test/aoj/2397.MinPoly.test.cpp

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• Last update: 2024-10-01 17:03:34+09:00
• Problem: https://onlinejudge.u-aizu.ac.jp/challenges/sources/JAG/Spring/2397

Depends on

• modintを扱うテンプレート (src/Internal/modint_traits.hpp)
• 剰余の高速化 (src/Internal/Remainder.hpp)
• 行列 (src/LinearAlgebra/Matrix.hpp)
• 行列の最小多項式 (src/LinearAlgebra/MinimalPolynomial.hpp)
• ベクトル (src/LinearAlgebra/Vector.hpp)
• Berlekamp-Massey (src/Math/berlekamp_massey.hpp)
• 逆元 ($\mathbb{Z}/m\mathbb{Z}$) (src/Math/mod_inv.hpp)
• ModInt (src/Math/ModInt.hpp)
• 疑似乱数 (src/Misc/rng.hpp)

Code

// competitive-verifier: PROBLEM https://onlinejudge.u-aizu.ac.jp/challenges/sources/JAG/Spring/2397
// competitive-verifier: TLE 0.5
// competitive-verifier: MLE 64
// 最小多項式を用いた行列累乗
#include <iostream>
#include <algorithm>
#include "src/LinearAlgebra/Matrix.hpp"
#include "src/LinearAlgebra/MinimalPolynomial.hpp"
#include "src/Math/ModInt.hpp"
using namespace std;
signed main() {
 cin.tie(0);
 ios::sync_with_stdio(0);
 using Mint= ModInt<int(1e9 + 9)>;
 long long W, H, N;
 for (int cnt= 0; cin >> W >> H >> N && W;) {
  pair<long long, int> obst[N];
  for (long long i= 0, x, y; i < N; i++) cin >> x >> y, obst[i]= make_pair(y - 1, x - 1);
  sort(obst, obst + N);
  Matrix<Mint> A(W, W);
  for (int i= 0; i < W; i++) {
   A[i][i]= 1;
   if (i) A[i][i - 1]= 1;
   if (i + 1 < W) A[i][i + 1]= 1;
  }
  Vector<Mint> b(W);
  b[0]= 1;
  long long y= 0;
  for (int i= 0; i < N; i++) {
   b= MinimalPolynomial(A, b).pow(obst[i].first - y);
   int j= i;
   while (j < N && obst[i].first == obst[j].first) b[obst[j++].second]= 0;
   i= j - 1;
   y= obst[i].first;
  }
  b= MinimalPolynomial(A, b).pow(H - 1 - y);
  cout << "Case " << ++cnt << ": " << b[W - 1] << '\n';
 }
 return 0;
}

#line 1 "test/aoj/2397.MinPoly.test.cpp"
// competitive-verifier: PROBLEM https://onlinejudge.u-aizu.ac.jp/challenges/sources/JAG/Spring/2397
// competitive-verifier: TLE 0.5
// competitive-verifier: MLE 64
// 最小多項式を用いた行列累乗
#include <iostream>
#include <algorithm>
#line 2 "src/LinearAlgebra/Matrix.hpp"
#include <cassert>
#include <vector>
#line 2 "src/LinearAlgebra/Vector.hpp"
#include <cstdint>
#line 4 "src/LinearAlgebra/Vector.hpp"
#include <valarray>
namespace _la_internal {
using namespace std;
template <class R> struct Vector {
 valarray<R> dat;
 Vector()= default;
 Vector(size_t n): dat(n) {}
 Vector(size_t n, const R &v): dat(v, n) {}
 Vector(const initializer_list<R> &v): dat(v) {}
 R &operator[](int i) { return dat[i]; }
 const R &operator[](int i) const { return dat[i]; }
 bool operator==(const Vector &r) const {
  if (dat.size() != r.dat.size()) return false;
  for (int i= dat.size(); i--;)
   if (dat[i] != r.dat[i]) return false;
  return true;
 }
 bool operator!=(const Vector &r) const { return !(*this == r); }
 explicit operator bool() const { return dat.size(); }
 Vector operator-() const { return Vector(dat.size())-= *this; }
