Problem: J - Tree Reconstruction

Problem Statement

You have a directed graph. Some non-negative value is assigned on each edge of the graph. You know that the value of the graph satisfies the flow conservation law That is, for every node v, the sum of values on edges incoming to v equals to the sum of values of edges outgoing from v. For example, the following directed graph satisfies the flow conservation law.

<image>


Suppose that you choose a subset of edges in the graph, denoted by E', and then you erase all the values on edges other than E'. Now you want to recover the erased values by seeing only remaining information. Due to the flow conservation law, it may be possible to recover all the erased values. Your task is to calculate the smallest possible size for such E'.

For example, the smallest subset E' for the above graph will be green edges in the following figure. By the flow conservation law, we can recover values on gray edges.

<image>


Input

The input consists of multiple test cases. The format of each test case is as follows.


N M
s_1 t_1
...
s_M t_M


The first line contains two integers N (1 \leq N \leq 500) and M (0 \leq M \leq 3,000) that indicate the number of nodes and edges, respectively. Each of the following M lines consists of two integers s_i and t_i (1 \leq s_i, t_i \leq N). This means that there is an edge from s_i to t_i in the graph.

You may assume that the given graph is simple:

* There are no self-loops: s_i \neq t_i.
* There are no multi-edges: for all i < j, \\{s_i, t_i\\} \neq \\{s_j, t_j\\}.



Also, it is guaranteed that each component of the given graph is strongly connected. That is, for every pair of nodes v and u, if there exist a path from v to u, then there exists a path from u to v.

Note that you are NOT given information of values on edges because it does not effect the answer.

Output

For each test case, print an answer in one line.

Sample Input 1


9 13
1 2
1 3
2 9
3 4
3 5
3 6
4 9
5 7
6 7
6 8
7 9
8 9
9 1


Output for the Sample Input 1


5


Sample Input 2


7 9
1 2
1 3
2 4
3 4
4 5
4 6
5 7
6 7
7 1


Output for the Sample Input 2


3


Sample Input 3


4 4
1 2
2 1
3 4
4 3


Output for the Sample Input 3


2






Example

Input

9 13
1 2
1 3
2 9
3 4
3 5
3 6
4 9
5 7
6 7
6 8
7 9
8 9
9 1


Output

5

Solution (in CPP):

Solution:
#include "bits/stdc++.h"
#include<unordered_map>
#include<unordered_set>
#pragma warning(disable:4996)
using namespace std;
using ld = long double;
template<class T>
using Table = vector<vector<T>>;
const ld eps = 1e-9;

// < "D:\D_Download\Visual Studio 2015\Projects\programing_contest_c++\Debug\a.txt" > "D:\D_Download\Visual Studio 2015\Projects\programing_contest_c++\Debug\b.txt"

/* verified AOJ1327 One-Dimensional Cellular Automaton
~Matrix~
??????????°????????¨?????????§??????
vector iostream cmath cassert ??????
Matrix a(VV), b(VV);
?????????:a*b
????????????:l*a
???:a+b
???:a-b
??¢???:a.transport()
?????????:a.cofactor()
?????????:a.det()
a???i???j??????:a.get(i,j)???
a???i??????j?????????k?????£??\:set(i,j,k)
????????????a[i][j] (a[i][j]=k)
a???x???:a.pow(x)
n*n?????¬??????E:Matrix(n)
m*n 0??????:Matrix(m,n)
m*n ??¨????´????p?????????:Matrix(m,n,p)
Row???row??§????????????(m*1??????):Matrix(row)
**(??´??°??????)**
????§????:a.triangulate()
?????????:a.rank()
?????????:a.inverse()
//(????????????a.det()?????????a.pre_inverse()??§?±??????? ??´??°???)
??¬??????????¶???????:a.rowReduction()
//??£???????¬?????¨????????§£??????
***
*/

#include <vector>
#include <iostream>
#include <cmath>
#include <cassert>
using namespace std;

typedef ld Elem;
typedef vector<Elem> Row;
typedef vector<Row> VV;

typedef long double ld;
const ld EPS = 1e-11;

const bool isZero(const Elem e) {
	return abs(e) < EPS;
}

struct Matrix {
	VV matrix;
	int n, m;

	Matrix(const VV &matrix_);
	explicit Matrix(int n_);
	explicit Matrix(const Row &row);
	Matrix(int m_, int n_);
	Matrix(int m_, int n_, Elem e);

	const Elem get(const int i, const int j) const;
	void set(const int x, const int y, const Elem k);

	const Matrix operator + (const Matrix &rhs) const;
	const Matrix operator * (const Matrix &rhs) const;
	const Matrix operator - (const Matrix &rhs) const;
	Matrix &operator += (const Matrix &rhs);
	Matrix &operator *= (const Matrix &rhs);
	Matrix &operator -= (const Matrix &rhs);

