Problem:
Score : 1200 points

Problem Statement

You are given a simple connected undirected graph G consisting of N vertices and M edges.
The vertices are numbered 1 to N, and the edges are numbered 1 to M.

Edge i connects Vertex a_i and b_i bidirectionally.
It is guaranteed that the subgraph consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 is a spanning tree of G.

An allocation of weights to the edges is called a good allocation when the tree consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 is a minimum spanning tree of G.

There are M! ways to allocate the edges distinct integer weights between 1 and M.
For each good allocation among those, find the total weight of the edges in the minimum spanning tree, and print the sum of those total weights modulo 10^{9}+7.

Constraints

All values in input are integers.

2 \leq N \leq 20

N-1 \leq M \leq N(N-1)/2

1 \leq a_i, b_i \leq N

G does not have self-loops or multiple edges.

The subgraph consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 is a spanning tree of G.

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
\vdots
a_M b_M

Output

Print the answer.

Sample Input 1

3 3
1 2
2 3
1 3

Sample Output 1

6

An allocation is good only if Edge 3 has the weight 3. For these good allocations, the total weight of the edges in the minimum spanning tree is 3, and there are two good allocations, so the answer is 6.

Sample Input 2

4 4
1 2
3 2
3 4
1 3

Sample Output 2

50

Sample Input 3

15 28
10 7
5 9
2 13
2 14
6 1
5 12
2 10
3 9
10 15
11 12
12 6
2 12
12 8
4 10
15 3
13 14
1 15
15 12
4 14
1 7
5 11
7 13
9 10
2 7
1 9
5 6
12 14
5 2

Sample Output 3

657573092

Print the sum of those total weights modulo 10^{9}+7.

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

#![allow(unused_imports, non_snake_case)]
use std::cmp::{max, min, Ordering};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

// {{{ input!
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($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, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}
//}}}

//{{{ read!
macro_rules! read {
	($($t:tt),*; $n:expr) => {{
		let stdin = ::std::io::stdin();
		let ret = ::std::io::BufRead::lines(stdin.lock()).take($n).map(|line| {
			let line = line.unwrap();
			let mut it = line.split_whitespace();
			_read!(it; $($t),*)
		}).collect::<Vec<_>>();
		ret
	}};
	($($t:tt),*) => {{
		let line = readln();
		let mut it = line.split_whitespace();
		_read!(it; $($t),*)
	}};
}

macro_rules! _read {
	($it:ident; [char]) => {
		_read!($it; String).chars().collect::<Vec<_>>()
	};
	($it:ident; [u8]) => {
		Vec::from(_read!($it; String).into_bytes())
	};
	($it:ident; [$t:ty]) => {
		$it.map(|s| s.parse::<$t>().unwrap_or_else(|e| panic!("{}", e))).collect::<Vec<_>>()
	};
	($it:ident; $t:ty) => {
		$it.next().unwrap_or_else(|| panic!("input mismatch")).parse::<$t>().unwrap_or_else(|e| panic!("{}", e))
	};
	($it:ident; $($t:ty),+) => {
		($(_read!($it; $t)),*)
	};
}
//}}}

//{{{ utility macros missing in 1.15.1
// https://atcoder.jp/contests/nikkei2019-qual/submissions/4098548
macro_rules! eprint {
	($($t:tt)*) => {{
            use ::std::io::Write;
            let _ = write!(::std::io::stderr(), $($t)*);
	}};
}
macro_rules! eprintln {
	() => { eprintln!(""); };
	($($t:tt)*) => {{
	    use ::std::io::Write;
            let _ = writeln!(::std::io::stderr(), $($t)*);
	}};
}
macro_rules! dbg {
    ($val:expr) => {
        match $val {
            tmp => {
                eprintln!(
                    "[{}:{}] {} = {:#?}",
                    file!(),
                    line!(),
                    stringify!($val),
                    &tmp
                );
                tmp
            }
        }
    };
}
//}}}

fn run() {
    input! {
        n: usize,
        m: usize,
        edges: [(usize1, usize1); m],
    }
    let mut sub_graph = vec![Vec::new(); n];
    for i in 0..n - 1 {
        let (a, b) = edges[i];
        sub_graph[a].push((b, i));
        sub_graph[b].push((a, i));
    }

    let mut parent = vec![0; n];
    let mut parent_edge = vec![0; n];
    let mut depth = vec![0; n];
    let mut queue = VecDeque::new();
    queue.push_back((0, usize::max_value()));
    while let Some((u, p)) = queue.pop_front() {
        for &(next, idx) in &sub_graph[u] {
            if next != p {
                depth[next] = depth[u] + 1;
                parent[next] = u;
                parent_edge[next] = idx;
                queue.push_back((next, u));
            }
        }
    }

    let total = m - (n - 1);
    let mut count = vec![0usize; (1 << (n - 1))];

    for i in (n - 1)..m {
        let (a, b) = edges[i];
        let mut state = 0;
        let mut u = a;
        let mut v = b;
        while u != v {
            if depth[u] > depth[v] {
                state |= (1 << parent_edge[u]);
                u = parent[u];
            } else {
                state |= (1 << parent_edge[v]);
                v = parent[v];
            }
        }
        count[state] += 1;
    }

