#include <iostream>
#include <list>
#include <queue>
#include <vector>
#include <stack>
#include <fstream>
#include <map>
#include <set>
#include <algorithm>
#include <climits>
#include <cstring>
#define INF ((1 << 30) - 1)
// 2^30-1 ca sa nu fie overflow daca faci INF + INF
const int nMax = 105;
using namespace std;
class DisjointSet {
private:
int m_parinte[nMax] = {}, m_dimensiune[nMax] = {};
public:
explicit DisjointSet(int n) {
for (int i = 1; i <= n; i++) {
m_parinte[i] = i;
m_dimensiune[i] = 1;
}
}
int cauta(int x) {
while (x != m_parinte[x]) {
x = m_parinte[x];
}
return x;
}
void uneste(int x, int y) {
int parinteX = cauta(x), parinteY = cauta(y);
// Uneste arborele mai mic la arborele mai mare, pentru o complexitate mai buna
if (m_dimensiune[parinteX] >= m_dimensiune[parinteY]) {
m_parinte[parinteY] = parinteX;
m_dimensiune[parinteX] += m_dimensiune[parinteY];
} else {
m_parinte[parinteX] = parinteY;
m_dimensiune[parinteY] += m_dimensiune[parinteX];
}
}
};
class Graf {
private:
int m_n, m_m;
vector<int> m_listaAd[nMax];
vector<pair<int, int>> m_ponderatListaAd[nMax];
int m_ponderatMatrice[105][105] = {};
vector<vector<int>> m_listaMuchii;
vector<pair<int, pair<int, int>>> m_listaMuchiiPonderat;
// DFS - https://www.infoarena.ro/problema/dfs
static constexpr int dfsMax = 1;
bool m_dfsViz[dfsMax] = {};
// BFS - https://www.infoarena.ro/problema/bfs
static constexpr int bfsMax = 1;
int m_bfsDist[bfsMax] = {};
queue<int> m_bfsQueue;
// CTC - https://www.infoarena.ro/problema/ctc
static constexpr int ctcMax = 1;
int m_ctcId[ctcMax] = {}, m_ctcLow[ctcMax] = {}, m_ctcUltId = 0;
bool m_ctcPeStiva[ctcMax] = {};
list<list<int>> m_ctc;
stack<int> m_ctcStack;
// Componente biconexe - https://www.infoarena.ro/problema/biconex
static constexpr int biconexMax = 1;
list<list<int>> m_biconexComps;
stack<int> m_biconexStack;
int m_biconexLow[biconexMax] = {};
// Muchii critice - https://leetcode.com/problems/critical-connections-in-a-network/
static constexpr int criticeMax = 1;
map<pair<int, int>, bool> m_criticeToRemove;
vector<vector<int>> m_critice;
int m_criticeLow[criticeMax] = {}; // Id-ul nodului minim in care te poti intoarce din nodul i
// Arbore partial de cost minim - https://www.infoarena.ro/problema/apm
int m_apcmCost = 0;
vector<pair<int, int>> m_apcmResult;
// Bellman-Ford - https://infoarena.ro/problema/bellmanford
static constexpr int bellmanMax = 1;
vector<int> m_bellmanDist = vector<int>(bellmanMax, INT_MAX);
int m_bellmanPuneriInCoada[bellmanMax] = {}, m_bellmanInQueue[bellmanMax] = {};
queue<int> m_bellmanQueue;
bool m_bellmanCircuitCostNegativ = false;
// Dijkstra - https://infoarena.ro/problema/dijkstra
static constexpr int dijkstraMax = 1;
vector<int> m_dijkstraDist = vector<int>(dijkstraMax, INT_MAX);
