proconlib

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:warning: tests/prim/unconnected.test.cpp

Depends on

Code

// competitive-verifier: PROBLEM https://atcoder.jp/contests/abc270/tasks/abc270_f
// competitive-verifier: IGNORE

#include <iostream>
#include <vector>
#include <limits>
#include "tools/prim.hpp"
#include "tools/popcount.hpp"
#include "tools/chmin.hpp"

using ll = long long;

int main() {
  std::cin.tie(nullptr);
  std::ios_base::sync_with_stdio(false);

  ll N, M;
  std::cin >> N >> M;
  std::vector<ll> X(N), Y(N);
  for (auto& X_i : X) std::cin >> X_i;
  for (auto& Y_i : Y) std::cin >> Y_i;
  std::vector<ll> A(M), B(M), Z(M);
  for (ll i = 0; i < M; ++i) {
    std::cin >> A[i] >> B[i] >> Z[i];
    --A[i], --B[i];
  }

  ll answer = std::numeric_limits<ll>::max();
  for (ll state = 0; state < 4; ++state) {
    tools::prim<ll> graph(N + tools::popcount(state));
    if (state & 1) {
      for (ll i = 0; i < N; ++i) {
        graph.add_edge(i, N, X[i]);
      }
    }
    if (state & 2) {
      for (ll i = 0; i < N; ++i) {
        graph.add_edge(i, N + (state & 1), Y[i]);
      }
    }
    for (ll i = 0; i < M; ++i) {
      graph.add_edge(A[i], B[i], Z[i]);
    }
    if (const auto groups = graph.query().first; groups.size() == 1) {
      tools::chmin(answer, groups[0].first);
    }
  }

  std::cout << answer << '\n';
  return 0;
}

#line 1 "tests/prim/unconnected.test.cpp"
// competitive-verifier: PROBLEM https://atcoder.jp/contests/abc270/tasks/abc270_f
// competitive-verifier: IGNORE

#include <iostream>
#include <vector>
#include <limits>
#line 1 "tools/prim.hpp"



#include <algorithm>
#include <cassert>
#include <cstddef>
#line 8 "tools/prim.hpp"
#include <queue>
#include <tuple>
#include <utility>
#line 1 "tools/greater_by.hpp"



namespace tools {

  template <class F>
  class greater_by {
  private:
    F selector;

  public:
    greater_by(const F& selector) : selector(selector) {
    }

    template <class T>
    bool operator()(const T& x, const T& y) const {
      return selector(x) > selector(y);
    }
  };
}


#line 13 "tools/prim.hpp"

namespace tools {

  template <typename T>
  class prim {
  public:
    struct edge {
      ::std::size_t id;
      ::std::size_t from;
      ::std::size_t to;
      T cost;
    };

  private:
    ::std::vector<edge> m_edges;
    ::std::vector<::std::vector<::std::size_t>> m_graph;

  public:
    prim() = default;
    prim(const ::tools::prim<T>&) = default;
    prim(::tools::prim<T>&&) = default;
    ~prim() = default;
    ::tools::prim<T>& operator=(const ::tools::prim<T>&) = default;
    ::tools::prim<T>& operator=(::tools::prim<T>&&) = default;

    explicit prim(const ::std::size_t n) : m_graph(n) {
    }

    ::std::size_t size() const {
      return this->m_graph.size();
    }

    ::std::size_t add_edge(::std::size_t u, ::std::size_t v, const T w) {
      assert(u < this->size());
      assert(v < this->size());
      ::std::tie(u, v) = ::std::minmax({u, v});
      this->m_edges.push_back(edge({this->m_edges.size(), u, v, w}));
      this->m_graph[u].push_back(this->m_edges.size() - 1);
      this->m_graph[v].push_back(this->m_edges.size() - 1);
      return this->m_edges.size() - 1;
    }

    const edge& get_edge(const ::std::size_t k) const {
      assert(k < this->m_edges.size());
      return this->m_edges[k];
    }

    const ::std::vector<edge>& edges() const {
      return this->m_edges;
    }

    ::std::pair<::std::vector<::std::pair<T, ::std::vector<::std::size_t>>>, ::std::vector<::std::size_t>> query() const {
      ::std::pair<::std::vector<::std::pair<T, ::std::vector<::std::size_t>>>, ::std::vector<::std::size_t>> res;
      auto& [groups, belongs_to] = res;
      belongs_to.resize(this->size());
      ::std::fill(belongs_to.begin(), belongs_to.end(), ::std::numeric_limits<::std::size_t>::max());

