xor basis walk (#193)

* first draft of walk for xor basis

* [auto-verifier] verify commit 8c0af3f

* changes

* [auto-verifier] verify commit 796f68e

* add assert to test it now

* asdf

* [auto-verifier] verify commit 504ae7d

* fix test now

* fix test

* fix

---------

Co-authored-by: GitHub <noreply@github.com>

Author: lrvideckis

.verify-helper/timestamps.remote.json

@@ -90,6 +90,7 @@
 "tests/library_checker_aizu_tests/handmade_tests/seg_tree_find.test.cpp": "2026-01-23 04:31:29 +0000",
 "tests/library_checker_aizu_tests/handmade_tests/seg_tree_find_small.test.cpp": "2026-01-23 04:31:29 +0000",
 "tests/library_checker_aizu_tests/handmade_tests/seg_tree_midpoint.test.cpp": "2026-02-27 11:16:07 -0700",
+"tests/library_checker_aizu_tests/handmade_tests/xor_basis_walk.test.cpp": "2026-03-08 13:19:49 -0600",
 "tests/library_checker_aizu_tests/loops/chooses.test.cpp": "2025-02-10 14:50:36 -0700",
 "tests/library_checker_aizu_tests/loops/quotients.test.cpp": "2024-11-17 14:04:03 -0600",
 "tests/library_checker_aizu_tests/loops/submasks.test.cpp": "2025-02-10 14:50:36 -0700",
@@ -106,7 +107,6 @@
 "tests/library_checker_aizu_tests/math/solve_linear_mod.test.cpp": "2025-09-07 16:12:35 -0600",
 "tests/library_checker_aizu_tests/math/tetration.test.cpp": "2024-11-17 14:04:03 -0600",
 "tests/library_checker_aizu_tests/math/totient.test.cpp": "2024-11-17 14:04:03 -0600",
-"tests/library_checker_aizu_tests/math/xor_basis_intersection.test.cpp": "2026-03-06 16:39:24 -0700",
 "tests/library_checker_aizu_tests/monotonic_stack_related/cartesian_binary_tree.test.cpp": "2025-02-10 14:50:36 -0700",
 "tests/library_checker_aizu_tests/monotonic_stack_related/cartesian_k_ary_tree.test.cpp": "2026-01-18 11:15:41 +0000",
 "tests/library_checker_aizu_tests/monotonic_stack_related/count_rectangles.test.cpp": "2026-01-18 11:15:41 +0000",

library/math/matrix_related/walk.hpp

@@ -0,0 +1,6 @@
+bitset<B> walk(bitset<B>& k) {
+  bitset<B> res;
+  for (int i = B, j = npivot; i--;)
+    if (basis[i][i] && res[i] ^ k[--j]) res ^= basis[i];
+  return res;
+}

library/math/matrix_related/xor_basis_ordered.hpp

@@ -5,24 +5,24 @@
 //!   basis_ordered<int> b1;
 //!   basis_ordered<ll> b2;
 //!   basis_ordered<bitset<lg>> b3;
-//!   b2.insert(x); // 0 <= x < (1LL<<lg)
+//!   b2.insert(x); // 0 <= x < (1LL<<B)
 //! @endcode
-//! @time O(lg)
-//! @space O(lg)
-const int lg = 60;
-template<class T> struct basis_ordered {
-  int siz = 0;
-  T b[lg]{};
-  int shrink(T& v) {
-    for (int i = lg - 1; i >= 0; i--)
-      if (((v >> i) & T(1)) != T(0)) {
-        if (b[i] == T(0)) return i;
-        v ^= b[i];
+//! @time O(q * B^2/32)
+//! @space O(B^2/32)
+template<int B> struct xor_basis {
+  bitset<B> basis[B];
+  int npivot = 0, nfree = 0;
+  int shrink(bitset<B>& v) {
+    for (int i = B; i--;)
+      if (v[i]) {
+        if (!basis[i][i]) return i;
+        v ^= basis[i];
       }
     return -1;
   }
-  bool insert(T v) {
+  bool insert(bitset<B>& v) {
     int i = shrink(v);
-    return i >= 0 ? b[i] = v, ++siz : 0;
+    return i >= 0 ? basis[i] = v, ++npivot : !++nfree;
   }
+#include "walk.hpp"
 };

tests/.config/.cppcheck_suppression_list

@@ -63,4 +63,3 @@ unusedFunction:../kactl/stress-tests/utilities/genTree.h:49
 containerOutOfBounds:../library/data_structures_[l,r)/uncommon/permutation_tree.hpp:85
 ctuOneDefinitionRuleViolation:../library/data_structures_[l,r)/bit.hpp:5
 ctuOneDefinitionRuleViolation:../library/data_structures_[l,r)/lazy_seg_tree.hpp:4
-shiftTooManyBits:../library/math/matrix_related/xor_basis_ordered.hpp:18

