Graph Ordering

input/code.cpp: In instantiation of 'void fast_print(const std::vector<_Tp>&) [with T = int]':
input/code.cpp:244:12:   required from 'std::ostream& operator,(std::ostream&, const T&) [with T = std::vector<int>; std::ostream = std::basic_ostream<char>]'
input/code.cpp:622:10:   required from here
input/code.cpp:181:20: warning: comparison between signed and unsigned integer expressions [-Wsign-compare]
  for (int i = 1; i < v.size(); i++) {
input/code.cpp: In function 'void fast_scan(int&)':
input/code.cpp:133:31: warning: ignoring return value of 'int scanf(const char*, ...)', declared with attribute warn_unused_result [-Wunused-result]
 void fast_scan(int &x) { scanf("%d", &x); }
                          ~~~~~^~~~~~~~~~
input/code.cpp: In function 'void fast_scan(long long int&)':
input/code.cpp:134:37: warning: ignoring return value of 'int scanf(const char*, ...)', declared with attribute warn_unused_result [-Wunused-result]
 void fast_scan(long long &x) { scanf("%lld", &x); }...

/*
                                                                                                     
                                             `-:://:::-                                             
                                           `//:-------:/:`                                          
                                          .+:--.......--:+`                                         
                                         `+:--..`````..--//`                                        
                                         .o:--..`` ``..--:o`                                        
                                         .o:--...```..---+/`                                        
                                       `/y+o/---....---:+o.                                         
                                   `...````-os+/:---:/+o/--.`                                       
              `-/+++++/:.      `...`       :h+d+oooo+/+-`   ...                                     
            `/++//:::://++-`....`         -.`//````````:`     `..`                                  
           `o+/::------://o/`           `-` -.          -`       `..`                               
 `---.-o/:./o/::-..``..-ЗАПУСКАЕМ      ..  ..            -`        `...       ``..``                
  `....o+:-++/:--.```..-://s.        `-`  .-              -`          `-o: .-//::::/:-`             
          `:s+/:--....-::/+s-`      .-   `-                -`           -///:--------:/:`           
           ./s+//:::::://oo-``..НЕЙРОННУЮ: СЕТЬ:::::::-`РАБОТЯГИ        `+:--........--:/`          
            .:ooo+++++osso-`    `.:-...`/` ./::-------:/:`   -`         :+--..``````.--:+:...-+:-`  
             `.-/+++++/+-.-`    -.   ``:so:/:--.......--:+`  `-```````o+/+--..`````..--:o/-..:s+:.  
                 ```````:``.. `-`     -` `+:--..`````..--/+-.../.`````..-o:--.......---/o.    `     
                        `:  `:-      -.  .o:--..`` ``..--:o`   `-`      `:o+:--------:+o-`          
                         `-`-...    ..   .o/--...```..--:+/`    `-`     `oy/so/////++o/.`           
                          -/`  `-` `- ``+s/o/:---...---:++.      `-`   .-../d://///:-.`             
                `.---..``-..-    .-/..`````-oo+/:::::/+o+-        `-``-`  `-.  ````                 
             `:++++/+++++-  ..``.-/:`      /y-:/++o++/:.`..`       ./.   `-                         
            -++/::::::://+/..:-``:` ..   `-.`  ```.```    `..`   `..`-` `-                          
       ``  -o//:--....-::/++` -.-`   `-`.-`                 `..`..`  `-.-                           
  -----ss+:++/:--.```..-://s.  /.     `::                    `-:.     ./`                           
  `````/:..+o/::-..``.--:/+s. ..-`   `-``-`                 ..` `-`  `-`-`                          
          `-s+/::-----::/+oo---``-` ..    .:-    ```      .-`     .-.-  `-`                         
           `:oo+//::://+os/..:`..-/:`      :y.-:::::::.`.-`        ./-`  `-`                        
            `./+oooooooo+/.`-    .-:...`.. .//:-------://`        `- `..` `:.                       
              ``.-::::-.``-/`  `-` `-  `oo:+:--.......--:/`      `-    `.:--h.``..```               
                          -.-`.-    .-   `+:--..`````..--//`    `-       /s-//::::::::.             
                         -` `/-      ..  .o:--..`` ``..--:o.```.-        `//:--------://`           
                        -` .-`.-`     -.`-o/--...```..--:+/.``-:....``:-.+:--....`...--:+`          
                       ..`-.   `-.   ``:os:o/:---...---:++.  `-     ``///+:-..``````.--:+-````-.`   
              `.:///////.-`      .:-..` -``-+o+/:::::/+o/.  `-         `:+:-..`````..--:o/:--/ys+-  
            `-++///////+o/. ``....`-.    :` `.:++++++/:.`  .-           -o/---......---/o.   `.`    
           `++//:-----::/+o:..`     .-`   :    ```````    .-           `+so+:--------:++-`          
  `````:-``:o/::-..`..--:/+o`         -.  `-             .-          `../../+o+////+o+:.`           
  -----syo/o+/:--.```..-://s.          .-` `-           .-        `...     ``-:////:-``             
       .` `/s//:--....-::/+s.            -. `-`        .-       `..`                                
           .+o+/:::--:://+s/-..`          .::+y  ```  .-     `..`                                   
            ./oo++////+oso-`   `....       :y-+:::::::/`   ...                                      
             `.:+oooooo/-`         `....-. .//:-------:/:-.`                                        
                ``...``                 /+:+:--.......--:+`                                         
                                         `+:--..`````..--//`                                        
                                         .o:--..`` ``..--:o`                                        
                                         .+/--...```..--:+/`                                        
                                         `-o/:---...---:++.                                         
                                          `-+o+/:---:/+o/.                                          
                                            `.:+oooo+/-.`                                           
                                               ``````                                               
*/