    // count'[s] = sum of count[t] s.t t is a subset of s
    for i in 0..(n - 1) {
        for s in 0..(1 << (n - 1)) {
            if 0 == (s & (1 << i)) {
                count[s | (1 << i)] += count[s];
            }
        }
    }

    use algonium::math::Mint107;
    let mut dp: Vec<Vec<Mint107>> = vec![vec![0.into(); total + 1]; (1 << (n - 1))];
    let mut cost: Vec<Vec<Mint107>> = vec![vec![0.into(); total + 1]; (1 << (n - 1))];
    dp[0][0] = 1.into();
    cost[0][0] = 0.into();
    for s in 0..(1 << (n - 1)) {
        for c in 0..total + 1 {
            if c < count[s] {
                dp[s][c + 1] += dp[s][c] * (count[s] - c);
                cost[s][c + 1] += cost[s][c] * (count[s] - c);
            }
            let order = s.count_ones() as usize + c + 1;
            for i in 0..n - 1 {
                if 0 == (s & (1 << i)) {
                    dp[s | (1 << i)][c] += dp[s][c];
                    cost[s | (1 << i)][c] += cost[s][c] + dp[s][c] * order;
                }
            }
        }
    }
    println!("{}", cost[(1 << (n - 1)) - 1][total]);
}

//{{{ template
fn main() {
    let _ = ::std::thread::Builder::new()
        .name("run".to_string())
        .stack_size(256 * 1024 * 1024)
        .spawn(run)
        .unwrap()
        .join()
        .unwrap();
}

pub trait SetMinMax {
    fn set_min(&mut self, v: Self) -> bool;
    fn set_max(&mut self, v: Self) -> bool;
}

impl<T> SetMinMax for T
where
    T: PartialOrd,
{
    fn set_min(&mut self, v: T) -> bool {
        *self > v && {
            *self = v;
            true
        }
    }
    fn set_max(&mut self, v: T) -> bool {
        *self < v && {
            *self = v;
            true
        }
    }
}

#[derive(PartialEq, Eq, Debug, Copy, Clone, Default, Hash)]
pub struct Reverse<T>(pub T);

impl<T: PartialOrd> PartialOrd for Reverse<T> {
    #[inline]
    fn partial_cmp(&self, other: &Reverse<T>) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }

    #[inline]
    fn lt(&self, other: &Self) -> bool {
        other.0 < self.0
    }
    #[inline]
    fn le(&self, other: &Self) -> bool {
        other.0 <= self.0
    }
    #[inline]
    fn ge(&self, other: &Self) -> bool {
        other.0 >= self.0
    }
    #[inline]
    fn gt(&self, other: &Self) -> bool {
        other.0 > self.0
    }
}

impl<T: Ord> Ord for Reverse<T> {
    #[inline]
    fn cmp(&self, other: &Reverse<T>) -> Ordering {
        other.0.cmp(&self.0)
    }
}

#[allow(unused)] // This is defined for read! macro
fn readln() -> String {
    let mut line = String::new();
    ::std::io::stdin()
        .read_line(&mut line)
        .unwrap_or_else(|e| panic!("{}", e));
    line
}
//}}}

