F2 determinant

Author: hitonanode

linear_algebra_matrix/linalg_bitset.hpp

@@ -43,6 +43,33 @@ template <int Wmax> int rank_gauss_jordan(int W, const std::vector<std::bitset<W
     return 0;
 }
 
+// determinant of F2 matrix.
+// Return 0 if the matrix is singular, otherwise return 1.
+// Complexity: O(W^3 / 64)
+template <int Wmax> int f2_determinant(const std::vector<std::bitset<Wmax>> &M) {
+    const int H = M.size();
+    if (H > Wmax) return 0;
+
+    auto tmp = M;
+    for (int h = 0; h < H; ++h) {
+        int piv = -1;
+        for (int j = h; j < H; ++j) {
+            if (tmp.at(j).test(h)) {
+                piv = j;
+                break;
+            }
+        }
+        if (piv == -1) return 0; // singular
+
+        if (piv != h) std::swap(tmp.at(piv), tmp.at(h));
+        for (int hh = h + 1; hh < H; ++hh) {
+            if (tmp.at(hh).test(h)) tmp.at(hh) ^= tmp.at(h);
+        }
+    }
+
+    return 1; // nonsingular
+}
+
 template <int W1, int W2>
 std::vector<std::bitset<W2>>
 matmul(const std::vector<std::bitset<W1>> &A, const std::vector<std::bitset<W2>> &B) {

linear_algebra_matrix/linalg_bitset.md

@@ -16,10 +16,14 @@ vector<bool> b;
 
 // Solve Ax = b (x: F_2^W)
 auto [feasible, x0, freedoms] = system_of_linear_equations<Wmax, vector<bool>>(A, b, W);
+
+// Calc determinant (or check whether A is regular)
+int det = f2_determinant<dim>(mat);
 ```
 
 ## 問題例
 
 - [AOJ 2624: Graph Automata Player](https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=2624)
 - [No.1421 国勢調査 (Hard) - yukicoder](https://yukicoder.me/problems/no/1421)
 - [AtCoder Beginner Contest 276 Ex - Construct a Matrix](https://atcoder.jp/contests/abc276/tasks/abc276_h) $2000 \times 8000$ 行列の線型方程式を解く．
+- [Library Checker: Determinant of Matrix (Mod 2)](https://judge.yosupo.jp/problem/matrix_det_mod_2)

linear_algebra_matrix/test/linalg_bitset_det.test.cpp

@@ -0,0 +1,28 @@
+#define PROBLEM "https://judge.yosupo.jp/problem/matrix_det_mod_2"
+#include "../linalg_bitset.hpp"
+
+#include <bitset>
+#include <iostream>
+#include <string>
+#include <vector>
+
+using namespace std;
+
+int main() {
+    cin.tie(nullptr);
+    ios::sync_with_stdio(false);
+
+    constexpr int dim = 1 << 12;
+    using BS = bitset<dim>;
+
+    int N;
+    cin >> N;
+    vector<BS> mat(N);
+    for (auto &v : mat) {
+        string s;
+        cin >> s;
+        v = BS(s);
+    }
+
+    cout << f2_determinant<dim>(mat) << '\n';
+}