#ifdef aimbot
#pragma comment(linker, "/stack:200000000")
#pragma GCC optimize("Ofast")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
#pragma GCC optimize("unroll-loops")
#endif

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <map>
#include <set>
#include <queue>
#include <ostream>
#include <istream>
#include <typeinfo>
#include <iomanip>
#include <cstdio>
#include <cstdlib>
#include <cassert>
#include <limits>
#include <fstream>
#include <array>
#include <list>
#include <bitset>
#include <functional>
#include <random>
#include <cstring>
#include <chrono>

#define random escape__from__random__aetuhoetnuhshe
#define mt make_tuple
#define x first
#define y second
#define pb push_back
#define ppb pop_back
#define mp make_pair
#define umap unordered_map
#define uset unordered_set
#define elif else if
#define len(v) ((int)v.size())
#define f(i, n) for (int i = 0; i < (n); i++)
#define rof(i, n) for (int i = ((n) - 1); i >= 0; i--) 
#define apply(v, act) for (auto &x : v) { act; }
#define log(args...) {string s = #args;deque<string> deq;\
string buf = "";int bal = 0;for (char c : s) {\
if (c == '(' || c == '[' || c == '{') {bal++;\
} else if (c == ')' || c == ']' || c == '}') {\
bal--;} else {if (bal == 0) {if (c == ',') {\
deq.pb(buf);buf = "";} else {if (c != ' ') {\
buf += c;}}}}}if (!buf.empty()) {deq.pb(buf);}\
smart_io::precall_print();smart_io::_print(deq, args);}

#define print    \
smart_io::precall_print(); \
cout,

#define scan cin,

#ifdef fast_allocator
const int MAXMEM = 200 * 1000 * 1024;
char _memory[MAXMEM];
size_t _ptr = 0;
void* operator new(size_t _x) { _ptr += _x; assert(_ptr < MAXMEM); return _memory + _ptr - _x; }
void operator delete (void*) noexcept {}
#endif

using namespace std;

char string_in_buffer[(int)260];


void fast_scan(int &x) { scanf("%d", &x); }
void fast_scan(long long &x) { scanf("%lld", &x); }
void fast_scan(unsigned long long &x) { scanf("%llu", &x); }
void fast_scan(double &x) { scanf("%lf", &x); }
void fast_scan(long double &x) { scanf("%Lf", &x); }
void fast_scan(char &x) { 
	scanf("%c", &x); 
	if (x == '\n') {
		fast_scan(x);
	}
}
void fast_scan(string &x) {
	scanf("%s", string_in_buffer);
	x = string(string_in_buffer);
}

template<class TFirst, class TSecond>
void fast_scan(pair<TFirst, TSecond> &p) {
	fast_scan(p.first);
	fast_scan(p.second);
}

template <class T>
void fast_scan(vector<T> &v) {
	for (auto &x : v) fast_scan(x);
}

void fast_print(const int &x) { printf("%d", x); }
void fast_print(const unsigned int &x) { printf("%u", x); }
void fast_print(const long long &x) { printf("%lld", x); }
void fast_print(const unsigned long long &x) { printf("%llu", x); }
void fast_print(const double &x) { printf("%.15lf", x); }
void fast_print(const long double &x) { printf("%.15Lf", x); }
void fast_print(const char &x) { printf("%c", x); };
void fast_print(const string &x) { printf("%s", x.c_str());}
void fast_print(const char v[]) { fast_print((string)v); }

