//a Priority queue in reverse order, e.g. the least cost is at the top
  priority_queue<Edge, vector<Edge>, std::greater_equal<Edge> > pq;

  //initialize visited - false, distances - infinite, previous vertices - -1
  for (int i = 0; i < G.vertices(); ++i) {
    m_visited[i] = false;
    m_dist[i] = -1;
    m_prev[i] = -1;
  }
  m_dist[m_start] = 0;

  pq.push(Edge(m_start,0)); //start with the 'start' vertex
  while (!pq.empty()) {
    //pop the min-cost edge from the PQ
    Edge e = pq.top();
    pq.pop();
    m_visited[e.end()] = true;
    if (e.end() == m_end) break; //reached the destination

    //iterate through its neighbors
    Edges neighbors = G.neighbors(e.end());
    for (Edges::const_iterator it = neighbors.begin(); it != neighbors.end(); ++it) {
      int neighbor = (*it).end();  //the current neighbor
      int cost = (*it).cost(); //cost of that neighbor
      int alt = m_dist[e.end()] + cost; //distance from the start

      //if the neighbor is not already visited, and the new distance is smaller than the already calculated
      //or it is not calculated yet
      //then set the new distance, and put this neighbor into the PQ
      if (!m_visited[neighbor] && (m_dist[neighbor] == -1 || alt < m_dist[neighbor])) {
        m_dist[neighbor] = alt;
        m_prev[neighbor] = e.end();
        pq.push(Edge(neighbor,alt));
      }
    }
  }