//{{{ https://github.com/ichyo/rust-algorithms.git
pub mod algonium {
    #![doc = " This library provides a collection of classic algorithms,"]
    #![doc = " focusing on programming contests such as codeforces and atcoder."]
    pub mod data_structure {
        mod bit {
            #[doc = " Binary Indexed Tree (0-indexed)"]
            #[doc = ""]
            #[doc = " This data structure supports these two queries in O(log n)"]
            #[doc = ""]
            #[doc = " 1. add w to v[at]"]
            #[doc = " 2. the sum of v[begin], v[begin+1], .., v[end-1]"]
            pub struct BIT {
                tree: Vec<i64>,
            }
            #[doc = " Binary Indexed Tree (range-version) (0-indexed)"]
            #[doc = ""]
            #[doc = " This data structure two queries in O(log n)"]
            #[doc = ""]
            #[doc = " 1. add w to v[begin], v[begin+1], ..., v[end-1]"]
            #[doc = " 2. get the sum of v[begin], v[begin+1], ..., v[end-1]"]
            pub struct BITRange {
                bit0: BIT,
                bit1: BIT,
            }
            impl BIT {
                #[doc = " Constructs a new BIT of length `len`."]
                #[doc = " All values are initialized zero."]
                #[doc = ""]
                #[doc = " # Examples"]
                #[doc = " ```"]
                #[doc = " use algonium::data_structure::BIT;"]
                #[doc = " # #[warn(unused_mut)]"]
                #[doc = " let mut bit = BIT::new(100);"]
                #[doc = " ```"]
                pub fn new(len: usize) -> BIT {
                    BIT { tree: vec![0; len] }
                }
                #[doc = " Returns the number of elements in the BIT."]
                #[doc = ""]
                #[doc = " # Examples"]
                #[doc = ""]
                #[doc = " ```"]
                #[doc = " use algonium::data_structure::BIT;"]
                #[doc = " let mut bit = BIT::new(100);"]
                #[doc = " assert_eq!(bit.len(), 100);"]
                #[doc = " ```"]
                pub fn len(&self) -> usize {
                    self.tree.len()
                }
                #[doc = " Add a value `value` to a element of index `index`."]
                #[doc = " v[index] += value"]
                #[doc = ""]
                #[doc = " # Panics"]
                #[doc = ""]
                #[doc = " Panics if `index > len`"]
                #[doc = ""]
                #[doc = " # Examples"]
                #[doc = ""]
                #[doc = " ```"]
                #[doc = " use algonium::data_structure::BIT;"]
                #[doc = " let mut bit = BIT::new(10);"]
                #[doc = " bit.add(5, 100);"]
                #[doc = " assert_eq!(bit.get(3, 6), 100);"]
                #[doc = " bit.add(5, 10);"]
                #[doc = " assert_eq!(bit.get(3, 6), 110);"]
                #[doc = " ```"]
                pub fn add(&mut self, index: usize, value: i64) {
                    assert!(index < self.tree.len());
                    let mut idx = index;
                    while idx < self.tree.len() {
                        self.tree[idx] += value;
                        idx |= idx + 1;
                    }
                }
                #[doc = " sum of v[idx] such that begin <= idx < end"]
                pub fn get(&self, begin: usize, end: usize) -> i64 {
                    if begin >= end {
                        return 0;
                    }
                    let a = if end > 0 { self.cum(end - 1) } else { 0 };
                    let b = if begin > 0 { self.cum(begin - 1) } else { 0 };
                    a - b
                }
                #[doc = " v[0] + ... + v[last]"]
                fn cum(&self, last: usize) -> i64 {
                    assert!(last < self.tree.len());
                    let mut res = 0;
                    let mut idx = last as i64;
                    while idx >= 0 {
                        res += self.tree[idx as usize];
                        idx = (idx & (idx + 1)) - 1;
                    }
                    res
                }
            }
            impl BITRange {
                pub fn new(n: usize) -> BITRange {
                    BITRange {
                        bit0: BIT::new(n + 1),
                        bit1: BIT::new(n + 1),
                    }
                }
                #[doc = " v[begin], v[begin+1], ..., v[end-1] += by"]
                pub fn add(&mut self, begin: usize, end: usize, by: i64) {
                    if begin >= end {
                        return;
                    }
                    let a = begin as i64;
                    let b = end as i64;
                    self.bit0.add(begin, -by * a);
                    self.bit0.add(end, by * b);
                    self.bit1.add(begin, by);
                    self.bit1.add(end, -by);
                }
                #[doc = " v[begin] + ... + v[end-1]"]
                pub fn get(&self, begin: usize, end: usize) -> i64 {
                    if begin >= end {
                        return 0;
                    }
                    self.cum(end) - self.cum(begin)
                }
                #[doc = " v[0] + ... + v[end-1]"]
                fn cum(&self, end: usize) -> i64 {
                    let a = self.bit0.get(0, end);
                    let b = self.bit1.get(0, end);
                    a + b * end as i64
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                #[should_panic]
                fn test_out_of_bound_add() {
                    let mut bit = BIT::new(10);
                    bit.add(100, 1);
                }
                #[test]
                #[should_panic]
                fn test_out_of_bound_range_add() {
                    let mut bit = BITRange::new(10);
                    bit.add(5, 100, 1);
                }
                #[test]
                #[should_panic]
                fn test_out_of_bound_get() {
                    let bit = BIT::new(10);
                    bit.get(0, 1000);
                }
                #[test]
                fn test_empty_range() {
                    let mut bit = BITRange::new(10);
                    bit.add(9, 0, 100);
