You are given $n$ pairwise non-collinear two-dimensional vectors. You can make shapes in the two-dimensional plane with these vectors in the following fashion:
- Start at the origin $(0, 0)$.
- Choose a vector and add the segment of the vector to the current point. For example, if your current point is at $(x, y)$ and you choose the vector $(u, v)$, draw a segment from your current point to the point at $(x + u, y + v)$ and set your current point to $(x + u, y + v)$.
- Repeat step 2 until you reach the origin again.
You can reuse a vector as many times as you want.
Count the number of different, non-degenerate (with an area greater than $0$) and convex shapes made from applying the steps, such that the shape can be contained within a $m \times m$ square, and the vectors building the shape are in counter-clockwise fashion. Since this number can be too large, you should calculate it by modulo $998244353$.
Two shapes are considered the same if there exists some parallel translation of the first shape to another.
A shape can be contained within a $m \times m$ square if there exists some parallel translation of this shape so that every point $(u, v)$ inside or on the border of the shape satisfies $0 \leq u, v \leq m$.Input
The first line contains two integers $n$ and $m$ — the number of vectors and the size of the square ($1 \leq n \leq 5$, $1 \leq m \leq 10^9$).
Each of the next $n$ lines contains two integers $x_i$ and $y_i$ — the $x$-coordinate and $y$-coordinate of the $i$-th vector ($|x_i|, |y_i| \leq 4$, $(x_i, y_i) \neq (0, 0)$).
It is guaranteed, that no two vectors are parallel, so for any two indices $i$ and $j$ such that $1 \leq i < j \leq n$, there is no real value $k$ such that $x_i \cdot k = x_j$ and $y_i \cdot k = y_j$.Output
Output a single integer — the number of satisfiable shapes by modulo $998244353$.Examplesinput
3 3 -1 0 1 1 0 -1
output
3
input
3 3 -1 0 2 2 0 -1
output
1
input
3 1776966 -1 0 3 3 0 -2
output
296161
input
4 15 -4 -4 -1 1 -1 -4 4 3
output
1
input
5 10 3 -4 4 -3 1 -3 2 -3 -3 -4
output
0
input
5 1000000000 -2 4 2 -3 0 -4 2 4 -1 -3
output
9248783
Note
The shapes for the first sample are:
The only shape for the second sample is:
The only shape for the fourth sample is:
Solution:
#include <bits/stdc++.h> using namespace std; template <typename T> T inverse(T a, T m) { T u = 0, v = 1; while (a != 0) { T t = m / a; m -= t * a; swap(a, m); u -= t * v; swap(u, v); } assert(m == 1); return u; } template <typename T> class Modular { public: using Type = typename decay<decltype(T::value)>::type; constexpr Modular() : value() {} template <typename U> Modular(const U& x) { value = normalize(x); } template <typename U> static Type normalize(const U& x) { Type v; if (-mod() <= x && x < mod()) v = static_cast<Type>(x); else v = static_cast<Type>(x % mod()); if (v < 0) v += mod(); return v; } const Type& operator()() const { return value; } template <typename U> explicit operator U() const { return static_cast<U>(value); } constexpr static Type mod() { return T::value; } Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; } Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; } template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); } template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); } Modular& operator++() { return *this += 1; } Modular& operator--() { return *this -= 1; } Modular operator++(int) { Modular result(*this); *this += 1; return result; } Modular operator--(int) { Modular result(*this); *this -= 1; return result; } Modular operator-() const { return Modular(-value); } template <typename U = T> typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) { #ifdef _WIN32 uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value); uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m; asm( "divl %4; \n\t" : "=a" (d), "=d" (m) : "d" (xh), "a" (xl), "r" (mod()) ); value = m; #else value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value)); #endif return *this; } template <typename U = T> typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) { int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod()); value = normalize(value * rhs.value - q * mod()); return *this; } template <typename U = T> typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) { value = normalize(value * rhs.value); return *this; } Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); } template <typename U> friend const Modular<U>& abs(const Modular<U>& v) { return v; } template <typename U> friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs); template <typename U> friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs); template <typename U> friend std::istream& operator>>(std::istream& stream, Modular<U>& number); private: Type value; }; template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; } template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); } template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; } template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); } template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); } template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); } template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; } template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; } template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; } template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; } template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; } template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; } template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; } template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; } template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; } template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; } template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; } template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; } template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; } template<typename T, typename U> Modular<T> power(const Modular<T>& a, const U& b) { assert(b >= 0); Modular<T> x = a, res = 1; U p = b; while (p > 0) { if (p & 1) res *= x; x *= x; p >>= 1; } return res; } template <typename T> bool IsZero(const Modular<T>& number) { return number() == 0; } template <typename T> string to_string(const Modular<T>& number) { return to_string(number()); } template <typename T> std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) { return stream << number(); } template <typename T> std::istream& operator>>(std::istream& stream, Modular<T>& number) { typename common_type<typename Modular<T>::Type, int64_t>::type x; stream >> x; number.value = Modular<T>::normalize(x); return stream; } /* using ModType = int; struct VarMod { static ModType value; }; ModType VarMod::value; ModType& md = VarMod::value; using Mint = Modular<VarMod>; */ constexpr int md = 998244353; using Mint = Modular<std::integral_constant<decay<decltype(md)>::type, md>>; const int MAX = 20; Mint dp[MAX][MAX][MAX][MAX][2][2]; Mint new_dp[MAX][MAX][MAX][MAX][2][2]; int main() { ios::sync_with_stdio(false); cin.tie(0); int n, m; cin >> n >> m; vector<int> x(n), y(n); for (int i = 0; i < n; i++) { cin >> x[i] >> y[i]; } memset(dp, 0, sizeof(dp)); dp[0][0][0][0][0][0] = 1; while (m > 0) { int bit = m & 1; memset(new_dp, 0, sizeof(new_dp)); for (int cpx = 0; cpx < MAX; cpx++) { for (int cmx = 0; cmx < MAX; cmx++) { for (int cpy = 0; cpy < MAX; cpy++) { for (int cmy = 0; cmy < MAX; cmy++) { for (int fx = 0; fx < 2; fx++) { for (int fy = 0; fy < 2; fy++) { Mint ft = dp[cpx][cmx][cpy][cmy][fx][fy]; if (ft == 0) { continue; } for (int mask = 0; mask < (1 << n); mask++) { int dpx = cpx; int dpy = cpy; int dmx = cmx; int dmy = cmy; for (int i = 0; i < n; i++) { if (mask & (1 << i)) { dpx += max(x[i], 0); dmx -= min(x[i], 0); dpy += max(y[i], 0); dmy -= min(y[i], 0); } } if ((dpx & 1) != (dmx & 1)) { continue; } if ((dpy & 1) != (dmy & 1)) { continue; } int new_fx = ((dpx & 1) > bit ? 1 : ((dpx & 1) < bit ? 0 : fx)); int new_fy = ((dpy & 1) > bit ? 1 : ((dpy & 1) < bit ? 0 : fy)); dpx >>= 1; dmx >>= 1; dpy >>= 1; dmy >>= 1; assert(dpx < MAX && dpy < MAX && dmx < MAX && dmy < MAX); new_dp[dpx][dmx][dpy][dmy][new_fx][new_fy] += ft; } } } } } } } swap(dp, new_dp); m >>= 1; } cout << dp[0][0][0][0][0][0] - 1 << '\n'; return 0; }