set<pair<int, int>> m_dijkstraMinHeap; // "min".. doar e un ordered set (crescator)
// Diametru arbore - https://www.infoarena.ro/problema/darb
static constexpr int diametruVizMax = 1;
bool m_diametruViz[diametruVizMax] = {};
int m_diametruNodMax = 0, m_diametruDistMax = 0;
// RoyFloyd - https://www.infoarena.ro/problema/royfloyd
static constexpr int royFloydMax = 1;
int m_royFloydDists[royFloydMax][royFloydMax] = {};
// Flux maxim - https://infoarena.ro/problema/maxflow
static constexpr int fluxMaximMax = 1;
int m_fluxMaximCapacitate[fluxMaximMax][fluxMaximMax] = {}, m_fluxMaximFlux[fluxMaximMax][fluxMaximMax] = {},
m_fluxMaximParinti[fluxMaximMax] = {};
queue<int> m_fluxMaximQueue;
// ---------------- Functii private ----------------
void orientatCtcDFS(int x) {
m_ctcStack.push(x);
m_ctcPeStiva[x] = true;
m_ctcId[x] = m_ctcLow[x] = ++m_ctcUltId;
for (auto y: m_listaAd[x]) {
// Nu am explorat nodul pana acum (neavand vreun id)
if (m_ctcId[y] == 0) {
orientatCtcDFS(y);
}
// Am intalnit un nod care inca nu a fost atribuit unei componente conexe.
// Poate nodul curent face parte din viitoarea componenta conexa, a carei (posibila) sursa
// a fost gasita de y.
if (m_ctcPeStiva[y]) {
m_ctcLow[x] = min(m_ctcLow[x], m_ctcLow[y]);
}
}
// Am ajuns la nodul de start al ctc-ului explorat in prezent
if (m_ctcId[x] == m_ctcLow[x]) {
list<int> compCurr;
while (true) {
auto y = m_ctcStack.top();
m_ctcStack.pop();
m_ctcPeStiva[y] = false;
compCurr.push_back(y);
if (y == x) break;
}
m_ctc.push_back(compCurr);
}
}
void neorientatBiconexAdd(int x, int y) {
// Creeaza o noua componenta pentru afisare
list<int> comp;
// Adauga in componenta toate nodurile pana la y, inclusiv y
while (m_biconexStack.top() != y) {
comp.push_back(m_biconexStack.top());
m_biconexStack.pop();
}
comp.push_back(y);
m_biconexStack.pop();
// Adauga in componenta si pe x, separat (in caz ca e un gap in stack intre y si x)
// ^ gap-ul poate aparea daca intalnim mai multe componente biconexe ce se intorc in acelasi nod
comp.push_back(x);
m_biconexComps.push_back(comp);
}
void neorientatBiconexDfs(int x, int prev, int id) {
// Initializam low-ul (nodul cel mai de sus din parcurgerea DFS in care putem ajunge)
// si punem nodul curent pe stack
m_biconexLow[x] = id;
m_biconexStack.push(x);
for (auto y: m_listaAd[x]) {
// Ignoram cazul in care ne intoarcem din nodul in care am plecat
if (y == prev) continue;
// Nodul y nu a fost vizitat => viziteaza-l
if (!m_biconexLow[y]) {
// Viziteaza-l si actualizeaza low
neorientatBiconexDfs(y, x, id + 1);
m_biconexLow[x] = min(m_biconexLow[x], m_biconexLow[y]);
// Am ajuns la originea ciclului / am dat peste un nod de mai jos din parcurgerea
// DFS la care nu mai putem ajunge altfel (=> componenta biconexa)
if (m_biconexLow[y] >= id) {