                    bit.add(7, 3, 100);
                    assert_eq!(0, bit.get(0, 9));
                    assert_eq!(0, bit.get(0, 5));
                    assert_eq!(0, bit.get(2, 5));
                }
                #[test]
                fn test_simple() {
                    let mut bit = BIT::new(10);
                    bit.add(2, 1);
                    bit.add(3, 3);
                    bit.add(5, 10);
                    bit.add(0, -4);
                    bit.add(9, -5);
                    assert_eq!(5, bit.get(0, 10));
                    assert_eq!(4, bit.get(2, 4));
                    assert_eq!(1, bit.get(2, 3));
                    assert_eq!(0, bit.get(2, 2));
                    assert_eq!(0, bit.get(2, 0));
                    assert_eq!(-4, bit.get(0, 1));
                    assert_eq!(5, bit.get(5, 10));
                    let mut bit = BITRange::new(10);
                    bit.add(1, 3, 1);
                    bit.add(2, 5, 2);
                    assert_eq!(1, bit.get(1, 2));
                    assert_eq!(4, bit.get(1, 3));
                    assert_eq!(6, bit.get(1, 4));
                    assert_eq!(6, bit.get(0, 4));
                }
            }
        }
        mod segment_tree {
            use std::cmp;
            use std::marker::PhantomData;
            pub trait Monoid<T> {
                fn id() -> Option<T> {
                    None
                }
                fn op(l: &Option<T>, r: &Option<T>) -> Option<T>;
            }
            pub struct MinOp<T: Ord> {
                phantom: PhantomData<T>,
            }
            impl<T: Ord + Clone> Monoid<T> for MinOp<T> {
                #[inline]
                fn op(l: &Option<T>, r: &Option<T>) -> Option<T> {
                    match (l.clone(), r.clone()) {
                        (Some(l), Some(r)) => Some(cmp::min(l, r)),
                        (Some(l), None) => Some(l),
                        (None, Some(r)) => Some(r),
                        (None, None) => None,
                    }
                }
            }
            pub struct SegmentTree<M: Monoid<T>, T: Clone> {
                phantom: PhantomData<M>,
                data: Vec<Option<T>>,
                size: usize,
                size_p2: usize,
            }
            impl<M: Monoid<T>, T: Clone> SegmentTree<M, T> {
                pub fn from_vec(v: Vec<T>) -> SegmentTree<M, T> {
                    let size = v.len();
                    let mut size_p2 = 1;
                    while size_p2 < v.len() {
                        size_p2 *= 2;
                    }
                    let mut data = vec![None; size_p2 * 2];
                    for (i, x) in v.into_iter().enumerate() {
                        data[size_p2 + i] = Some(x);
                    }
                    for i in (0..size_p2).rev() {
                        data[i] = M::op(&data[i * 2 + 0], &data[i * 2 + 1]);
                    }
                    SegmentTree {
                        phantom: PhantomData,
                        data: data,
                        size: size,
                        size_p2: size_p2,
                    }
                }
                pub fn size(&self) -> usize {
                    self.size
                }
                pub fn update(&mut self, mut pos: usize, value: T) {
                    assert!(pos < self.size);
                    pos += self.size_p2;
                    self.data[pos] = Some(value);
                    loop {
                        pos /= 2;
                        if pos == 0 {
                            break;
                        }
                        self.data[pos] = M::op(&self.data[pos * 2], &self.data[pos * 2 + 1]);
                    }
                }
                pub fn query(&self, mut l: usize, mut r: usize) -> Option<T> {
                    assert!(l <= r && r <= self.size);
                    l += self.size_p2;
                    r += self.size_p2;
                    let mut res1 = M::id();
                    let mut res2 = M::id();
                    while l < r {
                        if (l & 1) != 0 {
                            res1 = M::op(&res1, &self.data[l]);
                            l += 1;
                        }
                        if (r & 1) != 0 {
                            r -= 1;
                            res2 = M::op(&self.data[r], &res2);
                        }
                        l >>= 1;
                        r >>= 1;
                    }
                    M::op(&res1, &res2)
                }
            }
            #[doc = " segment tree to get minimum value in a range"]
            pub type RMQ<T> = SegmentTree<MinOp<T>, T>;
            #[cfg(test)]
            mod tests {
                use super::RMQ;
                #[test]
                fn test_rms() {
                    let mut test = vec![1, 5, 4, 8, 6, 9, 2, 0, 8, 1];
                    let mut rms = RMQ::from_vec(test.clone());
                    for i in 0..test.len() {
                        for j in i..test.len() + 1 {
                            assert_eq!(test[i..j].iter().cloned().min(), rms.query(i, j));
                        }
                    }
                    rms.update(7, 5);
                    test[7] = 5;
                    for i in 0..test.len() {
                        for j in i..test.len() + 1 {
                            assert_eq!(test[i..j].iter().cloned().min(), rms.query(i, j));
                        }
                    }
                }
            }
        }
        mod union_find {
            #[doc = " Disjoint-set data structure"]
            #[doc = ""]
            #[doc = " This provides operations for disjoint sets."]
            #[doc = " They runs in nearly constatant time."]
            #[doc = " (the actual time is `O(A(n))` where `A(n)` is the inverse of ackermann function.)"]
            #[doc = ""]
            #[doc = " 1. unite(x, y) - unite a set including x and another set including y into one."]
            #[doc = " 2. same(x, y) - determine if x and y are in the same set."]
            #[doc = " 3. size(x) - calculate the number of elements of the set including x."]
            #[doc = ""]