 Vector &operator+=(const Vector &r) { return dat+= r.dat, *this; }
 Vector &operator-=(const Vector &r) { return dat-= r.dat, *this; }
 Vector &operator*=(const R &r) { return dat*= r, *this; }
 Vector operator+(const Vector &r) const { return Vector(*this)+= r; }
 Vector operator-(const Vector &r) const { return Vector(*this)-= r; }
 Vector operator*(const R &r) const { return Vector(*this)*= r; }
 size_t size() const { return dat.size(); }
 friend R dot(const Vector<R> &a, const Vector<R> &b) { return assert(a.size() == b.size()), (a.dat * b.dat).sum(); }
};
using u128= __uint128_t;
using u64= uint64_t;
using u8= uint8_t;
class Ref {
 u128 *ref;
 u8 i;
public:
 Ref(u128 *ref, u8 i): ref(ref), i(i) {}
 Ref &operator=(const Ref &r) { return *this= bool(r); }
 Ref &operator=(bool b) { return *ref&= ~(u128(1) << i), *ref|= u128(b) << i, *this; }
 Ref &operator|=(bool b) { return *ref|= u128(b) << i, *this; }
 Ref &operator&=(bool b) { return *ref&= ~(u128(!b) << i), *this; }
 Ref &operator^=(bool b) { return *ref^= u128(b) << i, *this; }
 operator bool() const { return (*ref >> i) & 1; }
};
template <> class Vector<bool> {
 size_t n;
public:
 valarray<u128> dat;
 Vector(): n(0) {}
 Vector(size_t n): n(n), dat((n + 127) >> 7) {}
 Vector(size_t n, bool b): n(n), dat(-u128(b), (n + 127) >> 7) {
  if (int k= n & 127; k) dat[dat.size() - 1]&= (u128(1) << k) - 1;
 }
 Vector(const initializer_list<bool> &v): n(v.size()), dat((n + 127) >> 7) {
  int i= 0;
  for (bool b: v) dat[i >> 7]|= u128(b) << (i & 127), ++i;
 }
 Ref operator[](int i) { return {begin(dat) + (i >> 7), u8(i & 127)}; }
 bool operator[](int i) const { return (dat[i >> 7] >> (i & 127)) & 1; }
 bool operator==(const Vector &r) const {
  if (dat.size() != r.dat.size()) return false;
  for (int i= dat.size(); i--;)
   if (dat[i] != r.dat[i]) return false;
  return true;
 }
 bool operator!=(const Vector &r) const { return !(*this == r); }
 explicit operator bool() const { return n; }
 Vector operator-() const { return Vector(*this); }
 Vector &operator+=(const Vector &r) { return dat^= r.dat, *this; }
 Vector &operator-=(const Vector &r) { return dat^= r.dat, *this; }
 Vector &operator*=(bool b) { return dat*= b, *this; }
 Vector operator+(const Vector &r) const { return Vector(*this)+= r; }
 Vector operator-(const Vector &r) const { return Vector(*this)-= r; }
 Vector operator*(bool b) const { return Vector(*this)*= b; }
 size_t size() const { return n; }
 friend bool dot(const Vector<bool> &a, const Vector<bool> &b) {
  assert(a.size() == b.size());
  u128 v= 0;
  for (int i= a.dat.size(); i--;) v^= a.dat[i] & b.dat[i];
  return __builtin_parityll(v >> 64) ^ __builtin_parityll(u64(v));
 }
};
template <class R> Vector<R> operator*(const R &r, const Vector<R> &v) { return v * r; }
template <class R> ostream &operator<<(ostream &os, const Vector<R> &v) {
 os << '[';
 for (int _= 0, __= v.size(); _ < __; ++_) os << (_ ? ", " : "") << v[_];
 return os << ']';
}
}
using _la_internal::Vector;
