Koala and Notebook - VietMX's Blog

Koala Land consists of $m$ bidirectional roads connecting $n$ cities. The roads are numbered from $1$ to $m$ by order in input. It is guaranteed, that one can reach any city from every other city.

Koala starts traveling from city $1$. Whenever he travels on a road, he writes its number down in his notebook. He doesn’t put spaces between the numbers, so they all get concatenated into a single number.

Before embarking on his trip, Koala is curious about the resulting number for all possible destinations. For each possible destination, what is the smallest number he could have written for it?

Since these numbers may be quite large, print their remainders modulo $10^9+7$. Please note, that you need to compute the remainder of the minimum possible number, not the minimum possible remainder.

Input

The first line contains two integers $n$ and $m$ ($2 \le n \le 10^5, n – 1 \le m \le 10^5$), the number of cities and the number of roads, respectively.

The $i$-th of the following $m$ lines contains integers $x_i$ and $y_i$ ($1 \le x_i, y_i \le n$, $x_i \ne y_i$), representing a bidirectional road between cities $x_i$ and $y_i$.

It is guaranteed, that for any pair of cities there is at most one road connecting them, and that one can reach any city from every other city.

Output

Print $n – 1$ integers, the answer for every city except for the first city.

The $i$-th integer should be equal to the smallest number he could have written for destination $i+1$. Since this number may be large, output its remainder modulo $10^9+7$.

Examples

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output

input

output

input

output

Solution:

#include <bits/stdc++.h>

using namespace std;

template <typename A, typename B>
string to_string(pair<A, B> p);

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);

string to_string(const string& s) {
return '"' + s + '"';
}

string to_string(const char* s) {
return to_string((string) s);
}

string to_string(bool b) {
return (b ? "true" : "false");
}

string to_string(vector<bool> v) {
bool first = true;
string res = "{";
for (int i = 0; i < static_cast<int>(v.size()); i++) {
if (!first) {
res += ", ";
}
first = false;
res += to_string(v[i]);
}
res += "}";
return res;
}

template <size_t N>
string to_string(bitset<N> v) {
string res = "";
for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
}
return res;
}

template <typename A>
string to_string(A v) {
bool first = true;
string res = "{";
for (const auto &x : v) {
if (!first) {
res += ", ";
}
first = false;
res += to_string(x);
}
res += "}";
return res;
}

template <typename A, typename B>
string to_string(pair<A, B> p) {
return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";
}

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << to_string(H);
  debug_out(T...);
}

#ifdef LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif

template <typename T>
T inverse(T a, T m) {
T u = 0, v = 1;
while (a != 0) {
T t = m / a;
m -= t * a; swap(a, m);
u -= t * v; swap(u, v);
}
assert(m == 1);
return u;
}

template <typename T>
class Modular {
public:
using Type = typename decay<decltype(T::value)>::type;

constexpr Modular() : value() {}
template <typename U>
Modular(const U& x) {
value = normalize(x);
}

template <typename U>
static Type normalize(const U& x) {
Type v;
if (-mod() <= x && x < mod()) v = static_cast<Type>(x);
else v = static_cast<Type>(x % mod());
if (v < 0) v += mod();
    return v;
  }

  const Type& operator()() const { return value; }
  template <typename U>
explicit operator U() const { return static_cast<U>(value); }
constexpr static Type mod() { return T::value; }

Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; }
Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; }
  template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); }
template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); }
Modular& operator++() { return *this += 1; }
Modular& operator--() { return *this -= 1; }
Modular operator++(int) { Modular result(*this); *this += 1; return result; }
Modular operator--(int) { Modular result(*this); *this -= 1; return result; }
Modular operator-() const { return Modular(-value); }

template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) {
#ifdef _WIN32
uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;
asm(
"divl %4; \n\t"
: "=a" (d), "=d" (m)
: "d" (xh), "a" (xl), "r" (mod())
);
value = m;
#else
value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
#endif
return *this;
}
template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) {
int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod());
value = normalize(value * rhs.value - q * mod());
return *this;
}
template <typename U = T>
typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) {
value = normalize(value * rhs.value);
return *this;
}

Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); }

template <typename U>
friend const Modular<U>& abs(const Modular<U>& v) { return v; }

template <typename U>
friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs);

template <typename U>
friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs);

template <typename U>
friend std::istream& operator>>(std::istream& stream, Modular<U>& number);

private:
Type value;
};

template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; }
template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); }
template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; }

template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); }

template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; }

template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }

template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }

template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }

template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U>
Modular<T> power(const Modular<T>& a, const U& b) {
assert(b >= 0);
Modular<T> x = a, res = 1;
U p = b;
while (p > 0) {
if (p & 1) res *= x;
x *= x;
p >>= 1;
}
return res;
}

template <typename T>
bool IsZero(const Modular<T>& number) {
return number() == 0;
}

template <typename T>
string to_string(const Modular<T>& number) {
return to_string(number());
}

template <typename T>
std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) {
return stream << number();
}

template <typename T>
std::istream& operator>>(std::istream& stream, Modular<T>& number) {
typename common_type<typename Modular<T>::Type, int64_t>::type x;
stream >> x;
number.value = Modular<T>::normalize(x);
return stream;
}

/*
using ModType = int;

struct VarMod { static ModType value; };
ModType VarMod::value;
ModType& md = VarMod::value;
using Mint = Modular<VarMod>;
*/

constexpr int md = (int) 1e9 + 7;
using Mint = Modular<std::integral_constant<decay<decltype(md)>::type, md>>;

int main() {
ios::sync_with_stdio(false);
cin.tie(0);
int n, m;
cin >> n >> m;
vector<vector<pair<int, int>>> g(n);
for (int i = 0; i < m; i++) {
    int x, y;
    cin >> x >> y;
--x; --y;
g[x].emplace_back(y, i + 1);
g[y].emplace_back(x, i + 1);
}
const int ALPHA = 10;
const int LOG = 22;
vector<vector<int>> trie(1, vector<int>(ALPHA, -1));
vector<vector<int>> pr(1, vector<int>(LOG, -1));
vector<Mint> res(1, 0);
vector<int> depth(1, 0);
vector<int> pc(1, -1);
auto Go = [&](int v, int num) {
static vector<int> digits;
digits.clear();
while (num > 0) {
digits.push_back(num % 10);
num /= 10;
}
reverse(digits.begin(), digits.end());
for (int d : digits) {
if (trie[v][d] == -1) {
int t = (int) trie.size();
trie[v][d] = t;
trie.emplace_back(ALPHA, -1);
pr.emplace_back(LOG, -1);
pc.push_back(d);
depth.push_back(depth[v] + 1);
res.push_back(res[v] * 10 + d);
pr[t][0] = v;
for (int j = 1; j < LOG; j++) {
          pr[t][j] = pr[pr[t][j - 1]][j - 1];
          if (pr[t][j] == -1) {
            break;
          }
        }
      }       
      v = trie[v][d];
    }
    return v;
  };
  auto Compare = [&](int x, int y) {
    if (y == -1) {
      return true;
    }
    if (depth[x] != depth[y]) {
      return (depth[x] < depth[y]);
    }
    for (int j = LOG - 1; j >= 0; j--) {
if (pr[x][j] != pr[y][j]) {
x = pr[x][j];
y = pr[y][j];
}
}
return (pc[x] < pc[y]);
  };
  const int inf = (int) 1e9;
  vector<int> dist(n, inf);
vector<int> ver(n, -1);
set<pair<int, int>> s;
s.emplace(0, 0);
dist[0] = 0;
ver[0] = 0;
while (!s.empty()) {
int i = s.begin()->second;
s.erase(s.begin());
for (auto& p : g[i]) {
int to = p.first;
int num = p.second;
int new_ver = Go(ver[i], num);
if (Compare(new_ver, ver[to])) {
int new_dist = depth[new_ver];
if (new_dist < dist[to]) {
          if (dist[to] != inf) {
            s.erase(make_pair(dist[to], to));
          }
          dist[to] = new_dist;
          s.emplace(dist[to], to);
        }
        ver[to] = new_ver;
      }
    }
  }
  for (int i = 1; i < n; i++) {
    cout << res[ver[i]] << '\n';
  }
  return 0;
}

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