            #[doc = " [`UnionFind::new(n)`](#method.new) creates n disjoint sets. `i`-th set contains single element `i` (0-indexed)."]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::data_structure::UnionFind;"]
            #[doc = ""]
            #[doc = " let mut uf = UnionFind::new(4);"]
            #[doc = " assert!(!uf.same(0, 1));"]
            #[doc = ""]
            #[doc = " uf.unite(0, 1);"]
            #[doc = " assert!(uf.same(0, 1));"]
            #[doc = " assert_eq!(uf.size(0), 2);"]
            #[doc = ""]
            #[doc = " uf.unite(1, 2);"]
            #[doc = " assert!(uf.same(0, 2));"]
            #[doc = " assert_eq!(uf.size(0), 3);"]
            #[doc = " ```"]
            #[doc = ""]
            #[doc = " # Panics"]
            #[doc = " panics if `x` or `y` is not less than number of elements given in the initialization method."]
            #[doc = ""]
            pub struct UnionFind {
                data: Vec<i32>,
            }
            impl UnionFind {
                #[doc = " Creates a object with n disjoint sets. `i`-th set is `{ i }`."]
                pub fn new(n: usize) -> UnionFind {
                    UnionFind { data: vec![-1; n] }
                }
                #[doc = " Unite a set including `x` and another set including y into one."]
                #[doc = " Returns `true` only if they were in different set."]
                pub fn unite(&mut self, x: usize, y: usize) -> bool {
                    let x = self.root(x);
                    let y = self.root(y);
                    if x != y {
                        let (x, y) = if self.data[x] <= self.data[y] {
                            (x, y)
                        } else {
                            (y, x)
                        };
                        self.data[x] += self.data[y];
                        self.data[y] = x as i32;
                    }
                    x != y
                }
                #[doc = " Returns `true` only if `x` and `y` are in a same set."]
                pub fn same(&mut self, x: usize, y: usize) -> bool {
                    self.root(x) == self.root(y)
                }
                #[doc = " Returns the number of elements of a set including `x`."]
                pub fn size(&mut self, x: usize) -> u32 {
                    let r = self.root(x);
                    (-self.data[r]) as u32
                }
                #[doc = " internal method to return representative element of a set including `x`."]
                pub fn root(&mut self, x: usize) -> usize {
                    if self.data[x] < 0 {
                        x
                    } else {
                        let nx = self.data[x] as usize;
                        let r = self.root(nx);
                        self.data[x] = r as i32;
                        r
                    }
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let mut uf = UnionFind::new(3);
                    assert_eq!(1, uf.size(0));
                    assert_eq!(1, uf.size(1));
                    assert_eq!(1, uf.size(2));
                    assert_eq!(0, uf.root(0));
                    assert_eq!(1, uf.root(1));
                    assert_eq!(2, uf.root(2));
                    assert!(!uf.same(1, 2));
                    uf.unite(1, 2);
                    assert_ne!(uf.root(0), uf.root(2));
                    assert_eq!(uf.root(1), uf.root(2));
                    assert_eq!(1, uf.size(0));
                    assert_eq!(2, uf.size(1));
                    assert_eq!(2, uf.size(2));
                    assert!(uf.same(1, 2));
                    uf.unite(0, 1);
                    assert_eq!(uf.root(0), uf.root(2));
                    assert_eq!(uf.root(1), uf.root(2));
                    assert_eq!(3, uf.size(0));
                    assert_eq!(3, uf.size(1));
                    assert_eq!(3, uf.size(2));
                    assert!(uf.same(1, 2));
                    assert!(uf.same(0, 2));
                }
            }
        }
        pub use self::bit::{BITRange, BIT};
        pub use self::segment_tree::{MinOp, Monoid, SegmentTree, RMQ};
        pub use self::union_find::UnionFind;
    }
    pub mod math {
        mod comb {
            use super::mint::{Mint, Module};
            #[doc = " Useful struct to compute combinations"]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::math::{Comb, Mod107, Mint107};"]
            #[doc = " let comb: Comb<Mod107> = Comb::new(100);"]
            #[doc = " assert_eq!(Mint107::from(24), comb.fact(4));"]
            #[doc = " assert_eq!(Mint107::from(1), comb.fact(4) * comb.factinv(4));"]
            #[doc = " assert_eq!(Mint107::from(12), comb.perm(4, 2));"]
            #[doc = " assert_eq!(Mint107::from(6), comb.comb(4, 2));"]
            #[doc = " assert_eq!(Mint107::from(10), comb.multi_comb(4, 2));"]
            #[doc = " ```"]
            pub struct Comb<M: Module> {
                fact: Vec<Mint<M>>,
                factinv: Vec<Mint<M>>,
            }
            impl<M: Module> Comb<M> {
                #[doc = " Create a object that provides effiecint computation of combinations"]
                #[doc = " for input smaller than `n`."]
                #[doc = ""]
                #[doc = " This requires `O(n)` time."]
                pub fn new(n: usize) -> Comb<M> {
                    let mut fact: Vec<Mint<M>> = vec![0.into(); n + 1];
                    let mut factinv: Vec<Mint<M>> = vec![0.into(); n + 1];
                    fact[0] = 1.into();
                    for i in 0..n {
                        fact[i + 1] = fact[i] * (i + 1);
                    }
                    factinv[n] = fact[n].inv();
                    for i in (0..n).rev() {
                        factinv[i] = factinv[i + 1] * (i + 1);
                    }
                    Comb {
                        fact: fact,
                        factinv: factinv,
                    }
                }
                #[doc = " `n! = 1 * 2 * ... * n`"]
                #[doc = ""]
                #[doc = " `O(1)` if n is smaller than input in `new` method."]
                pub fn fact(&self, n: u64) -> Mint<M> {
                    if let Some(x) = self.fact.get(n as usize) {
                        *x