#line 5 "src/LinearAlgebra/Matrix.hpp"
namespace _la_internal {
template <class R, class D> struct Mat {
 Mat(): W(0) {}
 Mat(size_t h, size_t w): W(w), dat(h * w) {}
 Mat(size_t h, size_t w, R v): W(w), dat(v, h * w) {}
 Mat(initializer_list<initializer_list<R>> v): W(v.size() ? v.begin()->size() : 0), dat(v.size() * W) {
  auto it= begin(dat);
  for (const auto &r: v) {
   assert(r.size() == W);
   for (R x: r) *it++= x;
  }
 }
 size_t width() const { return W; }
 size_t height() const { return W ? dat.size() / W : 0; }
 auto operator[](int i) { return begin(dat) + i * W; }
 auto operator[](int i) const { return begin(dat) + i * W; }
protected:
 size_t W;
 valarray<R> dat;
 void add(const Mat &r) { assert(dat.size() == r.dat.size()), assert(W == r.W), dat+= r.dat; }
 D mul(const Mat &r) const {
  const size_t h= height(), w= r.W, l= W;
  assert(l == r.height());
  D ret(h, w);
  auto a= begin(dat);
  auto c= begin(ret.dat);
  for (int i= h; i--; c+= w) {
   auto b= begin(r.dat);
   for (int k= l; k--; ++a) {
    auto d= c;
    auto v= *a;
    for (int j= w; j--; ++b, ++d) *d+= v * *b;
   }
  }
  return ret;
 }
 Vector<R> mul(const Vector<R> &r) const {
  assert(W == r.size());
  const size_t h= height();
  Vector<R> ret(h);
  auto a= begin(dat);
  for (size_t i= 0; i < h; ++i)
   for (size_t k= 0; k < W; ++k, ++a) ret[i]+= *a * r[k];
  return ret;
 }
};
template <class D> struct Mat<bool, D> {
 struct Array {
  u128 *bg;
  Array(u128 *it): bg(it) {}
  Ref operator[](int i) { return Ref{bg + (i >> 7), u8(i & 127)}; }
  bool operator[](int i) const { return (bg[i >> 7] >> (i & 127)) & 1; }
 };
 struct ConstArray {
  const u128 *bg;
  ConstArray(const u128 *it): bg(it) {}
  bool operator[](int i) const { return (bg[i >> 7] >> (i & 127)) & 1; }
 };
 Mat(): H(0), W(0), m(0) {}
 Mat(size_t h, size_t w): H(h), W(w), m((w + 127) >> 7), dat(h * m) {}
 Mat(size_t h, size_t w, bool b): H(h), W(w), m((w + 127) >> 7), dat(-u128(b), h * m) {
  if (size_t i= h, k= w & 127; k)
   for (u128 s= (u128(1) << k) - 1; i--;) dat[i * m]&= s;
 }
 Mat(const initializer_list<initializer_list<bool>> &v): H(v.size()), W(H ? v.begin()->size() : 0), m((W + 127) >> 7), dat(H * m) {
  auto it= begin(dat);
  for (const auto &r: v) {
   assert(r.size() == W);
   int i= 0;
   for (bool b: r) it[i >> 7]|= u128(b) << (i & 127), ++i;
   it+= m;
  }
 }
 size_t width() const { return W; }
 size_t height() const { return H; }
 Array operator[](int i) { return {begin(dat) + i * m}; }
 ConstArray operator[](int i) const { return {begin(dat) + i * m}; }
 ConstArray get(int i) const { return {begin(dat) + i * m}; }
protected:
 size_t H, W, m;
 valarray<u128> dat;
 void add(const Mat &r) { assert(H == r.H), assert(W == r.W), dat^= r.dat; }
 D mul(const Mat &r) const {
  assert(W == r.H);
  D ret(H, r.W);
  valarray<u128> tmp(r.m << 8);
  auto y= begin(r.dat);
  for (size_t l= 0; l < W; l+= 8) {
   auto t= begin(tmp) + r.m;