      for (::std::size_t root = 0; root < this->size(); ++root) {
        if (belongs_to[root] < ::std::numeric_limits<::std::size_t>::max()) continue;

        const auto greater_by_cost = ::tools::greater_by([&](const auto& pair) { return this->m_edges[pair.first].cost; });
        ::std::priority_queue<::std::pair<::std::size_t, ::std::size_t>, ::std::vector<::std::pair<::std::size_t, ::std::size_t>>, decltype(greater_by_cost)> pq(greater_by_cost);
        groups.emplace_back(0, ::std::vector<::std::size_t>());
        auto& [total_cost, edge_ids] = groups.back();

        belongs_to[root] = groups.size() - 1;
        for (const auto e : this->m_graph[root]) {
          const auto next = this->m_edges[e].from ^ this->m_edges[e].to ^ root;
          if (belongs_to[next] < ::std::numeric_limits<::std::size_t>::max()) continue;

          pq.emplace(e, next);
        }

        while (!pq.empty()) {
          const auto [from_edge_id, here] = pq.top();
          pq.pop();

          if (belongs_to[here] < ::std::numeric_limits<::std::size_t>::max()) continue;

          belongs_to[here] = belongs_to[root];
          total_cost += this->m_edges[from_edge_id].cost;
          edge_ids.push_back(from_edge_id);
          for (const auto e : this->m_graph[here]) {
            const auto next = this->m_edges[e].from ^ this->m_edges[e].to ^ here;
            if (belongs_to[next] < ::std::numeric_limits<::std::size_t>::max()) continue;

            pq.emplace(e, next);
          }
        }
      }

      return res;
    }
  };
}


#line 1 "tools/popcount.hpp"



#include <bit>
#line 6 "tools/popcount.hpp"
#include <type_traits>
#line 1 "tools/is_integral.hpp"



#line 5 "tools/is_integral.hpp"

namespace tools {
  template <typename T>
  struct is_integral : ::std::is_integral<T> {};

  template <typename T>
  inline constexpr bool is_integral_v = ::tools::is_integral<T>::value;
}


#line 1 "tools/is_signed.hpp"



#line 5 "tools/is_signed.hpp"

namespace tools {
  template <typename T>
  struct is_signed : ::std::is_signed<T> {};

  template <typename T>
  inline constexpr bool is_signed_v = ::tools::is_signed<T>::value;
}


#line 1 "tools/make_unsigned.hpp"



#line 5 "tools/make_unsigned.hpp"

namespace tools {
  template <typename T>
  struct make_unsigned : ::std::make_unsigned<T> {};

  template <typename T>
  using make_unsigned_t = typename ::tools::make_unsigned<T>::type;
}


#line 1 "tools/uint128_t.hpp"



#line 1 "tools/detail/int128_t.hpp"



#line 7 "tools/detail/int128_t.hpp"
#include <cstdint>
#include <functional>
#line 11 "tools/detail/int128_t.hpp"
#include <string>
#include <string_view>
#line 1 "tools/abs.hpp"



namespace tools {
  constexpr float abs(const float x) {
    return x < 0 ? -x : x;
  }
  constexpr double abs(const double x) {
    return x < 0 ? -x : x;
  }
  constexpr long double abs(const long double x) {
    return x < 0 ? -x : x;
  }
  constexpr int abs(const int x) {
    return x < 0 ? -x : x;
  }
  constexpr long abs(const long x) {
    return x < 0 ? -x : x;
  }
  constexpr long long abs(const long long x) {
    return x < 0 ? -x : x;
  }
  constexpr unsigned int abs(const unsigned int x) {
    return x;
  }
  constexpr unsigned long abs(const unsigned long x) {
    return x;
  }
  constexpr unsigned long long abs(const unsigned long long x) {
    return x;
  }
}


#line 1 "tools/bit_ceil.hpp"



#line 10 "tools/bit_ceil.hpp"

namespace tools {
  template <typename T>
  constexpr T bit_ceil(T) noexcept;

  template <typename T>
  constexpr T bit_ceil(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_ceil<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_ceil(x);
    }
  }
}


#line 1 "tools/bit_floor.hpp"



#line 10 "tools/bit_floor.hpp"

namespace tools {
  template <typename T>
  constexpr T bit_floor(T) noexcept;

  template <typename T>
  constexpr T bit_floor(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_floor<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_floor(x);
    }
  }
}