                    } else if n >= M::module() as u64 {
                        Mint::from(0)
                    } else {
                        let mut res = 1.into();
                        for a in 1..(n + 1) {
                            res *= a;
                        }
                        res
                    }
                }
                #[doc = " returns `y` such that `n! * y == 1`."]
                #[doc = ""]
                #[doc = " `O(1)` if n is smaller than input in `new` method."]
                pub fn factinv(&self, n: u64) -> Mint<M> {
                    if let Some(x) = self.factinv.get(n as usize) {
                        *x
                    } else {
                        self.fact(n).inv()
                    }
                }
                #[doc = " `nPr = n! / (n - r)!`"]
                #[doc = ""]
                #[doc = " `O(1)` if n and r are smaller than input in `new` method."]
                pub fn perm(&self, n: u64, r: u64) -> Mint<M> {
                    if n >= r {
                        self.fact(n) * self.factinv((n - r) as u64)
                    } else {
                        0.into()
                    }
                }
                #[doc = " `nCr = n! / (n - r)! / r!`."]
                #[doc = ""]
                #[doc = " `O(1)` if n and r are smaller than input in `new` method."]
                pub fn comb(&self, n: u64, r: u64) -> Mint<M> {
                    let m = M::module() as u64;
                    if n >= m {
                        self.comb(n % m, r % m) * self.comb(n / m, r / m)
                    } else if n >= r {
                        self.fact(n) * self.factinv(n - r) * self.factinv(r)
                    } else {
                        Mint::from(0)
                    }
                }
                #[doc = " `(n + k - 1)! / k!`."]
                #[doc = ""]
                #[doc = " `O(1)` if n and r are smaller than input in `new` method."]
                pub fn multi_comb(&self, n: u64, r: u64) -> Mint<M> {
                    if r == 0 {
                        Mint::from(1)
                    } else {
                        self.comb(n + r - 1, r)
                    }
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    #[derive(Clone, Copy, Debug)]
                    struct Mod;
                    impl Module for Mod {
                        fn module() -> u32 {
                            1000000007
                        }
                    }
                    let c = Comb::<Mod>::new(100);
                    assert_eq!(Mint::from(336), c.perm(8, 3));
                    assert_eq!(Mint::from(56), c.comb(8, 3));
                    assert_eq!(Mint::from(10), c.multi_comb(3, 3));
                }
                #[test]
                fn test_fact() {
                    #[derive(Clone, Copy, Debug)]
                    struct Mod;
                    impl Module for Mod {
                        fn module() -> u32 {
                            1000000007
                        }
                    }
                    let c = Comb::<Mod>::new(100);
                    let p = 8721234;
                    let mut f = Mint::from(1);
                    for i in 1..(p + 1) {
                        f *= i;
                    }
                    assert_eq!(f, c.fact(p));
                }
            }
        }
        mod mint {
            use std::fmt;
            use std::marker::PhantomData;
            use std::mem;
            use std::ops;
            #[doc = " Trait for `Mint`. `module()` should return prime number."]
            pub trait Module: Copy + Clone {
                fn module() -> u32;
            }
            #[doc = " One of famous numbers in programming contest. `10^9 + 7`"]
            pub const MOD_107: u32 = 1_000_000_007;
            #[doc = " One of famous numbers in programming contest. `10^9 + 9`"]
            pub const MOD_109: u32 = 1_000_000_009;
            #[doc = " One of famous numbers in programming contest. `998_244_353`"]
            pub const MOD_998: u32 = 998_244_353;
            #[doc = " struct to implement Module trait. it returns `MOD_107`."]
            #[derive(Debug, Copy, Clone)]
            pub struct Mod107;
            impl Module for Mod107 {
                fn module() -> u32 {
                    MOD_107
                }
            }
            #[doc = " struct to implement Module trait. it returns `MOD_109`."]
            #[derive(Debug, Copy, Clone)]
            pub struct Mod109;
            impl Module for Mod109 {
                fn module() -> u32 {
                    MOD_109
                }
            }
            #[doc = " struct to implement Module trait. it returns `MOD_998`."]
            #[derive(Debug, Copy, Clone)]
            pub struct Mod998;
            impl Module for Mod998 {
                fn module() -> u32 {
                    MOD_998
                }
            }
            #[doc = " Wrapper class to compute mod `1_000_000_007` automatically."]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::math::{Mint107, MOD_107};"]
            #[doc = " let x: Mint107 = 1234567.into();"]
            #[doc = " let y: Mint107 = 2345678.into();"]
            #[doc = " let z = x * y;"]
            #[doc = " # // TODO: implement convert to u64"]
            #[doc = " assert_eq!(z.val as u64, 1234567u64 * 2345678u64 % MOD_107 as u64);"]
            #[doc = " ```"]
            #[doc = ""]
            pub type Mint107 = Mint<Mod107>;
            #[doc = " Wrapper class to compute mod `1_000_000_009` automatically."]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::math::{Mint109, MOD_109};"]
            #[doc = " let x: Mint109 = 1234567.into();"]
            #[doc = " let y: Mint109 = 2345678.into();"]
            #[doc = " let z = x * y;"]
            #[doc = " assert_eq!(z.val as u64, 1234567u64 * 2345678u64 % MOD_109 as u64);"]
            #[doc = " ```"]
            #[doc = ""]
            pub type Mint109 = Mint<Mod109>;
            #[doc = " Wrapper class to compute mod `998_244_353` automatically."]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::math::{Mint998, MOD_998};"]
            #[doc = " let x: Mint998 = 1234567.into();"]