template<class TFirst, class TSecond>
void fast_print(const pair<TFirst, TSecond> &p) {
	fast_print(p.first);
	fast_print(' ');
	fast_print(p.second);
}

template <class T>
void fast_print(const vector<T> &v) {
	if (v.empty()) return;
	fast_print(v[0]);
	for (int i = 1; i < v.size(); i++) {
		fast_print(' ');
		fast_print(v[i]);
	}
}

template <class T>
void fast_print(const vector<vector<T>> &v) {
	if (v.empty()) return;
	fast_print(v[0]);
	for (int i = 1; i < v.size(); i++) {
		fast_print('\n');
		fast_print(v[i]);
	}
}

template <class T>
void fast_print(const T &v) {
	for (const auto &x : v) {
		fast_print(x);
		fast_print(' ');
	}
}


using namespace std;


namespace smart_io {
	string print_start = "";
	string sep = " ";
	bool first_print = false;

	void precall_print() {
		fast_print(print_start);
		print_start = "\n";
		first_print = true;
	}

	void _print(deque<string>) {}
	template<class T, class... Args>
	void _print(deque<string> names, T elem, Args... args) {
		if (!first_print) {
			fast_print("\n");
		} else {
			first_print = false;
		}
		fast_print(names.front());
		fast_print(" = ");
		fast_print(elem);
		names.pop_front();
		_print(names, args...);
	}
} //namespace smart_io


template <class T>
ostream &operator,(ostream &os, const T &object) {
	if (!smart_io::first_print) {
		fast_print(smart_io::sep);
	} else {
		smart_io::first_print = false;
	}
	fast_print(object);
	return os;
}

template <class T>
istream &operator,(istream &is, T &object) {
	fast_scan(object);
	return is;
}

namespace random {
	using namespace std::chrono;
	mt19937 rng(duration_cast< milliseconds >(
		system_clock::now().time_since_epoch()
	).count());
	uniform_real_distribution<> prob_dist(0.0, 1.0);
};

namespace typedefs {
	typedef long long ll;
	typedef unsigned long long ull;
	typedef pair<int, int> pii;
	typedef long double ld;
}

namespace numbers_operation {
	template<class T>
	T floor_mod(T a, T b) {
		if (a >= 0 && b >= 0) return a % b;
		if (a <= 0 && b <= 0) return a % b;
		return abs(b) - (abs(a) % abs(b));
	}
}

using namespace numbers_operation;
using namespace typedefs;
using namespace random;

struct Critical {
	int n;
	vector<vector<int>> g;
	vector<int> tin, uptime;
	vector<bool> used;
	int timer = 0;
	void dfs(int v) {
		used[v] = true;
		tin[v] = uptime[v] = timer++;
		for (int sub : g[v]) {
			if (used[sub]) continue;
			dfs(sub);
		}
	}
	void check(int v, int _forb) {
		used[v] = true;
		for (int sub : g[v]) {
			if (used[sub]) continue;
			if (sub == _forb) continue;
			check(sub, _forb);
		}
	}
	bool is_critical_slow(int v) {
		used = vector<bool>(n, false);
		int cnt = 0;
		for (int sub : g[v]) {
			if (used[sub]) continue;
			cnt++;
			check(sub, v);
		}
		return cnt >= 2;
	}
	vector<bool> crit;
	void go(int v, int _p) {
		used[v] = true;
		for (int sub : g[v]) {
			if (sub == _p) continue;
			if (used[sub]) {
				uptime[v] = min(uptime[v], tin[sub]);
			} else {
				go(sub, v);
				uptime[v] = min(uptime[v], uptime[sub]);
				if (uptime[sub] >= tin[v]) {
					crit[v] = true;
				}
			}
		}
	}
	Critical() {}
	Critical(int _n) {
		n = _n;
		g.resize(n);
	}
	void add_edge(int a, int b) {
		g[a].pb(b);
		g[b].pb(a);
	}
	void update_critical() {
		tin = vector<int>(n, 0);
		uptime = vector<int>(n, 0);
		used = vector<bool>(n, false);
		dfs(0);
		crit = vector<bool>(n, false);
		used = vector<bool>(n, false);
		go(0, -1);
		crit[0] = is_critical_slow(0);
	}
};

int n, m;
vector<vector<int>> g;
vector<pii> edges;
vector<int> uptime;
vector<int> tin;
vector<bool> used;
vector<int> buff;

vector<bool> pick;

vector<int> order;

int timer = 0;
vector<int> at;


int A = -1;

struct Node {
	Node *l = NULL, *r = NULL, *pref = NULL;
	int value, prior;
	int size = 1;

