#[allow(unused_imports)] use std::cmp::*; #[allow(unused_imports)] use std::collections::*; use std::io::{Write, BufWriter}; // https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes.by_ref().map(|r|r.unwrap() as char) .skip_while(|c|c.is_whitespace()) .take_while(|c|!c.is_whitespace()) .collect() }; input_inner!{next, $($r)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr, ) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, [graph1; $len:expr]) => {{ let mut g = vec![vec![]; $len]; let ab = read_value!($next, [(usize1, usize1)]); for (a, b) in ab { g[a].push(b); g[b].push(a); } g }}; ($next:expr, ( $($t:tt),* )) => { ( $(read_value!($next, $t)),* ) }; ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::>() }; ($next:expr, chars) => { read_value!($next, String).chars().collect::>() }; ($next:expr, usize1) => (read_value!($next, usize) - 1); ($next:expr, [ $t:tt ]) => {{ let len = read_value!($next, usize); read_value!($next, [$t; len]) }}; ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); } #[allow(unused)] macro_rules! debug { ($($format:tt)*) => (write!(std::io::stderr(), $($format)*).unwrap()); } #[allow(unused)] macro_rules! debugln { ($($format:tt)*) => (writeln!(std::io::stderr(), $($format)*).unwrap()); } /// Verified by https://atcoder.jp/contests/arc093/submissions/3968098 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl ::std::fmt::Debug for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { let (mut a, mut b, _) = red(self.x, M::m()); if b < 0 { a = -a; b = -b; } write!(f, "{}/{}", a, b) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } // Finds the simplest fraction x/y congruent to r mod p. // The return value (x, y, z) satisfies x = y * r + z * p. fn red(r: i64, p: i64) -> (i64, i64, i64) { if r.abs() <= 10000 { return (r, 1, 0); } let mut nxt_r = p % r; let mut q = p / r; if 2 * nxt_r >= r { nxt_r -= r; q += 1; } if 2 * nxt_r <= -r { nxt_r += r; q -= 1; } let (x, z, y) = red(nxt_r, r); (x, y - q * z, z) } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 1_000_000_007; define_mod!(P, MOD); type MInt = mod_int::ModInt

; fn dfs(g: &[Vec<(usize, i32)>], v: usize, vis: &mut [bool], pot: &mut [i32], cur: i32) -> Result { if vis[v] { return [Err(()), Ok(0)][usize::from(cur == pot[v])]; } vis[v] = true; pot[v] = cur; let mut sum = 1; for &(w, c) in &g[v] { let sub = dfs(g, w, vis, pot, cur * c)?; sum += sub; } Ok(sum) } // The author read the editorial. // Tags: dp-with-intervals, atypical-transitions, dp-optimization fn solve() { let out = std::io::stdout(); let mut out = BufWriter::new(out.lock()); macro_rules! puts { ($($format:tt)*) => (let _ = write!(out,$($format)*);); } #[allow(unused)] macro_rules! putvec { ($v:expr) => { for i in 0..$v.len() { puts!("{}{}", $v[i], if i + 1 == $v.len() {"\n"} else {" "}); } } } input! { n: usize, m: usize, lrp: [(usize1, usize1, i32); m], } // First, we find in what positions 0 cannot be filled. let mut non0 = vec![0; n + 1]; for &(l, r, p) in &lrp { if p != 0 { non0[l] += 1; non0[r + 1] -= 1; } } for i in 0..n { non0[i + 1] += non0[i]; } // Second, we need the array a and its cumulative max: // a[r[k]] = if p[k] == 0 { l[k] } else { -inf } // cum_a[i] := max { l[k] | r[k] < i, p[k] = 0} // bias: -1 let mut a = vec![0; n]; let mut cum_a = vec![0; n + 1]; for i in 0..m { let (l, r, p) = lrp[i]; a[r] = if p == 0 { l + 1 } else { 0 }; } for i in 0..n { cum_a[i + 1] = a[i]; cum_a[i + 1] = max(cum_a[i + 1], cum_a[i]); } // dp[i]: #{x | x[i] = 0, x is a 0-1 assignemnt satisfying certain conditions, with weight 2^{-count(x, 0)}} // origin: -1 let mut dp = vec![MInt::new(0); n + 2]; let mut acc = vec![MInt::new(0); n + 2]; dp[0] += 1; acc[0] += 1; let inv2 = MInt::new(2).inv(); for i in 0..n + 1 { if non0[i] > 0 { acc[i + 1] = acc[i]; continue; } let val = acc[i] - if cum_a[i] == 0 { MInt::new(0) } else { acc[cum_a[i] - 1] }; dp[i + 1] = val * inv2; acc[i + 1] = acc[i] + dp[i + 1]; } // For a 0-1 assignment x, the number of corresponding arrays is // 2^{-count(x, 0)} * 2^n * 2^{-#edges in mst of in constraints}, // if there are no contradictions in the constraints. // We obtain the sum of 2^{-count(x, 0)} * 2^n, which, if multiplied by // 2^{-#edges in mst of in constraints}, gives the overall answer. // We unnecessarily counted the 0 at the sentinel at n (dp[n + 1]), so we have to // multiply the answer by 2. let mut g = vec![vec![]; n + 1]; for &(l, r, p) in &lrp { if p != 0 { g[l].push((r + 1, p)); g[r + 1].push((l, p)); } } let mut vis = vec![false; n + 1]; let mut pot = vec![0; n + 1]; let mut edges = 0; for i in 0..n + 1 { if vis[i] { continue; } let res = dfs(&g, i, &mut vis, &mut pot, 1); if res.is_err() { puts!("0\n"); return; } let res = res.unwrap() - 1; edges += res; } puts!("{}\n", dp[n + 1] * MInt::new(2).pow(n as i64 + 1) * inv2.pow(edges)); } fn main() { // In order to avoid potential stack overflow, spawn a new thread. let stack_size = 104_857_600; // 100 MB let thd = std::thread::Builder::new().stack_size(stack_size); thd.spawn(|| solve()).unwrap().join().unwrap(); }