	Row &operator[](const int x);

	const Matrix transport() const;
	const Matrix pow(int x) const;
	const Matrix cofactor(int x, int y) const;
	const Elem det() const;

	const Matrix triangulate() const;
	const int rank() const;

	//?????????????????¨????????°???(?????????)*(?????????)?????????
	//A:matrix,return det A * A^-1
	const Matrix pre_inverse() const;
	const Matrix inverse() const;
	const Matrix rowReduction() const;
};

const Matrix operator * (const Elem lambda, const Matrix &rhs) {
	Matrix tmp(rhs);
	for (int i = 0; i < rhs.m; i++)
		for (int j = 0; j < rhs.n; j++)
			tmp.set(i, j, tmp.get(i, j) * lambda);
	return tmp;
}

Matrix::Matrix(const VV &matrix_) : matrix(matrix_) {
	m = matrix_.size();
	if (m == 0) n = 0;
	else n = matrix_[0].size();
}
Matrix::Matrix(int n_) : m(n_), n(n_) {
	matrix = VV(n, Row(n, 0));
	for (int i = 0; i < n; ++i)
		set(i, i, 1);
}
Matrix::Matrix(const Row &row) : m(1), n(row.size()), matrix(VV(1, row)) {
	//size???m???vector<Elem>??????mx1???????????????
	(*this) = transport();
}
Matrix::Matrix(int m_, int n_) : m(m_), n(n_) {
	matrix = VV(m, Row(n, 0));
}
Matrix::Matrix(int m_, int n_, Elem e) : m(m_), n(n_) {
	matrix = VV(m, Row(n, e));
}

const Elem Matrix::get(const int i, const int j) const {
	if (0 <= i && i < m && 0 <= j && j < n)
		return matrix[i][j];

	cerr << "get(" << i << "," << j << ")is not exist." << endl;
	throw;
}
void Matrix::set(const int i, const int j, const Elem k) {
	if (0 <= i && i < m && 0 <= j && j < n) {
		*(matrix[i].begin() + j) = k;
		return;
	}
	cerr << "set(" << i << "," << j << ")is not exist." << endl;
	throw;
}

const Matrix Matrix::operator + (const Matrix &rhs) const {
	assert(m == rhs.m && n == rhs.n);

	Matrix tmp(m, n, 0);
	for (int i = 0; i < m; i++) {
		for (int j = 0; j < n; j++) {
			tmp.set(i, j, get(i, j) + rhs.get(i, j));
		}
	}
	return tmp;
}

const Matrix Matrix::operator * (const Matrix &rhs) const {
	assert(n == rhs.m);

	Matrix tmp(m, rhs.n, 0);
	Elem sum;
	for (int i = 0; i < m; i++)
		for (int j = 0; j < rhs.n; j++) {
			sum = 0;
			for (int k = 0; k < n; k++) {
				sum += get(i, k) * rhs.get(k, j);
			}
			tmp.set(i, j, sum);
		}
	return tmp;
}

const Matrix Matrix::operator - (const Matrix &rhs) const {
	return *this + ((Elem)-1 * rhs);
}

Matrix &Matrix::operator += (const Matrix &rhs) {
	return *this = *this + rhs;
}

Matrix &Matrix::operator *= (const Matrix &rhs) {
	return *this = *this * rhs;;
}

Matrix &Matrix::operator -= (const Matrix &rhs) {
	return *this = *this - rhs;
}

Row &Matrix::operator[](const int x) {
	return matrix[x];
}

const Matrix Matrix::transport() const {
	VV tmp;
	for (int i = 0; i < n; i++) {
		Row row;
		for (int j = 0; j < m; j++)
			row.push_back(get(j, i));
		tmp.push_back(row);
	}
	return tmp;
}

const Matrix Matrix::pow(int x) const {
	Matrix tmp(*this), e(m);
	for (int i = 1; i <= x; i <<= 1) {
		if ((x & i) > 0)
			e = e * tmp;
		tmp = tmp * tmp;
	}
	return e;
}

const Matrix Matrix::cofactor(int x, int y) const {
	VV tmp;
	for (int i = 0; i < m; i++) {
		if (x == i) continue;
		Row row;
		for (int j = 0; j < n; j++) {
			if (y == j) continue;
			row.push_back(get(i, j));
		}
		tmp.push_back(row);
	}
	return Matrix(tmp);
}

const Elem Matrix::det() const {
	assert(n == m);

