adds gauss xor and it's respective tests

algorithms/math/gauss-xor.cpp

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+const int MAXXI = 2009;
+using Equation = bitset<MAXXI>;
+struct GaussXor {
+	vector<char> B;
+	vector<Equation> A;
+
+	void add(const Equation& ai, bool bi) {
+		A.push_back(ai);
+		B.push_back(bi);
+	}
+
+	pair<bool, Equation> solution() {
+		int cnt = 0, n = A.size();
+		Equation vis; vis.set();
+		Equation x;
+		for(int j = MAXXI-1, i; j >= 0; j--) {
+			for(i = cnt; i < n; i++) {
+				if (A[i][j]) break;
+			}
+			if (i == n) continue;
+			swap(A[i], A[cnt]), swap(B[i], B[cnt]);
+			i = cnt++; vis[j] = 0;
+			for(int k = 0; k < n; k++) {
+				if (i == k || !A[k][j]) continue;
+				A[k] ^= A[i]; B[k] ^= B[i];
+			}
+		}
+		x = vis;
+		for(int i = 0; i < n; i++) {
+			int acum = 0;
+			for(int j = 0; j < MAXXI; j++) {
+				if (!A[i][j]) continue;
+				if (!vis[j]) {
+					vis[j] = 1;
+					x[j] = acum^B[i];
+				}
+				acum ^= x[j];
+			}
+			if (acum != B[i]) return {false, Equation()};
+		}
+		return {true, x};
+	}
+
+};
+

algorithms/math/gauss-xor.tex

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+\subsection{Gauss XOR Elimination / XOR-SAT}
+
+Execute gaussian elimination with xor over the system $Ax = b$ in. The add method must receive a bitset indicating which variables are present in the equation, and the solution of the equation.
+
+Time complexitiy: $O(\frac{nm^{2}}{64})$

tests/math/gauss-xor/CF-GYM-102861-K/in/001.txt

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+4 4
+4 2
+1 3
+2 3
+1 4

tests/math/gauss-xor/CF-GYM-102861-K/in/002.txt

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+4 3
+4 2
+2 3
+1 2

tests/math/gauss-xor/CF-GYM-102861-K/in/003.txt

@@ -0,0 +1,6 @@
+5 5
+3 5
+3 1
+1 4
+2 5
+2 4

tests/math/gauss-xor/CF-GYM-102861-K/main.cpp

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+#include <bits/stdc++.h>
+using namespace std;
+
+#include "../../../../algorithms/math/gauss-xor.cpp"
+
+int32_t main() {
+	ios_base::sync_with_stdio(0);
+	cin.tie(0);
+
+	int n, f;
+	cin >> n >> f;
+
+	vector<vector<int>> adj(n);
+	while (f--) {
+		int u, v;
+		cin >> u >> v;
+		u--, v--;
+		adj[u].push_back(v);
+		adj[v].push_back(u);
+	}
+
+	GaussXor gauss;
+	for (int i = 0; i < n; i++) {
+		Equation eq;
+
+		for (auto v : adj[i]) {
+			eq.set(v);
+		}
+
+		if (adj[i].size() & 1) {
+			eq.set(i);
+			gauss.add(eq, 0);
+		}
+		else {
+			gauss.add(eq, 1);
+		}
+	}
+
+	cout << "NY\n"[gauss.solution().first] << '\n';
+}
+

tests/math/gauss-xor/CF-GYM-102861-K/out/001.txt

@@ -0,0 +1 @@
+Y

tests/math/gauss-xor/CF-GYM-102861-K/out/002.txt

@@ -0,0 +1 @@
+Y

tests/math/gauss-xor/CF-GYM-102861-K/out/003.txt

@@ -0,0 +1 @@
+N