neorientatBiconexAdd(x, y);
}
}
// Nodul y a fost vizitat => doar actualizeaza min-ul in caz ca e nevoie,
// fara sa risti sa afisezi o componenta biconexa de doua ori
else {
m_biconexLow[x] = min(m_biconexLow[x], m_biconexLow[y]);
}
}
}
void neorientatMuchiiCriticeDfs(int x, int prev, int id) {
m_criticeLow[x] = id;
for (auto y: m_listaAd[x]) {
// Nu te intoarce in nodul din care ai plecat
if (y == prev) continue;
// Ruleaza DFS in continuare, cu un id mai mare
if (m_criticeLow[y] == 0) neorientatMuchiiCriticeDfs(y, x, id + 1);
// Nodul vizitat din cel curent face parte dintr-un ciclu,
// asa ca trebuie sa excludem muchia x-y
if (m_criticeLow[y] < id + 1) {
m_criticeToRemove[{x, y}] = m_criticeToRemove[{y, x}] = true;
}
// Actualizeaza low-ul nodului curent
m_criticeLow[x] = min(m_criticeLow[x], m_criticeLow[y]);
}
}
void diametruDFS(int x, int dist) {
if (dist > m_diametruDistMax) {
m_diametruDistMax = dist;
m_diametruNodMax = x;
}
m_diametruViz[x] = 1;
for (auto &y: m_listaAd[x]) {
if (!m_diametruViz[y]) {
diametruDFS(y, dist + 1);
}
}
}
void orientatRoyFloydSetup() {
for (int i = 1; i <= m_n; i++) {
for (int j = 1; j <= m_n; j++) {
if (i == j) {
m_royFloydDists[i][j] = 0;
} else if (m_ponderatMatrice[i][j] == 0) {
m_royFloydDists[i][j] = INF;
} else {
m_royFloydDists[i][j] = m_ponderatMatrice[i][j];
}
}
}
}
public:
// ---------------- Interfata publica ----------------
explicit Graf(int n = 0, int m = 0) : m_n(n), m_m(m) {}
/*************** Algoritmi generali ***************/
void ponderatCitesteListaMuchii(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y, c;
in >> x >> y >> c;
m_listaMuchiiPonderat.push_back({c, {x, y}});
}
}
void DFS(int k) {
m_dfsViz[k] = true;
for (auto x: m_listaAd[k]) {
if (!m_dfsViz[x]) {
DFS(x);
}
}
}
const auto &BFS(int start) {
m_bfsQueue.push(start);
m_bfsDist[start] = 1;
while (!m_bfsQueue.empty()) {
int curr = m_bfsQueue.front();
m_bfsQueue.pop();
for (auto i: m_listaAd[curr]) {
if (m_bfsDist[i] == 0) {
m_bfsDist[i] = m_bfsDist[curr] + 1;
m_bfsQueue.push(i);
}
}
}
return m_bfsDist;
}
static bool potiConstruiGraf(vector<int> grade) {
// Algoritmul Havel-Hakimi
while (true) {
sort(grade.begin(), grade.end(), greater<int>());
if (grade[0] == 0) break;
if (grade[0] > grade.size() - 1) return false;
int maxVal = grade[0];
for (int i = 1; i <= maxVal; i++) {
grade[0]--;
grade[i]--;
if (grade[i] < 0) return false;
}
}
return true;
}
void construiesteApcm() {
// Algoritmul lui Kruskal: luam muchiile cu cel mai mic cost, cat timp nu
// se creeaza cicluri.
sort(m_listaMuchiiPonderat.begin(), m_listaMuchiiPonderat.end());
// Foloseste un disjoint set pentru a sti daca muchia de la x la y va crea
// un ciclu (valoarea fiecarui nod reprezentand componenta conexa din care
// face parte).