#line 1 "tools/bit_width.hpp"



#line 10 "tools/bit_width.hpp"

namespace tools {
  template <typename T>
  constexpr int bit_width(T) noexcept;

  template <typename T>
  constexpr int bit_width(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_width<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_width(x);
    }
  }
}


#line 1 "tools/countr_zero.hpp"



#line 12 "tools/countr_zero.hpp"

namespace tools {
  template <typename T>
  constexpr int countr_zero(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::std::min(::tools::countr_zero<::tools::make_unsigned_t<T>>(x), ::std::numeric_limits<T>::digits);
    } else {
      return ::std::countr_zero(x);
    }
  }
}


#line 1 "tools/is_unsigned.hpp"



#line 5 "tools/is_unsigned.hpp"

namespace tools {
  template <typename T>
  struct is_unsigned : ::std::is_unsigned<T> {};

  template <typename T>
  inline constexpr bool is_unsigned_v = ::tools::is_unsigned<T>::value;
}


#line 1 "tools/hash_combine.hpp"



#line 6 "tools/hash_combine.hpp"

// Source: https://github.com/google/cityhash/blob/f5dc54147fcce12cefd16548c8e760d68ac04226/src/city.h
// License: MIT
// Author: Google Inc.

// Copyright (c) 2011 Google, Inc.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

namespace tools {
  template <typename T>
  void hash_combine(::std::size_t& seed, const T& v) {
    static const ::std::hash<T> hasher;
    static constexpr ::std::size_t k_mul = 0x9ddfea08eb382d69ULL;
    ::std::size_t a = (hasher(v) ^ seed) * k_mul;
    a ^= (a >> 47);
    ::std::size_t b = (seed ^ a) * k_mul;
    b ^= (b >> 47);
    seed = b * k_mul;
  }
}


#line 1 "tools/make_signed.hpp"



#line 5 "tools/make_signed.hpp"

namespace tools {
  template <typename T>
  struct make_signed : ::std::make_signed<T> {};

  template <typename T>
  using make_signed_t = typename ::tools::make_signed<T>::type;
}


#line 1 "tools/now.hpp"



#include <chrono>

namespace tools {
  inline long long now() {
    return ::std::chrono::duration_cast<::std::chrono::nanoseconds>(::std::chrono::high_resolution_clock::now().time_since_epoch()).count();
  }
}


#line 25 "tools/detail/int128_t.hpp"

namespace tools {
  using uint128_t = unsigned __int128;
  using int128_t = __int128;

  namespace detail {
    namespace int128_t {
      constexpr ::tools::uint128_t parse_unsigned(const ::std::string_view s) noexcept {
        assert(!s.empty());
        ::tools::uint128_t x = 0;
        ::std::size_t i = s[0] == '+';
        if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
          for (i += 2; i < s.size(); ++i) {
            assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
            x <<= 4;
            if ('0' <= s[i] && s[i] <= '9') {
              x |= s[i] - '0';
            } else if ('a' <= s[i] && s[i] <= 'f') {
              x |= s[i] - 'a' + 10;
            } else {
              x |= s[i] - 'A' + 10;
            }
          }
        } else {
          for (; i < s.size(); ++i) {
            assert('0' <= s[i] && s[i] <= '9');
            x *= 10;
            x += s[i] - '0';
          }
        }
        return x;
      }

      constexpr ::tools::int128_t parse_signed(const ::std::string_view s) noexcept {
        assert(!s.empty());
        ::tools::int128_t x = 0;
        if (s[0] == '-') {
          ::std::size_t i = 1;
          if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
            for (i += 2; i < s.size(); ++i) {
              assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
              x *= 16;
              if ('0' <= s[i] && s[i] <= '9') {
                x -= s[i] - '0';
              } else if ('a' <= s[i] && s[i] <= 'f') {
                x -= s[i] - 'a' + 10;
              } else {
                x -= s[i] - 'A' + 10;
              }
            }
          } else {
            for (; i < s.size(); ++i) {
              assert('0' <= s[i] && s[i] <= '9');
              x *= 10;
              x -= s[i] - '0';
            }
          }
        } else {
          ::std::size_t i = s[0] == '+';
          if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
            for (i += 2; i < s.size(); ++i) {
              assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
              x <<= 4;
              if ('0' <= s[i] && s[i] <= '9') {
                x |= s[i] - '0';
              } else if ('a' <= s[i] && s[i] <= 'f') {
                x |= s[i] - 'a' + 10;
              } else {
                x |= s[i] - 'A' + 10;
              }
            }
          } else {
            for (; i < s.size(); ++i) {
              assert('0' <= s[i] && s[i] <= '9');
              x *= 10;
              x += s[i] - '0';
            }
          }
        }
        return x;
      }
    }
  }