            #[doc = " let y: Mint998 = 2345678.into();"]
            #[doc = " let z = x * y;"]
            #[doc = " assert_eq!(z.val as u64, 1234567u64 * 2345678u64 % MOD_998 as u64);"]
            #[doc = " ```"]
            #[doc = ""]
            pub type Mint998 = Mint<Mod998>;
            #[doc = " Wrapper class to compute modulo operation."]
            #[doc = " See examples"]
            #[doc = " [`Mint107`](type.Mint107.html),"]
            #[doc = " [`Mint109`](type.Mint109.html),"]
            #[doc = " [`Mint998`](type.Mint998.html)"]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::math::{Mint107, MOD_107};"]
            #[doc = " let x: Mint107 = 1234567.into();"]
            #[doc = " let y: Mint107 = 2345678.into();"]
            #[doc = " let z = x * y;"]
            #[doc = " assert_eq!(z.val as u64, 1234567u64 * 2345678u64 % MOD_107 as u64);"]
            #[doc = " ```"]
            #[derive(Debug, Copy, Clone, Eq)]
            pub struct Mint<M: Module> {
                #[doc = " internal value. this is always less than `self.module()`."]
                pub val: u32,
                m: PhantomData<M>,
            }
            impl<M: Module> Mint<M> {
                fn module(self) -> u32 {
                    M::module()
                }
                fn new(val: u32) -> Mint<M> {
                    assert!(val < M::module());
                    Mint {
                        val: val,
                        m: PhantomData,
                    }
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::Add<T> for Mint<M> {
                type Output = Mint<M>;
                fn add(self, other: T) -> Mint<M> {
                    let nval = self.val + other.into().val;
                    Mint::new(if nval >= self.module() {
                        nval - self.module()
                    } else {
                        nval
                    })
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::Sub<T> for Mint<M> {
                type Output = Mint<M>;
                fn sub(self, other: T) -> Mint<M> {
                    let nval = self.val + self.module() - other.into().val;
                    Mint::new(if nval >= self.module() {
                        nval - self.module()
                    } else {
                        nval
                    })
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::Mul<T> for Mint<M> {
                type Output = Mint<M>;
                fn mul(self, other: T) -> Mint<M> {
                    let nval = self.val as u64 * other.into().val as u64;
                    Mint::new((nval % (self.module() as u64)) as u32)
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::Div<T> for Mint<M> {
                type Output = Mint<M>;
                fn div(self, other: T) -> Mint<M> {
                    self * other.into().inv()
                }
            }
            impl<M: Module> Mint<M> {
                #[doc = " Returns number `y` that satisfies `x * y == 1` where `x` is the original value."]
                #[doc = ""]
                #[doc = " This assumes `module()` returns prime number."]
                pub fn inv(self) -> Mint<M> {
                    let mut a = self.val as i32;
                    let mut b = self.module() as i32;
                    let mut u = 1 as i32;
                    let mut v = 0 as i32;
                    while b != 0 {
                        let t = a / b;
                        a -= t * b;
                        mem::swap(&mut a, &mut b);
                        u -= t * v;
                        mem::swap(&mut u, &mut v);
                    }
                    Mint::new(if u < 0 { u + self.module() as i32 } else { u } as u32)
                }
            }
            impl<M: Module> PartialEq for Mint<M> {
                fn eq(&self, other: &Mint<M>) -> bool {
                    self.val == other.val
                }
            }
            impl<M: Module> fmt::Display for Mint<M> {
                fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
                    self.val.fmt(f)
                }
            }
            macro_rules ! impl_signed_mint { ( $ ( $ t : ty ) * ) => ( $ ( impl < M : Module > From <$ t > for Mint < M > { # [ inline ] fn from ( x : $ t ) -> Mint < M > { let t = ( x as i64 ) % ( M :: module ( ) as i64 ) ; if x >= 0 { Mint { val : t as u32 , m : PhantomData } } else { Mint { val : ( M :: module ( ) as i64 + t ) as u32 , m : PhantomData } } } } ) * ) }
            macro_rules ! impl_unsigned_mint { ( $ ( $ t : ty ) * ) => ( $ ( impl < M : Module > From <$ t > for Mint < M > { # [ inline ] fn from ( x : $ t ) -> Mint < M > { let t = x as u64 % M :: module ( ) as u64 ; Mint :: new ( t as u32 ) } } ) * ) }
            impl_signed_mint! { i8 i16 i32 i64 isize }
            impl_unsigned_mint! { u8 u16 u32 u64 usize }
            impl<T: Into<Mint<M>>, M: Module> ops::AddAssign<T> for Mint<M> {
                fn add_assign(&mut self, other: T) {
                    *self = *self + other.into();
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::SubAssign<T> for Mint<M> {
                fn sub_assign(&mut self, other: T) {
                    *self = *self - other.into();
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::MulAssign<T> for Mint<M> {
                fn mul_assign(&mut self, other: T) {
                    *self = *self * other.into();
                }
            }
            impl<T: Into<Mint<M>>, M: Module> ops::DivAssign<T> for Mint<M> {
                fn div_assign(&mut self, other: T) {
                    *self = *self / other.into();
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let a: Mint<Mod107> = Mint::from(3);
                    let b: Mint<Mod107> = Mint::from(1000000000);
                    assert_eq!(Mint::from(3000000000u64 % Mod107::module() as u64), a * b);
                }
            }
        }
        pub use self::comb::Comb;
        pub use self::mint::{Mint, Module};
        pub use self::mint::{Mint107, Mint109, Mint998};
        pub use self::mint::{Mod107, Mod109, Mod998};
        pub use self::mint::{MOD_107, MOD_109, MOD_998};
    }
    pub mod util {
        mod permutation {