	Node(int _value) {
		value = _value;
		prior = rng();
	} 
};

int get_size(Node *node) {
	return node ? node->size : 0;
}

void update(Node *node) {
	node->size = get_size(node->l) + get_size(node->r) + 1;
}

Node *merge(Node *l, Node *r) {
	if (!l) return r;
	if (!r) return l;
	if (l->prior > r->prior) {
		l->r = merge(l->r, r);
		l->r->pref = l;
		update(l);
		return l;
	} else {
		r->l = merge(l, r->l);
		r->l->pref = r;
		update(r);
		return r;
	}
}

pair<Node*, Node*> split(Node *node, int k) {
	if (!node) return mp(node, node);
	if (get_size(node->l) >= k) {
		if (node->l) node->l->pref = NULL;
		auto t = split(node->l, k);
		node->l = NULL;
		update(node);
		return mp(t.x, merge(t.y, node));
	} else {
		if (node->r) node->r->pref = NULL;
		auto t = split(node->r, k - get_size(node->l) - 1);
		node->r = NULL;
		update(node);
		return mp(merge(node, t.x), t.y);
	}
}

int get_id(Node *node) {
	int id = get_size(node->l);
	while (node->pref) {
		if (node->pref->r == node) {
			id += 1 + get_size(node->pref->l);
		}
		node = node->pref;
	}
	return id;
}

Node *cart = NULL;

vector<Node*> pres;

void dfs(int v, int _p, int end) {
	buff.pb(v);
	if (v == end) {
		for (int x : buff) {
			pres[x] = new Node(x);
			cart = merge(cart, pres[x]);
			pick[x] = true;
		}
	}
	at[timer] = v;
	uptime[v] = tin[v] = timer++;
	used[v] = true;
	for (int sub : g[v]) {
		if (sub == _p) continue;
		if (used[sub]) {
			uptime[v] = min(uptime[v], tin[sub]);
		} else {
			// log(v + 1, sub + 1);
			dfs(sub, v, end);
			uptime[v] = min(uptime[v], uptime[sub]);
		}
	}
	buff.pop_back();
}

void echo(int v, int _p) {
	used[v] = true;

	if (!pick[v]) {
		buff.pb(v);
	}

	int B = at[uptime[v]];
	if (!buff.empty() && pick[B] && B != A) {
		// auto itA = find(order.begin(), order.end(), A);
		// auto itB = find(order.begin(), order.end(), B);
		int itA = get_id(pres[A]);
		int itB = get_id(pres[B]);
		if (itA < itB) {
			auto t = split(cart, itA + 1);

			// order.insert(itA + 1, buff.begin(), buff.end());
			for (int x : buff) {
				pres[x] = new Node(x);
				t.x = merge(t.x, pres[x]);
			}
			cart = merge(t.x, t.y);
		} else {
			reverse(buff.begin(), buff.end());
			// order.insert(itB + 1, buff.begin(), buff.end());
			auto t = split(cart, itB + 1);
			for (int x : buff) {
				pres[x] = new Node(x);	
				t.x = merge(t.x, pres[x]);
			}
			cart = merge(t.x, t.y);
		}
		for (int x : buff) {
			pick[x] = true;
		}
		buff.clear();
	}

	for (int sub : g[v]) {
		if (sub == _p) continue;
		if (used[sub]) continue;
		if (pick[v]) {
			A = v;
		}
		echo(sub, v);
	}
}

Critical base;
set<int> finded_crit;
void _index(int v) {
	used[v] = true;
	for (int sub : g[v]) {
		if (used[sub]) continue;
		if (base.crit[sub]) {
			finded_crit.insert(sub);
		} else {
			_index(sub);
		}
	}
}

signed main(signed argc, char *argv[]) {
	{
		scan n, m;
		base = Critical(n);
		g.clear();
		edges.clear();
		uptime.clear();
		tin.clear();
		used.clear();
		buff.clear();
		at.clear();
		order.clear();

		timer = 0;

		g.resize(n);
		at.resize(2 * n);
		uptime.resize(n);
		tin.resize(n);
		
		f(i, m) {
			int a, b;
			scan a, b;
			a--;b--;
			g[a].pb(b);
			g[b].pb(a);
			base.add_edge(a, b);
			edges.emplace_back(a, b);
		}