  constexpr ::tools::uint128_t abs(const ::tools::uint128_t& x) noexcept {
    return x;
  }
  constexpr ::tools::int128_t abs(const ::tools::int128_t& x) {
    return x >= 0 ? x : -x;
  }
}

#define UINT128_C(c) ::tools::detail::int128_t::parse_unsigned(#c)
#define INT128_C(c) ::tools::detail::int128_t::parse_signed(#c)

inline ::std::istream& operator>>(::std::istream& is, ::tools::uint128_t& x) {
  ::std::string s;
  is >> s;
  x = ::tools::detail::int128_t::parse_unsigned(s);
  return is;
}
inline ::std::istream& operator>>(::std::istream& is, ::tools::int128_t& x) {
  ::std::string s;
  is >> s;
  x = ::tools::detail::int128_t::parse_signed(s);
  return is;
}

inline ::std::ostream& operator<<(::std::ostream& os, ::tools::uint128_t x) {
  ::std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else {
    s.push_back('0');
  }

  ::std::ranges::reverse(s);
  return os << s;
}
inline ::std::ostream& operator<<(::std::ostream& os, ::tools::int128_t x) {
  ::std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else if (x < 0) {
    while (x < 0) {
      s.push_back('0' + (-(x % 10)));
      x /= 10;
    }
    s.push_back('-');
  } else {
    s.push_back('0');
  }

  ::std::ranges::reverse(s);
  return os << s;
}

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
namespace std {
  template <>
  struct hash<::tools::uint128_t> {
    ::std::size_t operator()(const ::tools::uint128_t& x) const {
      static const ::std::size_t seed = ::tools::now();

      ::std::size_t hash = seed;
      ::tools::hash_combine(hash, static_cast<::std::uint64_t>(x >> 64));
      ::tools::hash_combine(hash, static_cast<::std::uint64_t>(x & ((UINT128_C(1) << 64) - 1)));
      return hash;
    }
  };
  template <>
  struct hash<::tools::int128_t> {
    ::std::size_t operator()(const ::tools::int128_t& x) const {
      static ::std::hash<::tools::uint128_t> hasher;
      return hasher(static_cast<::tools::uint128_t>(x));
    }
  };
}
#endif

namespace tools {
  template <>
  struct is_integral<::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_integral<const ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<const ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_integral<volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<volatile ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_integral<const volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_integral<const volatile ::tools::uint128_t> : ::std::true_type {};

  template <>
  struct is_signed<::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<::tools::uint128_t> : ::std::false_type {};
  template <>
  struct is_signed<const ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<const ::tools::uint128_t> : ::std::false_type {};
  template <>
  struct is_signed<volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<volatile ::tools::uint128_t> : ::std::false_type {};
  template <>
  struct is_signed<const volatile ::tools::int128_t> : ::std::true_type {};
  template <>
  struct is_signed<const volatile ::tools::uint128_t> : ::std::false_type {};

  template <>
  struct is_unsigned<::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_unsigned<const ::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<const ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_unsigned<volatile ::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<volatile ::tools::uint128_t> : ::std::true_type {};
  template <>
  struct is_unsigned<const volatile ::tools::int128_t> : ::std::false_type {};
  template <>
  struct is_unsigned<const volatile ::tools::uint128_t> : ::std::true_type {};

  template <>
  struct make_signed<::tools::int128_t> {
    using type = ::tools::int128_t;
  };
  template <>
  struct make_signed<::tools::uint128_t> {
    using type = ::tools::int128_t;
  };
  template <>
  struct make_signed<const ::tools::int128_t> {
    using type = const ::tools::int128_t;
  };
  template <>
  struct make_signed<const ::tools::uint128_t> {
    using type = const ::tools::int128_t;
  };
  template <>
  struct make_signed<volatile ::tools::int128_t> {
    using type = volatile ::tools::int128_t;
  };
  template <>
  struct make_signed<volatile ::tools::uint128_t> {
    using type = volatile ::tools::int128_t;
  };
  template <>
  struct make_signed<const volatile ::tools::int128_t> {
    using type = const volatile ::tools::int128_t;
  };
  template <>
  struct make_signed<const volatile ::tools::uint128_t> {
    using type = const volatile ::tools::int128_t;
  };