   for (int i= 0, n= min<size_t>(8, W - l); i < n; ++i, y+= r.m) {
    auto u= begin(tmp);
    for (int s= 1 << i; s--;) {
     auto z= y;
     for (int j= r.m; j--; ++u, ++t, ++z) *t= *u ^ *z;
    }
   }
   auto a= begin(dat) + (l >> 7);
   auto c= begin(ret.dat);
   for (int i= H; i--; a+= m) {
    auto u= begin(tmp) + ((*a >> (l & 127)) & 255) * r.m;
    for (int j= r.m; j--; ++c, ++u) *c^= *u;
   }
  }
  return ret;
 }
 Vector<bool> mul(const Vector<bool> &r) const {
  assert(W == r.size());
  Vector<bool> ret(H);
  auto a= begin(dat);
  for (size_t i= 0; i < H; ++i) {
   u128 v= 0;
   for (size_t j= 0; j < m; ++j, ++a) v^= *a & r.dat[j];
   ret[i]= __builtin_parityll(v >> 64) ^ __builtin_parityll(u64(v));
  }
  return ret;
 }
};
template <class R> struct Matrix: public Mat<R, Matrix<R>> {
 using Mat<R, Matrix<R>>::Mat;
 explicit operator bool() const { return this->W; }
 static Matrix identity(int n) {
  Matrix ret(n, n);
  for (; n--;) ret[n][n]= R(true);
  return ret;
 }
 Matrix submatrix(const vector<int> &rows, const vector<int> &cols) const {
  Matrix ret(rows.size(), cols.size());
  for (int i= rows.size(); i--;)
   for (int j= cols.size(); j--;) ret[i][j]= (*this)[rows[i]][cols[j]];
  return ret;
 }
 Matrix submatrix_rm(vector<int> rows, vector<int> cols) const {
  sort(begin(rows), end(rows)), sort(begin(cols), end(cols)), rows.erase(unique(begin(rows), end(rows)), end(rows)), cols.erase(unique(begin(cols), end(cols)), end(cols));
  const int H= this->height(), W= this->width(), n= rows.size(), m= cols.size();
  vector<int> rs(H - n), cs(W - m);
  for (int i= 0, j= 0, k= 0; i < H; ++i)
   if (j < n && rows[j] == i) ++j;
   else rs[k++]= i;
  for (int i= 0, j= 0, k= 0; i < W; ++i)
   if (j < m && cols[j] == i) ++j;
   else cs[k++]= i;
  return submatrix(rs, cs);
 }
 bool operator==(const Matrix &r) const {
  if (this->width() != r.width() || this->height() != r.height()) return false;
  for (int i= this->dat.size(); i--;)
   if (this->dat[i] != r.dat[i]) return false;
  return true;
 }
 bool operator!=(const Matrix &r) const { return !(*this == r); }
 Matrix &operator*=(const Matrix &r) { return *this= this->mul(r); }
 Matrix operator*(const Matrix &r) const { return this->mul(r); }
 Matrix &operator*=(R r) { return this->dat*= r, *this; }
 template <class T> Matrix operator*(T r) const {
  static_assert(is_convertible_v<T, R>);
  return Matrix(*this)*= r;
 }
 Matrix &operator+=(const Matrix &r) { return this->add(r), *this; }
 Matrix operator+(const Matrix &r) const { return Matrix(*this)+= r; }
 Vector<R> operator*(const Vector<R> &r) const { return this->mul(r); }
 Vector<R> operator()(const Vector<R> &r) const { return this->mul(r); }
 Matrix pow(uint64_t k) const {
  size_t W= this->width();
  assert(W == this->height());
  for (Matrix ret= identity(W), b= *this;; b*= b)
   if (k & 1 ? ret*= b, !(k>>= 1) : !(k>>= 1)) return ret;
 }
};
template <class R, class T> Matrix<R> operator*(const T &r, const Matrix<R> &m) { return m * r; }