tests/library_checker_aizu_tests/handmade_tests/xor_basis_walk.test.cpp

@@ -0,0 +1,46 @@
+#define PROBLEM \
+  "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A"
+#include "../template.hpp"
+#include "../../../library/contest/random.hpp"
+#include "../../../library/math/matrix_related/xor_basis_ordered.hpp"
+const int B = 18;
+vector<bitset<B>> get_all(const vector<bitset<B>>& basis) {
+  int n = ssize(basis);
+  vector<bitset<B>> span;
+  for (int mask = 0; mask < (1 << n); mask++) {
+    bitset<B> curr_xor;
+    assert(curr_xor.none());
+    for (int bit = 0; bit < n; bit++)
+      if ((mask >> bit) & 1) curr_xor ^= basis[bit];
+    span.push_back(curr_xor);
+  }
+  ranges::sort(span, {}, [&](const bitset<B>& x) -> long {
+    return x.to_ulong();
+  });
+  return span;
+}
+int main() {
+  cin.tie(0)->sync_with_stdio(0);
+  for (int num_tests = 0; num_tests < 1000; num_tests++) {
+    xor_basis<B> b;
+    int n = rnd(1, 18);
+    vector<bitset<B>> naive_basis;
+    for (int i = 0; i < n; i++) {
+      bitset<B> val = rnd(0, (1 << n) - 1);
+      if (b.insert(val)) naive_basis.push_back(val);
+      assert(b.npivot + b.nfree == i + 1);
+    }
+    assert(ssize(naive_basis) == b.npivot);
+    vector<bitset<B>> fast_basis;
+    for (int i = 0; i < B; i++)
+      if (b.basis[i][i]) fast_basis.push_back(b.basis[i]);
+    vector<bitset<B>> naive_span = get_all(naive_basis);
+    vector<bitset<B>> fast_span = get_all(fast_basis);
+    assert(naive_span == fast_span);
+    for (int i = 0; i < ssize(naive_span); i++) {
+      bitset<B> k = i;
+      assert(naive_span[i] == b.walk(k));
+    }
+  }
+  cout << "Hello World\n";
+}

tests/library_checker_aizu_tests/math/xor_basis_intersection.test.cpp

@@ -14,20 +14,16 @@ void check_condition_unordered(const basis& b) {
     or_bits |= b.b[i];
   }
 }
-void check_condition_ordered(
-  const basis_ordered<ll>& basis3,
-  const basis_ordered<bitset<lg>>& basis4) {
-  ll or_bits3 = 0;
-  bitset<lg> or_bits4 = 0;
-  for (int i = 0; i < lg; i++) {
-    assert(basis3.b[i] == 0 ||
-      bit_floor(1ULL * basis3.b[i]) == (1ULL << i));
-    assert(
-      (bit_floor(uint64_t(basis3.b[i])) & or_bits3) == 0);
-    assert((bit_floor(uint64_t(basis4.b[i].to_ullong())) &
-             or_bits4.to_ullong()) == 0);
-    or_bits3 |= basis3.b[i];
-    or_bits4 |= basis4.b[i];
+const int B = 30;
+void check_condition_ordered(const xor_basis<B>& basis2) {
+  bitset<B> or_bits2;
+  for (int i = 0; i < B; i++) {
+    assert(basis2.basis[i].none() ||
+      bit_floor(basis2.basis[i].to_ulong()) ==
+        (1ULL << i));
+    assert((bit_floor(basis2.basis[i].to_ulong()) &
+             or_bits2.to_ulong()) == 0);
+    or_bits2 |= basis2.basis[i];
   }
 }
 int main() {
@@ -38,43 +34,36 @@ int main() {
     int n;
     cin >> n;
     basis basis1;
-    basis_ordered<int> basis2; // to check that it compiles
-    assert(
-      sz(basis2.b) == lg); // remove unused var warning
-    basis_ordered<ll> basis3;
-    basis_ordered<bitset<lg>> basis4;
+    xor_basis<B> basis2;
+    for (int i = 0; i < B; i++)
+      assert(basis2.basis[i].none());
     for (int i = 0; i < n; i++) {
       int val;
       cin >> val;
+      bitset<B> v = val;
       assert(basis1.insert(val));
       assert(!basis1.insert(val));
-      assert(basis3.insert(val));
-      assert(!basis3.insert(val));
-      assert(basis4.insert(val));
-      assert(!basis4.insert(val));
+      assert(basis2.insert(v));
+      assert(!basis2.insert(v));
     }
     check_condition_unordered(basis1);
-    check_condition_ordered(basis3, basis4);
+    check_condition_ordered(basis2);
     int m;
     cin >> m;
-    basis basis5;
-    basis_ordered<int> basis6;
-    assert(sz(basis6.b) == lg);
-    basis_ordered<ll> basis7;
-    basis_ordered<bitset<lg>> basis8;
+    basis basis3;
+    xor_basis<B> basis4;
     for (int j = 0; j < m; j++) {
       int val;
       cin >> val;
-      assert(basis5.insert(val));
-      assert(!basis5.insert(val));
-      assert(basis7.insert(val));
-      assert(!basis7.insert(val));
-      assert(basis8.insert(val));
-      assert(!basis8.insert(val));
+      bitset<B> v = val;
+      assert(basis3.insert(val));
+      assert(!basis3.insert(val));
+      assert(basis4.insert(v));
+      assert(!basis4.insert(v));
     }
-    check_condition_unordered(basis5);
-    check_condition_ordered(basis7, basis8);
-    basis inter = intersection(basis1, basis5);
+    check_condition_unordered(basis3);
+    check_condition_ordered(basis4);
+    basis inter = intersection(basis1, basis3);
     check_condition_unordered(inter);
     cout << sz(inter.b) << " ";
     for (int val : inter.b) cout << val << " ";