17th Friday Fun Session – 12th May 2017
We use Bellman-Ford Algorithm to find the shortest path from a single source node/vertex (red color) to all destination nodes/vertices.
Let’s use our intuition
Given that distance from city-1 (city-1 is node 1 here) to city-2 is 5, and from city-2 to city-3 is 6, if I have travel from city-1 to city-2 and city-3 respectively what would be the cheapest way to do so?
We can start from city-1 and reach city-2 at cost 5. Now that we have arrived at city-2 at cost 5, we can add cost 6 to it and reach city-3 via city-2. Thus, the shortest path from city-1 to city-2 and city-3 are respectively 5 and 11.
Distance
Since city-1 is the source, we can add a self-loop on it with cost 0. That means, reaching city-1 from itself would cost 0. We also set that reaching city-2 and city-3 would cost infinity. We set so, because as of now we don’t know what would be the cost to reach there. So we put the maximum possible cost. Let’s call it distance. So we have distance [1] = 0, distance [2] = ∞ and distance [3] = ∞.
Predecessor
Let’s also maintain another array, called predecessor to indicate the last node from which we arrived here. We set predecessor [1] = 0, predecessor [2] = 0, predecessor [3] = 0. Since we have not arrived to city-2 and city-3 yet, we set the predecessor value for them to something invalid (0). For city-1, it is the source, can be set to 0 as well.
Relaxation
Now we take each path/edge. We have two edges here. First one is from city-1 to city-2 with cost 5 and the second edge is from city-2 to city-3 with cost 3. Now let’s do what is called relaxation on each of the edges.
We see that using first edge we can arrive at city-2 from city-1 at a cost of 5 (distance [1] + cost of first edge). Since 5 is less than the existing distance of city-2 that is ∞, we update distance [2] to 5. We also note that, we arrived here from city-1 and hence got this new distance, that means we also update predecessor [2] = 1.
Now let’s do relaxation on edge 2. We see that distance [3] that is ∞ as of now can be improved by using edge 2. We set distance [3] = distance [2] + cost of edge 2 = 5 + 6 = 11. Since we arrived here from city-2, let’s update predecessor [3] = 2.
Does order of edges for relaxation matter?
Now we are done with relaxation for all the edges once. First, we did the relaxation on first edge. Then we did the relaxation on the second edge. What would happen if the we changed the order? That means do the relaxation on the second edge first and then do it on the first edge.
Let’s do it. Start the relaxation afresh with new edge order. As of now we have predecessor [1] = 0, predecessor [2] = 0, predecessor [3] = 0. Also distance [1] = 0, distance [2] = ∞ and distance [3] = ∞.
We do relaxation on second edge (city-2 to city-3 at cost 6) first. We see that both distance [2] and distance [3] = ∞. Hence there is no chance to improve distance [3] since distance [2] + 6 = ∞ + 6 = ∞, that is no better than the existing distance [3] that also ∞. Hence relaxing second edge did not yield anything.
Let’s do relaxation on first edge. We know that would result in distance [2] = 5 and predecessor [2] = 1.
Well, at this point we see that we are done with relaxation on all the edges once and yet we have not found the shortest path to reach city-3.
So when shall we get the result?
Iteration
That brings us to the next concept called iteration. Relaxation on all the edges once is called an iteration. So how many iterations do we need to get the shortest path to all destination nodes? Let’s use our intuition on the example that we are working on. We got 3 nodes. So if we have to reach from one end (say node 1) to the other end (say node 3), the maximum edges we might have to travel is 2, that is, the number of nodes minus 1. If we do the relaxation on the edges in an order that would choose the furthest edge from source (or the closest to the destination, in this case, second edge that is going from city-2 to city-3 at cost 6), we see that at each iteration we would increase the path (source node to furthest node) by at least one edge. And hence at 2 iterations we will certainly reach all reachable (I am saying reachable because all destinations might not be reachable) destinations.
Let’s continue our workout from where we left. Let’s start iteration 2 for the cases when we did the relaxation on second edge first. At this 2nd iteration, we again start with second edge. This time we can update distance [3] = distance [2] + 6 = 5 + 6 = 11. Predecessor [3] = 2.
We are done with2 iterations and we have found the result to reach from city-1 to both city-2 and city-3.
Are all nodes reachable?
Let’s consider the below example, that is constructed by adding one more node 4.
We will see that distance [4] will remain ∞ and predecessor [4] will remain 0 (invalid) after |V| -1 = 4 – 1 = 3 iterations, where |V| is the number of nodes/vertices. This is because there is no incoming edge (path) to city-4. Hence city-4 is unreachable.