  template <>
  struct make_unsigned<::tools::int128_t> {
    using type = ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<::tools::uint128_t> {
    using type = ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const ::tools::int128_t> {
    using type = const ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const ::tools::uint128_t> {
    using type = const ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile ::tools::int128_t> {
    using type = volatile ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile ::tools::uint128_t> {
    using type = volatile ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile ::tools::int128_t> {
    using type = const volatile ::tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile ::tools::uint128_t> {
    using type = const volatile ::tools::uint128_t;
  };

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
  template <>
  constexpr ::tools::uint128_t bit_ceil<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    if (x <= 1) return 1;
    --x;
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
    x |= x >> 32;
    x |= x >> 64;
    return ++x;
  }

  template <>
  constexpr ::tools::uint128_t bit_floor<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    x |= x >> 1;
    x |= x >> 2;
    x |= x >> 4;
    x |= x >> 8;
    x |= x >> 16;
    x |= x >> 32;
    x |= x >> 64;
    return x & ~(x >> 1);
  }

  template <>
  constexpr int bit_width<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    int w = 0;
    if (x & UINT128_C(0xffffffffffffffff0000000000000000)) {
      x >>= 64;
      w += 64;
    }
    if (x & UINT128_C(0xffffffff00000000)) {
      x >>= 32;
      w += 32;
    }
    if (x & UINT128_C(0xffff0000)) {
      x >>= 16;
      w += 16;
    }
    if (x & UINT128_C(0xff00)) {
      x >>= 8;
      w += 8;
    }
    if (x & UINT128_C(0xf0)) {
      x >>= 4;
      w += 4;
    }
    if (x & UINT128_C(0xc)) {
      x >>= 2;
      w += 2;
    }
    if (x & UINT128_C(0x2)) {
      x >>= 1;
      w += 1;
    }
    w += x;
    return w;
  }

  namespace detail {
    namespace countr_zero {
      template <::std::size_t N>
      struct ntz_traits;

      template <>
      struct ntz_traits<128> {
        using type = ::tools::uint128_t;
        static constexpr int shift = 120;
        static constexpr type magic = UINT128_C(0x01061438916347932a5cd9d3ead7b77f);
        static constexpr int ntz_table[255] = {
          128,   0,   1,  -1,   2,  -1,   8,  -1,   3,  -1,  15,  -1,   9,  -1,  22,  -1,
            4,  -1,  29,  -1,  16,  -1,  36,  -1,  10,  -1,  43,  -1,  23,  -1,  50,  -1,
            5,  -1,  33,  -1,  30,  -1,  57,  -1,  17,  -1,  64,  -1,  37,  -1,  71,  -1,
           11,  -1,  60,  -1,  44,  -1,  78,  -1,  24,  -1,  85,  -1,  51,  -1,  92,  -1,
           -1,   6,  -1,  20,  -1,  34,  -1,  48,  31,  -1,  -1,  69,  58,  -1,  -1,  90,
           18,  -1,  67,  -1,  65,  -1,  99,  -1,  38,  -1, 101,  -1,  72,  -1, 106,  -1,
           -1,  12,  -1,  40,  -1,  61,  -1,  82,  45,  -1,  -1, 103,  79,  -1, 113,  -1,
           -1,  25,  -1,  74,  86,  -1,  -1, 116,  -1,  52,  -1, 108,  -1,  93,  -1, 120,
          127,  -1,  -1,   7,  -1,  14,  -1,  21,  -1,  28,  -1,  35,  -1,  42,  -1,  49,
           -1,  32,  -1,  56,  -1,  63,  -1,  70,  -1,  59,  -1,  77,  -1,  84,  -1,  91,
           -1,  19,  -1,  47,  -1,  68,  -1,  89,  -1,  66,  -1,  98,  -1, 100,  -1, 105,
           -1,  39,  -1,  81,  -1, 102,  -1, 112,  -1,  73,  -1, 115,  -1, 107,  -1, 119,
          126,  -1,  13,  -1,  27,  -1,  41,  -1,  -1,  55,  62,  -1,  -1,  76,  83,  -1,
           -1,  46,  -1,  88,  -1,  97,  -1, 104,  -1,  80,  -1, 111,  -1, 114,  -1, 118,
          125,  -1,  26,  -1,  54,  -1,  75,  -1,  -1,  87,  96,  -1,  -1, 110,  -1, 117,
          124,  -1,  53,  -1,  -1,  95, 109,  -1, 123,  -1,  94,  -1, 122,  -1, 121
        };
      };

