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A simple bfs example ... I do not understand

I am trying to understand how BFS works with a queue in order to figure out the shortest path. Let's say I have a grid:

1--2--3
|  |  |
4--5--6
|  |  |
7--8--9
|
0

The starting spot is "9" and the target is "0".

So ... I push the start ...

push 9 {9}
pop 9 {}
push 6 {6}
push 8 {6,8}
pop 6 {8}
push 3 {8,3}
push 5 {8,3,5}
pop 8 {3,5}
push 7 {3,5,7}
pop 3 {5,7}
push 2 {5,7,2}
pop 5 {7,2}
push 4 {7,2,4}
pop 7 {2,5}
found 0

How can I get the shortest path out of this mess? I do not understand how this gives me the shortest path. Am I thinking about it wrong?

Thank!

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2 answers

To find the shortest path, each node must also “remember” how you reached it during your BFS [this vertex led to its discovery].

In cpp for your example you can use map<int,int>for it.
A simple example:

map[9] = -1; //indicationg source
map[6] = 9;
map[8] = 9;
map[3] = 6;
map[7] = 8 ;
...
map[0] = 7;

, 0 [ -1].

+5

, node, . node ( ), , , node. ++, , ++. - :

#include <iostream>
#include <queue>
#include <stdexcept>
#include <vector>

struct graph {

  graph(size_t nodes)
    : m_adjacency_list(nodes) {
  }

  size_t number_of_nodes() const {
    return m_adjacency_list.size();
  }

  std::vector<size_t> const& neighbours_of(size_t node) const {
    return m_adjacency_list.at(node);
  }

  void add_edge(size_t from, size_t to) {
    std::vector<size_t>& al = m_adjacency_list.at(from);
    if (to >= m_adjacency_list.size())
      throw std::runtime_error("Tried to add edge to non-existant node");
    for (size_t i = 0; i < al.size(); ++i) if (al[i] == to) return;
    al.push_back(to);
  }

private:

  std::vector<std::vector<size_t>> m_adjacency_list;
};


int main() {

  // generate your grid
  graph g(10);
  g.add_edge(1, 2);
  g.add_edge(1, 4);
  g.add_edge(2, 1);
  g.add_edge(2, 3);
  g.add_edge(2, 5);
  g.add_edge(3, 2);
  g.add_edge(3, 6);
  g.add_edge(4, 1);
  g.add_edge(4, 5);
  g.add_edge(4, 7);
  g.add_edge(5, 2);
  g.add_edge(5, 4);
  g.add_edge(5, 6);
  g.add_edge(5, 8);
  g.add_edge(6, 3);
  g.add_edge(6, 5);
  g.add_edge(6, 9);
  g.add_edge(7, 4);
  g.add_edge(7, 8);
  g.add_edge(7, 0);
  g.add_edge(8, 5);
  g.add_edge(8, 7);
  g.add_edge(8, 9);
  g.add_edge(9, 6);
  g.add_edge(9, 8);
  g.add_edge(0, 7);

  // do the bfs
  std::vector<size_t> reached_by(g.number_of_nodes(), g.number_of_nodes());
  std::queue<size_t> q;
  size_t start = 9;
  size_t target = 0;
  reached_by[start] = start;
  q.push(start);
  while (!q.empty()) {
    size_t node = q.front();
    q.pop();
    for (size_t i = 0; i < g.neighbours_of(node).size(); ++i) {
      size_t candidate = g.neighbours_of(node)[i];
      if (reached_by[candidate] == g.number_of_nodes()) {
        reached_by[candidate] = node;
        if (candidate == target) break;
        q.push(candidate);
      }
    }
  }

  if (reached_by[target] == g.number_of_nodes())
    std::cout<<"No path to "<<target<<" found!"<<std::endl;
  else {
    std::cout<<"Path to "<<target<<": ";
    for (size_t node = target; node != start; node = reached_by[node])
      std::cout<<node<<" <- ";
    std::cout<<start<<std::endl;
  }

}

, _by, , node node. , .

Path to 0: 0 <- 7 <- 8 <- 9
+2

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