(a.front()[j], bit) ^ is_diff != KthBit(a[i][j], bit)) { std::cout << "-1\n"; return; } } group[i] |= is_diff << bit; } group[n] |= criteria << bit; for (int j = 1; j < m; ++j) { const int is_diff = (KthBit(a.front()[j], bit) != criteria ? 1 : 0); for (int i = 1; i < n; ++i) { if (KthBit(a[i].front(), bit) ^ is_diff != KthBit(a[i][j], bit)) { std::cout << "-1\n"; return; } } group[j + n] |= KthBit(a.front()[j], bit) << bit; } } const auto Operation = [&rc, &group]( const int bit, const int choice, const std::vector& unknown, const std::vector& or_01, const std::vector& or_12) -> std::tuple, std::vector, std::vector> { int cost = 0; std::vector next_unknown, next_or_01, next_or_12; for (const int i : or_01) { if (KthBit(group[i], bit) == choice) { ++cost; } else { next_or_01.emplace_back(i); } } for (const int i : or_12) { const int bit_of_i = (KthBit(group[i], bit) == choice ? 1 : 0); const int bit_of_rc = KthBit(rc[i], bit); if (bit_of_i > bit_of_rc) { ++cost; } else if (bit_of_i == bit_of_rc) { next_or_12.emplace_back(i); } } for (const int i : unknown) { if (KthBit(group[i], bit) == choice) { // 1 if (KthBit(rc[i], bit) == 0) { return {false, 0, std::vector{}, std::vector{}, std::vector{}}; } ++cost; next_or_12.emplace_back(i); } else { // 0 (KthBit(rc[i], bit) == 1 ? next_or_01 : next_unknown).emplace_back(i); } } return {true, cost, next_unknown, next_or_01, next_or_12}; }; const int argmin = std::distance(rc.begin(), std::ranges::min_element(rc)); const int need_to_choose = // これより上位のビットは自由に選べない std::bit_width(static_cast(rc[argmin])); std::uint32_t chosen_group = 0; int current_cost = 0; std::vector unknown(n + m), or_01, or_12; std::iota(unknown.begin(), unknown.end(), 0); for (int bit = kDigits - 1; bit >= need_to_choose; --bit) { const int choice = KthBit(group[argmin], bit) ^ 1; bool is_valid; int added_cost; std::tie(is_valid, added_cost, unknown, or_01, or_12) = Operation(bit, choice, unknown, or_01, or_12); if (!is_valid) { std::cout << "-1\n"; return; } chosen_group |= choice << bit; current_cost += added_cost; } int min_cost = (n + m) * 2 + 1; std::uint32_t achievable_group = 0; // 上位からグループを決定する const auto dfs = [Operation, &min_cost, &achievable_group]( auto dfs, const int bit, const std::uint32_t chosen_group, const int current_cost, const std::vector unknown, const std::vector or_01, const std::vector or_12) -> void { if (bit < 0) { if (current_cost < min_cost) { min_cost = current_cost; achievable_group = chosen_group; } return; } std::array next_cost{}; std::array, 2> next_unknown{}, next_or_01{}, next_or_12{}; for (const int choice : std::vector{0, 1}) { std::tie(std::ignore, next_cost[choice], next_unknown[choice], next_or_01[choice], next_or_12[choice]) = Operation(bit, choice, unknown, or_01, or_12); next_cost[choice] += current_cost; } const int early = (next_cost[0] + next_unknown[0].size() + next_or_01[0].size() < next_cost[1] + next_unknown[1].size() + next_or_01[1].size() ? 0 : 1); for (const int parity : std::vector{1, 0}) { const int choice = parity ^ early; if (next_cost[choice] >= min_cost) continue; dfs(dfs, bit - 1, chosen_group | (choice << bit), next_cost[choice], next_unknown[choice], next_or_01[choice], next_or_12[choice]); } }; dfs(dfs, need_to_choose - 1, chosen_group, current_cost, unknown, or_01, or_12); // 操作列の生成 std::vector> ans; for (int i = 0; i < n; ++i) { const int ans_i = group[i] ^ achievable_group ^ UINT32_C((1 << kDigits) - 1); if (ans_i > rc[i]) { assert(0 <= (ans_i ^ rc[i]) && (ans_i ^ rc[i]) <= rc[i]); ans.emplace_back('r', i, rc[i]); ans.emplace_back('r', i, ans_i ^ rc[i]); } else if (ans_i > 0) { ans.emplace_back('r', i, ans_i); } } for (int j = 0; j < m; ++j) { const int ans_j = group[j + n] ^ achievable_group ^ UINT32_C((1 << kDigits) - 1); if (ans_j > rc[j + n]) { assert(0 <= (ans_j ^ rc[j + n]) && (ans_j ^ rc[j + n]) <= rc[j + n]); ans.emplace_back('c', j, rc[j + n]); ans.emplace_back('c', j, ans_j ^ rc[j + n]); } else if (ans_j > 0) { ans.emplace_back('c', j, ans_j); } } const int k = ans.size(); assert(k == min_cost); std::cout << k << '\n'; for (const auto& [r_or_c, ij, x] : ans) { std::cout << r_or_c << ' ' << ij + 1 << ' ' << x << std::endl; } } int main() { constexpr int kMaxT = 50000; int t; std::cin >> t; assert(1 <= t && t <= kMaxT); while (t--) Solve(); return 0; }