gauss.cpp

#define _CRT_SECURE_NO_DEPRECATE
#pragma GCC optimize("O3")
#pragma GCC target("sse4")
#include "bits/stdc++.h"
//#include <ext/pb_ds/assoc_container.hpp>
//#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
//using namespace __gnu_pbds;
#define int int64_t
#define SYNC std::ios_base::sync_with_stdio(0);cin.tie(NULL);cout.tie(NULL);
#define FRE freopen("input.txt","r",stdin);freopen("output.txt","w",stdout);
typedef long double ld;
typedef pair<int,int> ii;
typedef pair<int,ii> iii;
typedef vector<int> vi;
typedef vector<ii>   vii;
//typedef tree<int, null_type, less<int>, rb_tree_tag,
//             tree_order_statistics_node_update>
//    ost;
#define rep(i,l,r)   for (int i = (l); i < (r); i++)
#define here cout << " Hey!!\n";
#define  pb push_back
#define  F  first
#define  S  second
#define all(v) (v).begin(),(v).end()
#define sz(a) (int)((a).size())
#define sq(x) ((x)*(x))
template <typename T>
void etch(T V) {
    for (auto x : V)    cout << x << ' ' ;
    cout << '\n';
}
vector<string> vec_splitter(string s) {
    s += ',';
    vector<string> res;
    while(!s.empty()) {
        res.push_back(s.substr(0, s.find(',')));
        s = s.substr(s.find(',') + 1);
    }
    return res;
}
void debug_out(
        vector<string> __attribute__ ((unused)) args,
        __attribute__ ((unused)) int idx,
        __attribute__ ((unused)) int LINE_NUM) { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(vector<string> args, int idx, int LINE_NUM, Head H, Tail... T) {
    if(idx > 0) cerr << ", "; else cerr << "Line(" << LINE_NUM << ") ";
    stringstream ss; ss << H;
    cerr << args[idx] << " = " << ss.str();
    debug_out(args, idx + 1, LINE_NUM, T...);
}
#ifndef ONLINE_JUDGE
#define debug(...) debug_out(vec_splitter(#__VA_ARGS__), 0, __LINE__, __VA_ARGS__)
#else
#define debug(...) 42
#endif
const int MOD = 2;
const int MOD1 = 998244353;
const int N = 2e5+5;
const ld PI = 3.141592653589793116;

int modular_exp(int base,int exp,int mod)
{
    int ans =1;
    while(exp)
    {
        if(exp%2)   ans = ans%mod*base%mod;
        base = (base%mod*base%mod)%mod;
        exp/=2;
        ans%=mod;
    }
    return ans;
}
int inv_mod(int base,int mod)
{
    return modular_exp(base,mod-2,mod);
}


const double EPS = 1e-9;
const int INF = 2; // it doesn't actually have to be infinity or a big number

int gauss (vector < vector<int> > a, vector<int> & ans) {
    int n = (int) a.size();
    int m = (int) a[0].size() - 1;

    vector<int> where (m, -1);
    for (int col=0, row=0; col<m && row<n; ++col) {
        int sel = row;
        for (int i=row; i<n; ++i)
            if (abs (a[i][col]) > abs (a[sel][col]))
                sel = i;
        if (abs (a[sel][col]) < EPS)
            continue;
        for (int i=col; i<=m; ++i)
            swap (a[sel][i], a[row][i]);
        where[col] = row;

        for (int i=0; i<n; ++i)
            if (i != row) {
                int c = (a[i][col] * inv_mod(a[row][col], MOD))%MOD;
                for (int j=col; j<=m; ++j) {
                    a[i][j] -= a[row][j] * c;
                    a[i][j] %= MOD;
                    if (a[i][j] < 0)    a[i][j] += MOD;
                }
            }
        ++row;
    }

    ans.assign (m, 0);
    for (int i=0; i<m; ++i)
        if (where[i] != -1)
            ans[i] = (a[where[i]][m] * inv_mod(a[where[i]][i], MOD))%MOD;
    for (int i=0; i<n; ++i) {
        int sum = 0;
        for (int j=0; j<m; ++j) {
            sum += ans[j] * a[i][j];
            sum %= MOD;
            if (sum < 0)    sum += MOD;
        }
        if (abs (sum - a[i][m]) > EPS)
            return 0;
    }

    for (int i=0; i<m; ++i)
        if (where[i] == -1)
            return INF;
    return 1;
}

int32_t main()
{
    SYNC
    int n, m; cin >> n >> m;
    vi g[n];
    for (int i = 0; i < m; i++) {
        int u, v; cin >> u >> v;
        u--, v--;
        g[u].pb(v);
        g[v].pb(u);
    }
    vector<vi> A(n, vi(n, 0));
    vi B(n, 1), X;
    for (int i = 0; i < n; i++) {
        int cnt = 0;
        for (int v : g[i]) {
            A[i][v] = 1;
            cnt++;
        }
        A[i][i] = cnt & 1LL;
    }
    rep(i,0,n)  A[i].pb(B[i]);
    cout << (gauss(A, X) > 0 ? "Y" : "N");
    return 0;
}