template <class R> ostream &operator<<(ostream &os, const Matrix<R> &m) {
 os << "\n[";
 for (int i= 0, h= m.height(); i < h; os << ']', ++i) {
  if (i) os << "\n ";
  os << '[';
  for (int j= 0, w= m.width(); j < w; ++j) os << (j ? ", " : "") << m[i][j];
 }
 return os << ']';
}
template <class K> static bool is_zero(K x) {
 if constexpr (is_floating_point_v<K>) return abs(x) < 1e-8;
 else return x == K();
}
}
using _la_internal::Matrix;
#line 2 "src/Internal/modint_traits.hpp"
#include <type_traits>
namespace math_internal {
struct m_b {};
struct s_b: m_b {};
}
template <class mod_t> constexpr bool is_modint_v= std::is_base_of_v<math_internal::m_b, mod_t>;
template <class mod_t> constexpr bool is_staticmodint_v= std::is_base_of_v<math_internal::s_b, mod_t>;
#line 3 "src/Math/berlekamp_massey.hpp"
// a[n] = c[0] * a[n-1] + c[1] * a[n-2] + ... + c[d-1] * a[n-d]
// return c
template <class K> std::vector<K> berlekamp_massey(const std::vector<K> &a) {
 size_t n= a.size(), d= 0, m= 0, i, j;
 if (n == 0) return {};
 std::vector<K> c(n), b(n), tmp;
 K x= 1, y, coef;
 for (c[0]= b[0]= 1, i= 0; i < n; ++i) {
  for (++m, y= a[i], j= 1; j <= d; ++j) y+= c[j] * a[i - j];
  if (y == K()) continue;
  for (tmp= c, coef= y / x, j= m; j < n; ++j) c[j]-= coef * b[j - m];
  if (2 * d <= i) d= i + 1 - d, b= tmp, x= y, m= 0;
 }
 c.resize(d + 1), c.erase(c.begin());
 for (auto &x: c) x= -x;
 return c;
}
#line 2 "src/Misc/rng.hpp"
#include <random>
#line 4 "src/Misc/rng.hpp"
uint64_t rng() {
 static uint64_t x= 10150724397891781847ULL * std::random_device{}();
 return x^= x << 7, x^= x >> 9;
}
uint64_t rng(uint64_t lim) { return rng() % lim; }
int64_t rng(int64_t l, int64_t r) { return l + rng() % (r - l); }
#line 7 "src/LinearAlgebra/MinimalPolynomial.hpp"
// c s.t. (c[d] * M^d + c[d-1] * M^(d-1)  + ... + c[1] * M + c[0]) * b = 0
template <class mod_t, class LinMap> class MinimalPolynomial {
 std::vector<mod_t> poly, rev;
 size_t dg, n;
 std::vector<Vector<mod_t>> bs;
 static inline int deg(const std::vector<mod_t> &p) {
  for (int d= p.size() - 1;; d--)
   if (d < 0 || p[d] != mod_t()) return d;
 }
 static inline std::vector<mod_t> bostan_mori_msb(const std::vector<mod_t> &q, uint64_t k) {
  int d= deg(q);
  assert(d >= 0), assert(q[0] != mod_t());
  std::vector<mod_t> ret(std::max(d, 1));
  if (k == 0) return ret.back()= mod_t(1), ret;
  std::vector<mod_t> v(d + 1);
  for (int i= 0; i <= d; i+= 2)
   for (int j= 0; j <= d; j+= 2) v[(i + j) >> 1]+= q[i] * q[j];
  for (int i= 1; i <= d; i+= 2)
   for (int j= 1; j <= d; j+= 2) v[(i + j) >> 1]-= q[i] * q[j];
  auto w= bostan_mori_msb(v, k >> 1);
  for (int i= 2 * d - 1 - (k & 1); i >= d; i-= 2)
   for (int j= 0; j <= d; j+= 2) ret[i - d]+= q[j] * w[(i - j) >> 1];
  for (int i= 2 * d - 1 - !(k & 1); i >= d; i-= 2)
   for (int j= 1; j <= d; j+= 2) ret[i - d]-= q[j] * w[(i - j) >> 1];
  return ret;
 }
 std::vector<mod_t> x_pow_mod(uint64_t k) const {
  assert(k >= n);