DisjointSet ds(m_n);
for (auto &m: m_listaMuchiiPonderat) {
int c = m.first, x = m.second.first, y = m.second.second;
if (ds.cauta(x) != ds.cauta(y)) {
ds.uneste(x, y);
// Adauga la rezultat muchia gasita
m_apcmCost += c;
m_apcmResult.push_back({x, y});
}
}
}
const auto &getApcmResult() {
return m_apcmResult;
}
auto getApcmCost() {
return m_apcmCost;
}
/*************** Grafuri neorientate ***************/
void neorientatCitesteListaMuchii(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y;
in >> x >> y;
m_listaMuchii.push_back({x, y});
}
}
void neorientatListaMuchiiToListaAdiacenta() {
for (auto &e: m_listaMuchii) {
m_listaAd[e[0]].push_back(e[1]);
m_listaAd[e[1]].push_back(e[0]);
}
}
void neorientatCitesteListaAdiacenta(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y;
in >> x >> y;
m_listaAd[x].push_back(y);
m_listaAd[y].push_back(x);
}
}
int neorientatNrCompConexe() {
int nrComp = 0;
for (int i = 1; i <= m_n; i++) {
if (!m_dfsViz[i]) {
nrComp++;
DFS(i);
}
}
return nrComp;
}
const auto &neorientatBiconexe() {
for (int i = 1; i <= m_n; i++) {
if (!m_biconexLow[i]) {
neorientatBiconexDfs(i, -1, 1);
}
}
return m_biconexComps;
}
const auto &neorientatMuchiiCritice() {
neorientatMuchiiCriticeDfs(0, -1, 1);
// In rezultat, punem muchiile ce nu au fost marcate ca trebuind sa fie sterse
for (auto &e: m_listaMuchii) {
if (!m_criticeToRemove[{e[0], e[1]}]) {
m_critice.push_back(e);
}
}
return m_critice;
}
int diametru() {
diametruDFS(1, 1);
m_diametruDistMax = 0;
memset(m_diametruViz, 0, sizeof(m_diametruViz));
diametruDFS(m_diametruNodMax, 1);
return m_diametruDistMax;
}
/*************** Grafuri orientate ***************/
void orientatCitesteListaAdiacenta(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y;
in >> x >> y;
m_listaAd[x].push_back(y);
}
}
void orientatPonderatCitesteListaAdiacenta(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y, c;
in >> x >> y >> c;
m_ponderatListaAd[x].push_back({y, c});
}
}
void orientatPonderatCitesteMatricePonderi(ifstream &in) {
for (int i = 1; i <= m_n; i++) {
for (int j = 1; j <= m_n; j++) {
int p;
in >> p;
m_ponderatMatrice[i][j] = p;
}
}
}
const auto &orientatCtc() {
// Algoritmul lui Tarjan
for (int i = 1; i <= m_n; i++) {
// Nu am explorat nodul pana acum (neavand vreun id)
if (m_ctcId[i] == 0) {
orientatCtcDFS(i);
}
}
return m_ctc;
}
void orientatRuleazaBellmanFord(int start) {
// Gaseste graful de costuri minime, plecand din start la celelalte n-1 noduri.
// Putem avea circuit de cost negativ -> va fi detectat.
// Incepem cu optimizarile plecand din nodul de start
m_bellmanQueue.push(start);
m_bellmanDist[start] = 0;
m_bellmanInQueue[start] = true;
m_bellmanPuneriInCoada[start] = 1;
// Ne oprim cand nu mai avem nimic de optimizat / am gasit un circuit cu cost negativ
while (!m_bellmanQueue.empty() && !m_bellmanCircuitCostNegativ) {
int x = m_bellmanQueue.front();
m_bellmanQueue.pop();
// Marcam nodul curent ca ne mai fiind in queue
m_bellmanInQueue[x] = false;
// Luam toate arcele la rand si incercam sa optimizam distante, folosindu-le
for (auto &e: m_ponderatListaAd[x]) {
int y = e.first, c = e.second;
// Am gasit un arc (de la x la y) ce optimizeaza costul lui y (= obtinem
// o distanta mai mica din start->y daca mergem prin x)
if (m_bellmanDist[y] > m_bellmanDist[x] + c) {
m_bellmanDist[y] = m_bellmanDist[x] + c;
// Daca y nu e deja in coada, pune-l (facem verificarea ca sa nu
// il adaugam de mai multe ori in coada), pentru ca, optimizand
// distanta pana la el, putem optimiza distante si plecand din el.