      template <typename T>
      constexpr int impl(const T x) noexcept {
        using tr = ::tools::detail::countr_zero::ntz_traits<::std::numeric_limits<T>::digits>;
        using type = typename tr::type;
        return tr::ntz_table[static_cast<type>(tr::magic * static_cast<type>(x & -x)) >> tr::shift];
      }
    }
  }

  template <>
  constexpr int countr_zero<::tools::uint128_t>(const ::tools::uint128_t x) noexcept {
    return ::tools::detail::countr_zero::impl(x);
  }
#endif
}


#line 5 "tools/uint128_t.hpp"


#line 11 "tools/popcount.hpp"

namespace tools {
  template <typename T>
  constexpr int popcount(T) noexcept;

  template <typename T>
  constexpr int popcount(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::popcount<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::popcount(x);
    }
  }

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
  template <>
  constexpr int popcount<::tools::uint128_t>(::tools::uint128_t x) noexcept {
    x = (x & UINT128_C(0x55555555555555555555555555555555)) + (x >> 1 & UINT128_C(0x55555555555555555555555555555555));
    x = (x & UINT128_C(0x33333333333333333333333333333333)) + (x >> 2 & UINT128_C(0x33333333333333333333333333333333));
    x = (x & UINT128_C(0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f)) + (x >> 4 & UINT128_C(0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f));
    x = (x & UINT128_C(0x00ff00ff00ff00ff00ff00ff00ff00ff)) + (x >> 8 & UINT128_C(0x00ff00ff00ff00ff00ff00ff00ff00ff));
    x = (x & UINT128_C(0x0000ffff0000ffff0000ffff0000ffff)) + (x >> 16 & UINT128_C(0x0000ffff0000ffff0000ffff0000ffff));
    x = (x & UINT128_C(0x00000000ffffffff00000000ffffffff)) + (x >> 32 & UINT128_C(0x00000000ffffffff00000000ffffffff));
    x = (x & UINT128_C(0x0000000000000000ffffffffffffffff)) + (x >> 64 & UINT128_C(0x0000000000000000ffffffffffffffff));
    return x;
  }
#endif
}


#line 1 "tools/chmin.hpp"



#line 6 "tools/chmin.hpp"

namespace tools {

  template <typename M, typename N>
  bool chmin(M& lhs, const N& rhs) {
    bool updated;
    if constexpr (::std::is_integral_v<M> && ::std::is_integral_v<N>) {
      updated = ::std::cmp_less(rhs, lhs);
    } else {
      updated = rhs < lhs;
    }
    if (updated) lhs = rhs;
    return updated;
  }
}


#line 10 "tests/prim/unconnected.test.cpp"

using ll = long long;

int main() {
  std::cin.tie(nullptr);
  std::ios_base::sync_with_stdio(false);

  ll N, M;
  std::cin >> N >> M;
  std::vector<ll> X(N), Y(N);
  for (auto& X_i : X) std::cin >> X_i;
  for (auto& Y_i : Y) std::cin >> Y_i;
  std::vector<ll> A(M), B(M), Z(M);
  for (ll i = 0; i < M; ++i) {
    std::cin >> A[i] >> B[i] >> Z[i];
    --A[i], --B[i];
  }

  ll answer = std::numeric_limits<ll>::max();
  for (ll state = 0; state < 4; ++state) {
    tools::prim<ll> graph(N + tools::popcount(state));
    if (state & 1) {
      for (ll i = 0; i < N; ++i) {
        graph.add_edge(i, N, X[i]);
      }
    }
    if (state & 2) {
      for (ll i = 0; i < N; ++i) {
        graph.add_edge(i, N + (state & 1), Y[i]);
      }
    }
    for (ll i = 0; i < M; ++i) {
      graph.add_edge(A[i], B[i], Z[i]);
    }
    if (const auto groups = graph.query().first; groups.size() == 1) {
      tools::chmin(answer, groups[0].first);
    }
  }

  std::cout << answer << '\n';
  return 0;
}
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