  std::vector<mod_t> ret(n), u= bostan_mori_msb(rev, k - n + dg);
  for (int i= dg; i--;)
   for (int j= i + 1; j--;) ret[n - 1 - i]+= u[j] * rev[i - j];
  return ret;
 }
public:
 MinimalPolynomial(const LinMap &M, Vector<mod_t> b): n(b.size()), bs(n) {
  static_assert(is_modint_v<mod_t>);
  Vector<mod_t> a(n);
  for (size_t i= n; i--;) a[i]= rng(1, mod_t::mod() - 1);
  std::vector<mod_t> v((n + 1) << 1);
  for (size_t i= v.size(), j= 0;; b= M(b)) {
   if (j < n) bs[j]= b;
   if (v[j++]= dot(a, b); !(--i)) break;
  }
  rev= berlekamp_massey(v);
  for (auto &x: rev) x= -x;
  rev.insert(rev.begin(), 1), poly.assign(rev.rbegin(), rev.rend()), rev.erase(rev.begin() + (dg= deg(rev)) + 1, rev.end());
 }
 Vector<mod_t> pow(uint64_t k) const {  // M^k * b
  if (k < n) return bs[k];
  auto r= x_pow_mod(k);
  Vector<mod_t> ret= r[0] * bs[0];
  for (int i= r.size(); --i;) ret+= r[i] * bs[i];
  return ret;
 }
 const mod_t &operator[](size_t k) const { return poly[k]; }
 const auto begin() const { return poly.begin(); }
 const auto end() const { return poly.end(); }
 size_t degree() const { return dg; }
};
template <class mod_t, class LinMap> mod_t linear_map_det(const LinMap &M, int n) {
 Vector<mod_t> b(n);
 for (int i= n; i--;) b[i]= rng(1, mod_t::mod() - 1);
 std::vector<mod_t> D(n);
 for (auto &x: D) x= rng(1, mod_t::mod() - 1);
 auto f= [&](Vector<mod_t> a) {
  for (int i= n; i--;) a[i]*= D[i];
  return M(a);
 };
 mod_t ret= MinimalPolynomial(f, b)[0], den= 1;
 if (n & 1) ret= -ret;
 for (const auto &x: D) den*= x;
 return ret / den;
}
#line 2 "src/Math/mod_inv.hpp"
#include <utility>
#line 5 "src/Math/mod_inv.hpp"
template <class Uint> constexpr inline Uint mod_inv(Uint a, Uint mod) {
 std::make_signed_t<Uint> x= 1, y= 0, z= 0;
 for (Uint q= 0, b= mod, c= 0; b;) z= x, x= y, y= z - y * (q= a / b), c= a, a= b, b= c - b * q;
 return assert(a == 1), x < 0 ? mod - (-x) % mod : x % mod;
}
#line 2 "src/Internal/Remainder.hpp"
namespace math_internal {
using namespace std;
using u8= unsigned char;
using u32= unsigned;
using i64= long long;
using u64= unsigned long long;
using u128= __uint128_t;
struct MP_Na {  // mod < 2^32
 u32 mod;
 constexpr MP_Na(): mod(0) {}
 constexpr MP_Na(u32 m): mod(m) {}
 constexpr inline u32 mul(u32 l, u32 r) const { return u64(l) * r % mod; }
 constexpr inline u32 set(u32 n) const { return n; }
 constexpr inline u32 get(u32 n) const { return n; }
 constexpr inline u32 norm(u32 n) const { return n; }
 constexpr inline u32 plus(u64 l, u32 r) const { return l+= r, l < mod ? l : l - mod; }
 constexpr inline u32 diff(u64 l, u32 r) const { return l-= r, l >> 63 ? l + mod : l; }
};
template <class u_t, class du_t, u8 B> struct MP_Mo {  // mod < 2^32, mod < 2^62
 u_t mod;
 constexpr MP_Mo(): mod(0), iv(0), r2(0) {}
 constexpr MP_Mo(u_t m): mod(m), iv(inv(m)), r2(-du_t(mod) % mod) {}
 constexpr inline u_t mul(u_t l, u_t r) const { return reduce(du_t(l) * r); }
 constexpr inline u_t set(u_t n) const { return mul(n, r2); }