            #[doc = " It contains same methods as `std::next_permutaion` in C++."]
            #[doc = " these methods should rearrange the elements into the next/prev lexicographically greater permutation."]
            #[doc = ""]
            #[doc = " This is implemented in [T] where T is Ord."]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::util::Permutation;"]
            #[doc = ""]
            #[doc = " let mut perm = [5, 0, 9, 2];"]
            #[doc = " assert!(perm.next_permutation());"]
            #[doc = " assert_eq!(perm, [5, 2, 0, 9]);"]
            #[doc = " assert!(perm.next_permutation());"]
            #[doc = " assert_eq!(perm, [5, 2, 9, 0]);"]
            #[doc = " assert!(perm.prev_permutation());"]
            #[doc = " assert_eq!(perm, [5, 2, 0, 9]);"]
            #[doc = ""]
            #[doc = " let mut perm = [4, 3, 2, 1, 0];"]
            #[doc = " assert!(!perm.next_permutation());"]
            #[doc = " assert_eq!(perm, [4, 3, 2, 1, 0]);"]
            #[doc = " ```"]
            pub trait Permutation {
                #[doc = " It rearranges the elements into the next lexicographically greater permutation."]
                #[doc = ""]
                #[doc = " Returns `false` without modifying any state only if current is the last permutation."]
                fn next_permutation(&mut self) -> bool;
                #[doc = " It rearranges the elements into the previous lexicographically greater permutation."]
                #[doc = ""]
                #[doc = " Returns `false` without modifying any state only if current is the first permutation."]
                fn prev_permutation(&mut self) -> bool;
            }
            impl<T> Permutation for [T]
            where
                T: Ord,
            {
                fn next_permutation(&mut self) -> bool {
                    if self.len() <= 1 {
                        return false;
                    }
                    let mut i = self.len() - 1;
                    while i > 0 && self[i - 1] >= self[i] {
                        i -= 1;
                    }
                    if i == 0 {
                        return false;
                    }
                    let mut j = self.len() - 1;
                    while self[j] <= self[i - 1] {
                        j -= 1;
                    }
                    self.swap(j, i - 1);
                    self[i..].reverse();
                    true
                }
                fn prev_permutation(&mut self) -> bool {
                    if self.len() <= 1 {
                        return false;
                    }
                    let mut i = self.len() - 1;
                    while i > 0 && self[i - 1] <= self[i] {
                        i -= 1;
                    }
                    if i == 0 {
                        return false;
                    }
                    self[i..].reverse();
                    let mut j = self.len() - 1;
                    while self[j - 1] < self[i - 1] {
                        j -= 1;
                    }
                    self.swap(i - 1, j);
                    true
                }
            }
            #[cfg(test)]
            mod tests {
                use super::*;
                #[test]
                fn test_simple() {
                    let mut data = [0, 0, 1, 1, 2];
                    data.next_permutation();
                    assert_eq!(&data, &[0, 0, 1, 2, 1]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 0, 2, 1, 1]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 1, 0, 1, 2]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 1, 0, 2, 1]);
                    data.next_permutation();
                    assert_eq!(&data, &[0, 1, 1, 0, 2]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 1, 0, 2, 1]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 1, 0, 1, 2]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 0, 2, 1, 1]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 0, 1, 2, 1]);
                    data.prev_permutation();
                    assert_eq!(&data, &[0, 0, 1, 1, 2]);
                    assert!(!data.prev_permutation());
                }
            }
        }
        mod random {
            use std::num::Wrapping;
            #[doc = " Not secure but fast random generator."]
            #[doc = ""]
            #[doc = " # Examples"]
            #[doc = " ```"]
            #[doc = " use algonium::util::XorShift;"]
            #[doc = " let mut xorshift = XorShift::new();"]
            #[doc = " assert_eq!(3701687786, xorshift.next_u32());"]
            #[doc = " assert_eq!(458299110, xorshift.next_u32());"]
            #[doc = " ```"]
            #[derive(Clone, Debug, Default)]
            pub struct XorShift {
                x: Wrapping<u32>,
                y: Wrapping<u32>,
                z: Wrapping<u32>,
                w: Wrapping<u32>,
            }
            impl XorShift {
                #[doc = " Create object with default seed."]
                pub fn new() -> XorShift {
                    XorShift {
                        x: Wrapping(123_456_789),
                        y: Wrapping(362_436_069),
                        z: Wrapping(521_288_629),
                        w: Wrapping(88_675_123),
                    }
                }
                #[doc = " Get next random u32 number."]
                pub fn next_u32(&mut self) -> u32 {
                    let x = self.x;
                    let t = x ^ (x << 11);
                    self.x = self.y;
                    self.y = self.z;
                    self.z = self.w;
                    let v = self.w;
                    self.w = v ^ (v >> 19) ^ (t ^ (t >> 8));
                    self.w.0
                }
            }
            #[cfg(test)]
            mod tests {
                use super::XorShift;
                #[test]
                fn test_basic() {
                    let mut xorshift = XorShift::new();
                    assert_eq!(3701687786, xorshift.next_u32());
                    assert_eq!(458299110, xorshift.next_u32());
                }
            }
        }
        pub use self::permutation::Permutation;
        pub use self::random::XorShift;
    }
}

//}}}