if (!m_bellmanInQueue[y]) {
m_bellmanQueue.push(y);
m_bellmanInQueue[y] = true;
// Numara de cate ori au fost puse in coada nodurile. Daca un nod
// a fost pus de >= n ori (=> n optimizari), inseamna ca am gasit
// un circuit de cost negativ.
m_bellmanPuneriInCoada[y]++;
if (m_bellmanPuneriInCoada[y] >= m_n) {
m_bellmanCircuitCostNegativ = true;
}
}
}
}
}
}
bool circuitNegativBellmanFord() {
return m_bellmanCircuitCostNegativ;
}
const auto &getBellmanFordDists() {
return m_bellmanDist;
}
const auto &orientatRuleazaDijkstra(int start) {
// Incepem algoritmul din nodul de start
m_dijkstraDist[start] = 0;
// Punem in set perechea {0, [nod start]}, 0 fiind distanta de la start pana la el insusi
m_dijkstraMinHeap.insert({0, start});
while (!m_dijkstraMinHeap.empty()) {
// Procesam nodul de la distanta cea mai mica fata de start
auto x = m_dijkstraMinHeap.begin()->second;
m_dijkstraMinHeap.erase(m_dijkstraMinHeap.begin());
for (auto &e: m_ponderatListaAd[x]) {
auto y = e.first, c = e.second;
// Obtinem un drum mai scurt (fata de cel gasit) daca trecem prin x
if (m_dijkstraDist[y] > m_dijkstraDist[x] + c) {
// Nodul y e deja marcat ca trebuind sa fie procesat => il scoatem din set, ca sa il readaugam
// cu noua distanta, mai mica (ca sa nu pierdem timp incercand sa-l optimizam cu distanta veche
// mai tarziu) -- 90p->100p
if (m_dijkstraMinHeap.count({m_dijkstraDist[y], y}) > 0) {
m_dijkstraMinHeap.erase(m_dijkstraMinHeap.find({m_dijkstraDist[y], y}));
}
// Actualizam distanta lui y si il punem la procesat
m_dijkstraDist[y] = m_dijkstraDist[x] + c;
m_dijkstraMinHeap.insert({m_dijkstraDist[y], y});
}
}
}
return m_dijkstraDist;
}
void orientatRoyFloyd() {
orientatRoyFloydSetup();
// Nodul k = nodul pe care incercam sa il integram in drumul de la i la j
for (int k = 1; k <= m_n; k++) {
// Verificam toate drumurile daca pot fi scurtate folosind k drept nod intermediar
// (i -> ... -> k -> ... -> j)
for (int i = 1; i <= m_n; i++) {
for (int j = 1; j <= m_n; j++) {
if (m_royFloydDists[i][k] + m_royFloydDists[k][j] < m_royFloydDists[i][j]) {
m_royFloydDists[i][j] = m_royFloydDists[i][k] + m_royFloydDists[k][j];
}
}
}
}
}
const auto &orientatRoyFloydGetDists() {
return m_royFloydDists;
}
void citesteInputFluxMaxim(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y, c;
in >> x >> y >> c;
m_fluxMaximCapacitate[x][y] = c;
m_listaAd[x].push_back(y);
m_listaAd[y].push_back(x);
}
}
int orientatFluxMaximBFS(int start = 1) {
memset(m_fluxMaximParinti, 0, sizeof(m_fluxMaximParinti));
m_fluxMaximQueue.push(start);
m_fluxMaximParinti[start] = -1;
while (!m_fluxMaximQueue.empty()) {
int x = m_fluxMaximQueue.front();
m_fluxMaximQueue.pop();
// Din moment ce am facut graful unul neorientat (pentru a avea acces la vecinii nodului destinatie),