 constexpr inline u_t get(u_t n) const { return n= reduce(n), n >= mod ? n - mod : n; }
 constexpr inline u_t norm(u_t n) const { return n >= mod ? n - mod : n; }
 constexpr inline u_t plus(u_t l, u_t r) const { return l+= r, l < (mod << 1) ? l : l - (mod << 1); }
 constexpr inline u_t diff(u_t l, u_t r) const { return l-= r, l >> (B - 1) ? l + (mod << 1) : l; }
private:
 u_t iv, r2;
 static constexpr u_t inv(u_t n, int e= 6, u_t x= 1) { return e ? inv(n, e - 1, x * (2 - x * n)) : x; }
 constexpr inline u_t reduce(const du_t &w) const { return u_t(w >> B) + mod - ((du_t(u_t(w) * iv) * mod) >> B); }
};
using MP_Mo32= MP_Mo<u32, u64, 32>;
using MP_Mo64= MP_Mo<u64, u128, 64>;
struct MP_Br {  // 2^20 < mod <= 2^41
 u64 mod;
 constexpr MP_Br(): mod(0), x(0) {}
 constexpr MP_Br(u64 m): mod(m), x((u128(1) << 84) / m) {}
 constexpr inline u64 mul(u64 l, u64 r) const { return rem(u128(l) * r); }
 static constexpr inline u64 set(u64 n) { return n; }
 constexpr inline u64 get(u64 n) const { return n >= mod ? n - mod : n; }
 constexpr inline u64 norm(u64 n) const { return n >= mod ? n - mod : n; }
 constexpr inline u64 plus(u64 l, u64 r) const { return l+= r, l < (mod << 1) ? l : l - (mod << 1); }
 constexpr inline u64 diff(u64 l, u64 r) const { return l-= r, l >> 63 ? l + (mod << 1) : l; }
private:
 u64 x;
 constexpr inline u128 quo(const u128 &n) const { return (n * x) >> 84; }
 constexpr inline u64 rem(const u128 &n) const { return n - quo(n) * mod; }
};
template <class du_t, u8 B> struct MP_D2B1 {  // mod < 2^63, mod < 2^64
 u64 mod;
 constexpr MP_D2B1(): mod(0), s(0), d(0), v(0) {}
 constexpr MP_D2B1(u64 m): mod(m), s(__builtin_clzll(m)), d(m << s), v(u128(-1) / d) {}
 constexpr inline u64 mul(u64 l, u64 r) const { return rem((u128(l) * r) << s) >> s; }
 constexpr inline u64 set(u64 n) const { return n; }
 constexpr inline u64 get(u64 n) const { return n; }
 constexpr inline u64 norm(u64 n) const { return n; }
 constexpr inline u64 plus(du_t l, u64 r) const { return l+= r, l < mod ? l : l - mod; }
 constexpr inline u64 diff(du_t l, u64 r) const { return l-= r, l >> B ? l + mod : l; }
private:
 u8 s;
 u64 d, v;
 constexpr inline u64 rem(const u128 &u) const {
  u128 q= (u >> 64) * v + u;
  u64 r= u64(u) - (q >> 64) * d - d;
  if (r > u64(q)) r+= d;
  if (r >= d) r-= d;
  return r;
 }
};
using MP_D2B1_1= MP_D2B1<u64, 63>;
using MP_D2B1_2= MP_D2B1<u128, 127>;
template <class u_t, class MP> constexpr u_t pow(u_t x, u64 k, const MP &md) {
 for (u_t ret= md.set(1);; x= md.mul(x, x))
  if (k & 1 ? ret= md.mul(ret, x) : 0; !(k>>= 1)) return ret;
}
}
#line 6 "src/Math/ModInt.hpp"
namespace math_internal {
template <class MP, u64 MOD> struct SB: s_b {
protected:
 static constexpr MP md= MP(MOD);
};
template <class U, class B> struct MInt: public B {
 using Uint= U;
 static constexpr inline auto mod() { return B::md.mod; }
 constexpr MInt(): x(0) {}
 template <class T, typename= enable_if_t<is_modint_v<T> && !is_same_v<T, MInt>>> constexpr MInt(T v): x(B::md.set(v.val() % B::md.mod)) {}