	if (m == 1)
		return get(0, 0);
	Elem sum = 0;
	for (int i = 0; i < m; i++) {
		sum += ((i % 2 == 0 ? 1 : -1) * get(i, 0)) * Matrix(cofactor(i, 0)).det();
	}
	return sum;
}

const Matrix Matrix::triangulate() const {
	Matrix tmp(*this);
	Elem e;
	int p = 0;
	for (int i = 0; i < m && p < n; i++, p++) {
		if (isZero(tmp.get(i, p))) {
			tmp.set(i, p, 0);
			bool flag = true;
			for (int j = i + 1; j < m; j++)
				if (!isZero(tmp.get(j, p))) {
					for (int k = 0; k < n; k++)
						tmp.set(i, k, tmp.get(i, k) + tmp.get(j, k));
					//tmp[i].swap(tmp[j]);
					flag = false;
					break;
				}
			if (flag) {
				i--;
				continue;
			}
		}
		for (int j = i + 1; j < m; j++) {
			e = tmp.get(j, p) / tmp.get(i, p);
			for (int k = 0; k < n; k++)
				tmp.set(j, k, tmp.get(j, k) - tmp.get(i, k) * e);
		}
	}
	return tmp;
}

const int Matrix::rank() const {
	Matrix tmp(triangulate());
	for (int i = min(tmp.m - 1, tmp.n - 1); i >= 0; i--) {
		for (int j = tmp.n - 1; j >= i; j--)
			if (isZero(tmp.get(i, j)))
				continue;
			else
				return i + 1;
	}
	return 0;
}

const Matrix Matrix::pre_inverse() const {
	assert(m == n);

	Matrix tmp(m, n, 0);
	for (int i = 0; i < m; i++)
		for (int j = 0; j < n; j++)
			tmp.set(i, j, ((i + j) % 2 == 0 ? 1 : -1)*cofactor(i, j).det());
	return tmp.transport();
}

/*O(n!)????£?
const Matrix Matrix::inverse() const {
Matrix tmp(pre_inverse());
Elem e = det();
assert(!isZero(e));
tmp = 1 / e * tmp;
return tmp.transport();
}*/

const Matrix Matrix::inverse() const {
	assert(m == n);

	Matrix tmp(m, n * 2), tmp2(m, n);
	for (int i = 0; i < m; i++)
		for (int j = 0; j < n; j++)
			tmp.set(i, j, get(i, j));
	for (int i = 0; i < m; i++)
		tmp.set(i, i + n, 1);

	tmp = tmp.rowReduction();

	//?????????????????¨??????????????????????????§??????
	for (int i = 0; i < m; i++)
		assert(isZero(tmp.get(i, i) - 1));

	for (int i = 0; i < m; i++)
		for (int j = 0; j < n; j++)
			tmp2.set(i, j, tmp.get(i, j + n));

	return tmp2;
}

/*
a b c j
d e f k
g h i l

ax+by+cz=j;
dx+ey+fz=k;
gx+hy+iz=l;

|
V

1 0 0 x
0 1 0 y
0 0 1 z
???????????????
*/
const Matrix Matrix::rowReduction() const {
	Matrix tmp(*this);
	Elem e;
	int p = 0;
	for (int i = 0; i < m && p < n; i++, p++) {
		if (isZero(tmp.get(i, p))) {
			tmp.set(i, p, 0);
			bool flag = true;
			for (int j = i + 1; j < m; j++)
				if (!isZero(tmp.get(j, p))) {
					for (int k = 0; k < n; k++)
						tmp.set(i, k, tmp.get(i, k) + tmp.get(j, k));
					//tmp[i].swap(tmp[j]);
					flag = false;
					break;
				}
			if (flag) {
				i--;
				continue;
			}
		}
		e = 1 / tmp.get(i, p);
		tmp.set(i, p, 1);
		for (int k = i + 1; k < n; k++)
			tmp.set(i, k, tmp.get(i, k)*e);
		for (int j = 0; j < m; j++) {
			if (i == j) continue;
			e = tmp.get(j, p);
			for (int k = 0; k < n; k++)
				tmp.set(j, k, tmp.get(j, k) - tmp.get(i, k) * e);
		}
	}
	return tmp;
}

int main() {
	int N, M; cin >> N >> M;
	vector<vector<ld>>edges(N, vector<ld>(M));
	for (int i = 0; i < M; ++i) {
		int a, b; cin >> a >> b; a--; b--;
		edges[a][i]++;
		edges[b][i]--;
	}
	Matrix mat(edges);
	int num=mat.rank();
	int ans = M - num;
	cout << ans << endl;
	return 0;
}