// am putea sa ne intoarcem din nodul final in alte noduri inca nevizitate, dar nu vrem asta
if (x == m_n) {
continue;
}
for (auto &y: m_listaAd[x]) {
// Daca nu am vizitat nodul y in BFS-ul curent si daca inca mai putem pompa
// flux prin lantul x-y, viziteaza-l
if (!m_fluxMaximParinti[y] && m_fluxMaximCapacitate[x][y] - m_fluxMaximFlux[x][y] > 0) {
m_fluxMaximQueue.push(y);
m_fluxMaximParinti[y] = x;
}
}
}
// Returnam daca am ajuns la destinatie cu parcurgerea curenta
return m_fluxMaximParinti[m_n];
}
int orientatFluxMaxim() {
// Algoritmul Edmonds-Karp
int fluxMaximTotal = 0;
while (orientatFluxMaximBFS()) {
// Am creat arborele BFS -> ne intoarcem pe fiecare ruta posibila, folosindu-ne de vecinii
// nodului final
for (auto &x: m_listaAd[m_n]) {
if (!m_fluxMaximParinti[x] || m_fluxMaximCapacitate[x][m_n] - m_fluxMaximFlux[x][m_n] == 0) {
continue;
}
m_fluxMaximParinti[m_n] = x;
// Gaseste fluxul minim ce poate fi pompat pe lantul gasit de BFS
int fluxMinimLant = INT_MAX, curr = m_n;
while (m_fluxMaximParinti[curr] != -1) {
int prev = m_fluxMaximParinti[curr];
fluxMinimLant = min(fluxMinimLant, m_fluxMaximCapacitate[prev][curr] - m_fluxMaximFlux[prev][curr]);
curr = prev;
}
if (fluxMinimLant == 0) {
continue;
}
// Actualizeaza fluxurile
curr = m_n;
while (m_fluxMaximParinti[curr] != -1) {
int prev = m_fluxMaximParinti[curr];
m_fluxMaximFlux[prev][curr] += fluxMinimLant;
m_fluxMaximFlux[curr][prev] -= fluxMinimLant;
curr = prev;
}
fluxMaximTotal += fluxMinimLant;
}
}
return fluxMaximTotal;
}
static const int hamiltonMinimMax = 19;
int m_hamiltonMinimDP[1 << hamiltonMinimMax][hamiltonMinimMax] = {};
void citesteInputCicluHamiltonian(ifstream &in) {
for (int i = 0; i < m_m; i++) {
int x, y, c;
in >> x >> y >> c;
m_ponderatListaAd[y].push_back({x, c});
}
}
int orientatCostMinimCicluHamiltonian() {
for (int k = 0; k < 1 << hamiltonMinimMax; k++) {
for (int i = 0; i < m_n; i++) {
m_hamiltonMinimDP[k][i] = INF;
}
}
m_hamiltonMinimDP[1][0] = 0;
for (int k = 0; k <= 1 << (m_n + 1); k++) {
for (int i = 0; i < m_n; i++) {
if (k & (1 << i)) {
for (auto &pairJ: m_ponderatListaAd[i]) {
int j = pairJ.first, c = pairJ.second;
if (k & (1 << j)) {
m_hamiltonMinimDP[k][i] = min(
m_hamiltonMinimDP[k][i],
m_hamiltonMinimDP[k & ~(1 << i)][j] + c
);
}
}
}
}
}
int res = INF;
for (auto &pair0: m_ponderatListaAd[0]) {
int i = pair0.first, c = pair0.second;
res = min(res, m_hamiltonMinimDP[((1 << m_n) - 1)][i] + c);
}
return (res != INF) ? res : -1;
}
};
int main() {
// Input rapid
ios_base::sync_with_stdio(false);
cin.tie(nullptr);
// I/O
ifstream in("hamilton.in");
ofstream out("hamilton.out");
int n, m;
in >> n >> m;
Graf g(n, m);
g.citesteInputCicluHamiltonian(in);
in.close();
int costMinim = g.orientatCostMinimCicluHamiltonian();
if (costMinim < 0) out << "Nu exista solutie";
else out << costMinim;
out.close();
return 0;
}