 constexpr MInt(__int128_t n): x(B::md.set((n < 0 ? ((n= (-n) % B::md.mod) ? B::md.mod - n : n) : n % B::md.mod))) {}
 constexpr MInt operator-() const { return MInt() - *this; }
#define FUNC(name, op) \
 constexpr MInt name const { \
  MInt ret; \
  return ret.x= op, ret; \
 }
 FUNC(operator+(const MInt & r), B::md.plus(x, r.x))
 FUNC(operator-(const MInt & r), B::md.diff(x, r.x))
 FUNC(operator*(const MInt & r), B::md.mul(x, r.x))
 FUNC(pow(u64 k), math_internal::pow(x, k, B::md))
#undef FUNC
 constexpr MInt operator/(const MInt &r) const { return *this * r.inv(); }
 constexpr MInt &operator+=(const MInt &r) { return *this= *this + r; }
 constexpr MInt &operator-=(const MInt &r) { return *this= *this - r; }
 constexpr MInt &operator*=(const MInt &r) { return *this= *this * r; }
 constexpr MInt &operator/=(const MInt &r) { return *this= *this / r; }
 constexpr bool operator==(const MInt &r) const { return B::md.norm(x) == B::md.norm(r.x); }
 constexpr bool operator!=(const MInt &r) const { return !(*this == r); }
 constexpr bool operator<(const MInt &r) const { return B::md.norm(x) < B::md.norm(r.x); }
 constexpr inline MInt inv() const { return mod_inv<U>(val(), B::md.mod); }
 constexpr inline Uint val() const { return B::md.get(x); }
 friend ostream &operator<<(ostream &os, const MInt &r) { return os << r.val(); }
 friend istream &operator>>(istream &is, MInt &r) {
  i64 v;
  return is >> v, r= MInt(v), is;
 }
private:
 Uint x;
};
template <u64 MOD> using MP_B= conditional_t < (MOD < (1 << 30)) & MOD, MP_Mo32, conditional_t < MOD < (1ull << 32), MP_Na, conditional_t<(MOD < (1ull << 62)) & MOD, MP_Mo64, conditional_t<MOD<(1ull << 41), MP_Br, conditional_t<MOD<(1ull << 63), MP_D2B1_1, MP_D2B1_2>>>>>;
template <u64 MOD> using ModInt= MInt < conditional_t<MOD<(1 << 30), u32, u64>, SB<MP_B<MOD>, MOD>>;
}
using math_internal::ModInt;
#line 10 "test/aoj/2397.MinPoly.test.cpp"
using namespace std;
signed main() {
 cin.tie(0);
 ios::sync_with_stdio(0);
 using Mint= ModInt<int(1e9 + 9)>;
 long long W, H, N;
 for (int cnt= 0; cin >> W >> H >> N && W;) {
  pair<long long, int> obst[N];
  for (long long i= 0, x, y; i < N; i++) cin >> x >> y, obst[i]= make_pair(y - 1, x - 1);
  sort(obst, obst + N);
  Matrix<Mint> A(W, W);
  for (int i= 0; i < W; i++) {
   A[i][i]= 1;
   if (i) A[i][i - 1]= 1;
   if (i + 1 < W) A[i][i + 1]= 1;
  }
  Vector<Mint> b(W);
  b[0]= 1;
  long long y= 0;
  for (int i= 0; i < N; i++) {
   b= MinimalPolynomial(A, b).pow(obst[i].first - y);
   int j= i;
   while (j < N && obst[i].first == obst[j].first) b[obst[j++].second]= 0;
   i= j - 1;
   y= obst[i].first;
  }
  b= MinimalPolynomial(A, b).pow(H - 1 - y);
  cout << "Case " << ++cnt << ": " << b[W - 1] << '\n';
 }
 return 0;
}

Test cases

Env	Name	Status	Elapsed	Memory
g++-13	testcase_00	AC	107 ms	4 MB
clang++-18	testcase_00	AC	93 ms	4 MB

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