Did order of edges for relaxation really matter?
We have seen that intermediate results (distance and predecessor values) might vary based on the order of edges we chose for relaxation but final result after all the iterations will still be the same. Hence the order of edges on which we do relaxation does not really matter (as far as final result is concerned).
So after |V| – 1 iterations we have got the correct result?
Unfortunately not! Well, we did get the final result. But as of now we don’t know whether the result is valid or not. That sounds interesting. So, we have found the result and still we don’t know whether the result is correct/valid or not. So what is the issue? Well, let’s consider the below case.
I have added a third edge from city-3 to city-1. And the cost is -12. Negative cost? Why? We don’t answer the why question but let’s answer the what question. Cost -12 means reaching city-1 from city-4 would cost -1.
Let’s continue our workout where we have finished 2 iterations by considering the second edge first. Let’s assume we also considered the third edge and that third edge was considered at first for relaxation at each iteration. Distance [3] got a less than infinity value after 2nd iteration. Since we used 3rd edge at first, that means third edge was not used till 2nd iteration to update any distance for any node. That means the values (distance and predecessor) we got last time would be the same value even with the presence of third edge after 2 iterations.
Since we just added an extra edge (3rd edge) but no new nodes, the number of iterations we have to do still remains 2. That also means the result we found so far still valid in this case. But is the result (distance [2] = 5 and distance [3] = 11) correct with this new situation?
Negative cycle
Now that you arrived at city-3 at cost 11, you can go to city-1 at cost 11 + (-12) = -1 and then city-2 at -4 and so on. The more you travel, the less cost you would incur. And hence the shortest path found after 2 iterations are not valid.
So how do we find that the result is invalid? Well after we are done with |V|-1 iterations, we have to do one more iteration that is the |V|th one (a cycle involving |V| nodes can be found with |V| edges). If that changes distance value for any node that means there exists a negative cycle (a cycle whose edges (costs) sum to a negative value). When there is negative cycle present in a graph then the answer found is invalid.
Negative edge vs. negative cycle
Does negative edge means negative cycle? Does presence of negative edge mean no answer can be found?
Negative edge is fine with Bellman-Ford as in the above example. A correct solution can still be found. A correct solution cannot be found when there is a negative cycle. But Bellman-Ford can detect a negative cycle and in that case it can indicate that a correct solution is not found.
Are all iterations required?
Not really. When we did the relaxation on the first edge first, we already found the shortest paths to both city-2 and city-3. How do we know? Well, at iteration 2 we would have found that no distance got updated. If an iteration does not change any distance value then we can terminate the algorithm there and return valid result. Because in that case, subsequent iterations are not going to change anything. It also means there is no negative cycle.
The shortest path sequence
We can use the predecessor array recursively to get the shortest path sequence. For example, earlier after 2 iterations we got the following result.
distance [1] = 0, distance [2] = 5, distance [3] = 11
predecessor [1] = 0, predecessor [2] = 1, predecessor [3] = 2
If we want to find the shortest path sequence to city-3, we can find the predecessor [3] that is 2, recursively we can check the predecessor [2] that is 1 and that equals to source node, that is city-1. So we stop and the sequence is city-1 to city-2 to city-3.
Distance for a particular node can be updated more than once in an iteration
In this example above we have two nodes. We have to do one iteration. We have two edges: first with cost 5, second with cost 2. Source is node 1. If we relax the first edge then distance [2] will be 5. Subsequent relaxation on second edge would result the node 2 distance to be updated again with 2 (because, existing distance 5 > (0 + 2)). We see that node 2 distance got updated twice within the same iteration.
The algorithm
Now that we have done with the workout, let’s write down the algorithm.
Function BellmanFord()
{
input = G {V, E};
distance[] = ∞;
predecessor[] = -1;
distance[sourceNode] = 0;
for i = 1 to |V|-1
{
valueChanged = false;
for j = 1 to |E|
valueChanged = Relax (E[j]) || valueChanged;
if(!valueChanged)
return Result();
}
for j = 1 to |E|
if(Relax(E[j])
print ‘negative cycle detected, solution not possible’;
}
Function Relax (e)
{
if(distance[e.to]) > distance[e.from] + e.cost)
{
distance[e.to] = distance[e.from] + e.cost;
predecessor[e.to] = e.from;
return true;
}
return false;
}
Function Result()
{
print ‘success’;
print distance[];
print predecessor[];
}
The complexity
For each iteration (number of vertices – 1), we are iterating over all the edges. That means the complexity is O(|V|. |E|).
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