		used = vector<bool>(n, false);
		base.update_critical();
		vector<int> pos;
		f(i, n) {
			if (base.crit[i]) continue;
			if (!used[i]) {
				if (len(g[i]) == 1) {
					pos.pb(i);
					used[i] = true;
				} else {
					finded_crit.clear();
					_index(i);
					if (len(finded_crit) <= 1) {
						pos.pb(i);
					}
				}
			}
		}
		if (len(pos) <= 1) {
			pos.pb(n - 1);
		}
		if (len(pos) != 2) {
			print "IMPOSSIBLE";
			return 0;
		}

		int start = pos[0];
		int end = pos[1];
		shuffle(pos.begin(), pos.end(), rng);

		used = vector<bool>(n, false);
		pick = vector<bool>(n, false);
		pres.resize(n, NULL);
		dfs(start, -1, end);
		buff.clear();
		used = vector<bool>(n, false);
		echo(start, -1);
		vector<int> p(n);
		f(i, n) {
			if (pres[i]) {
				p[i] = get_id(pres[i]);
			} else {
				print "IMPOSSIBLE";
				return 0;
			}
		}
		vector<int> in(n), out(n);
		for (pii edge : edges) {
			if (p[edge.x] > p[edge.y]) {
				swap(edge.x, edge.y);
			}
			in[edge.y]++;
			out[edge.x]++;
		}
		bool fail = false;
		f(i, n) {
			if (i != start && in[i] == 0) {
				fail = true;
			}
			if (i != end && out[i] == 0) {
				fail = true;
			}
		}
		if (!fail) {
			vector<int> order(n);
			f(i, n) {
				order[p[i]] = i;
			}
			apply(order, x++);
			print order;
		} else {
			print "IMPOSSIBLE";
		}
	}
}

Test 1

Group: 1, 4, 5

Verdict: ACCEPTED

Test 2

Group: 1, 5

Verdict: ACCEPTED

Test 3

Group: 1, 5

Verdict: ACCEPTED

Test 4

Group: 1, 5

Verdict: ACCEPTED

Test 5

Group: 1, 5

Verdict: ACCEPTED

Test 6

Group: 2, 3, 5

Verdict: ACCEPTED

Test 7

Group: 2, 3, 5

Verdict: ACCEPTED

Test 8

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 9

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 10

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 11

Group: 2, 3, 5

Verdict: ACCEPTED

Test 12

Group: 2, 3, 5

Verdict: ACCEPTED

Test 13

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 14

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 15

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 16

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 17

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 18

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 19

Group: 3, 4, 5

Verdict: ACCEPTED

Test 20

Group: 3, 4, 5

Verdict: ACCEPTED

Test 21

Group: 3, 4, 5

Verdict: ACCEPTED

Test 22

Group: 3, 4, 5

Verdict: ACCEPTED

Test 23

Group: 3, 5

Verdict: ACCEPTED

Test 24

Group: 3, 5

Verdict: ACCEPTED

Test 25

Group: 3, 5

Verdict: ACCEPTED

Test 26

Group: 3, 5

Verdict: ACCEPTED

Test 27

Group: 3, 5

Verdict: ACCEPTED

Test 28

Group: 5

Verdict: ACCEPTED

Test 29

Group: 5

Verdict: ACCEPTED

Test 30

Group: 4, 5

Verdict: ACCEPTED

Test 31

Group: 4, 5

Verdict: ACCEPTED

Test 32

Group: 4, 5

Verdict: ACCEPTED

Test 33

Group: 4, 5

Verdict: ACCEPTED

Test 34

Group: 5

Verdict: ACCEPTED

Test 35

Group: 5

Verdict: ACCEPTED

Test 36

Group: 5

Verdict: ACCEPTED

Test 37

Group: 1, 2, 3, 4, 5

Verdict: ACCEPTED

Test 38

Group: 2, 3, 5

Verdict: ACCEPTED

Test 39

Group: 2, 3, 5

Verdict: ACCEPTED

Test 40

Group: 2, 3, 5

Verdict: ACCEPTED

Test 41

Group: 1, 2, 3, 5

Verdict: ACCEPTED

Test 42

Group: 2, 3, 5

Verdict: ACCEPTED

Test 43

Group: 3, 4, 5

Verdict: ACCEPTED

Test 44

Group: 3, 4, 5

Verdict: ACCEPTED

Test 45

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 46

Group: 2, 3, 4, 5

Verdict: ACCEPTED

Test 47

Group: 2, 3, 5

Verdict: ACCEPTED

Test 48

Group: 3, 4, 5

Verdict: ACCEPTED

Test 49

Group: 4, 5

Verdict: ACCEPTED