We saw how to find the shortest path in a graph with positive edges using the Dijkstra’s algorithm. We also know how to find the shortest paths from a given source node to all other nodes even when there are negative edges using the Bellman-Ford algorithm. Now we’ll see that there’s a faster algorithm running in linear time that can find the shortest paths from a given source node to all other reachable vertices in a directed acyclic graph, also known as a DAG.

Because the DAG is acyclic we don’t have to worry about negative cycles. As we already know it’s pointless to speak about shortest path in the presence of negative cycles because we can “loop” over these cycles and practically our path will become shorter and shorter.

Thus we have two problems to overcome with Dijkstra and the Bellman-Ford algorithms. First of all we needed only positive weights and on the second place we didn’t want cycles. Well, we can handle both cases in this algorithm.

Overview

The first thing we know about DAGs is that they can easily be topologically sorted. Topological sort can be used in many practical cases, but perhaps the mostly used one is when trying to schedule dependent tasks.

After a topological sort we end with a list of vertices of the DAG and we’re sure that if there’s an edge (u, v), u will precede v in the topologically sorted list.

This information is precious and the only thing we need to do is to pass through this sorted list and to calculate distances for a shortest paths just like the algorithm of Dijkstra.

OK, so let’s summarize this algorithm:
– First we must topologically sort the DAG;
– As a second step we set the distance to the source to 0 and infinity to all other vertices;
– Then for each vertex from the list we pass through all its neighbors and we check for shortest path;

It’s pretty much like the Dijkstra’s algorithm with the main difference that we used a priority queue then, while this time we use the list from the topological sort.

Code

This time the code is actually a pseudocode. Altough all the examples so far was in PHP, perhaps pseudocode is easier to understand and doesn’t bind you in a specific language implementation. Also if you don’t feel comforatable with the given programming language it can be more difficult for you to understand the code than by reading pseudocode.

1. Topologically sort G into L;
2. Set the distance to the source to 0;
3. Set the distances to all other vertices to infinity;
4. For each vertex u in L
5.    - Walk through all neighbors v of u;
6.    - If dist(v) > dist(u) + w(u, v) 
7.       - Set dist(v) <- dist(u) + w(u, v);

Application

It’s clear why and where we must use this algorithm. The only problem is that we must be sure that the graph doesn’t have cycles. However if we’re aware of how the graph is created we may have some additional information if there are cycles or not – then this linear time algorithm can be very applicable.

Let’s assume we have a list of tasks to accomplish. Some of the tasks depend on others, so we must be very careful with the order of their execution. If the relationship between these tasks were simple enough we could represent them as a linked list, which would be great, and we would know the exact order of their execution. The problem is that sometimes the relations between the different tasks are more complex and some tasks depend on two or more other tasks, which in their turn depend on one or more tasks, etc.

Thus we can’t model this problem using linked lists or trees. The only rational solution is to model the problem using a graph. What kind of graph do we need? Well, we definitely need a directed graph, to desribe the relations, and this graph shouldn’t have cycles. So we need the so called directed acyclic graph (DAG).

Why we don’t what a cycle in the graph? The answer of this question is simple and obvious. In case of cyclic graph, we wouldn’t be able to determine the priority of task execution, thus we won’t be able to sort the tasks properly.

Now the solution we want is to sort the vertices of the graph in some order so for each edge (u, v) u will precede v. Then we’ll have a linear order of all tasks and by starting their execution we’ll know that everything will be OK.

This kind of sort is also known as “topological” sort (or topsort) and it is one of the very basic graph algorithms.

Overview

OK, so we have an acyclic directed graph, how do we proceed to get a linked list with all the vertices sorted? Since it’s an acyclic graph we know that there is at least one vertex without predecessor. Thus at first place, we can put all the vertices without predecessors into our linked list.

This approach answers the question – is there a possibility to have more than one valid topological sort of a graph? Indeed, the only thing we’d like to do is to put all the vertices in the correct order, but since there might be vertices with no predecessors any combination of them will be a valid topological sort for a graph.

As we can see from the picture above even for vertices with predecessors our topological sort can vary. Thus [9, 6, 2, 7, 4, 1] is a valid topological sorted graph, but [6, 9, 2, 7, 4, 1] is also a valid topological sort out of the same graph!

Now we can generalize the algorithm in some basic steps.

1. Make an empty list L and an empty list S;
2. Put all the vertices with no predecessors in L;
3. While L has items in it;
3.1. Pop an item from L – n, and push it to S;
3.2. For each vertex m adjacent to n;
3.2.1. Remove (n, m);
3.2.2. If m has no predecessors – push it to L;

Code

Here’s the very basic PHP implementation. As you can see the short implementation shows us how easy this algorithm is. However its importance to computer science and programming is enormous.

class G
{
    protected $_g = array(
        array(0, 1, 1, 0, 0, 0, 0),
        array(0, 0, 0, 1, 0, 0, 0),
        array(0, 0, 0, 0, 1, 0, 0),
        array(0, 0, 0, 0, 1, 0, 0),
        array(0, 0, 0, 0, 0, 0, 1),
        array(0, 0, 0, 0, 0, 0, 1),
        array(0, 0, 0, 0, 0, 0, 0),
    );
    protected $_list = array();
    protected $_ts   = array();
    protected $_len  = null;
    
    public function __construct()
    {
        $this->_len = count($this->_g);
        
        // finds the vertices with no predecessors
        $sum = 0;
        for ($i = 0; $i < $this->_len; $i++) {
            for ($j = 0; $j < $this->_len; $j++) {
                $sum += $this->_g[$j][$i];
            }
            
            if (!$sum) {
                // append to list
                array_push($this->_list, $i);
            }
            $sum = 0;
        }
    }
    
    public function topologicalSort() 
    {
        while ($this->_list) {
            $t = array_shift($this->_list);
            array_push($this->_ts, $t);
            
            foreach ($this->_g[$t] as $key => $vertex) {
                if ($vertex == 1) {
                    $this->_g[$t][$key] = 0;
                    
                    $sum = 0;
                    for ($i = 0; $i < $this->_len; $i++) {
                        $sum += $this->_g[$i][$key];
                    }
                    
                    if (!$sum) {
                        array_push($this->_list, $key);
                    }
                }
                $sum = 0;
            }
        }
        
        print_r($this->_ts);
    }
}

$g = new G();
/*
Array
(
    [0] => 0
    [1] => 5
    [2] => 1
    [3] => 2
    [4] => 3
    [5] => 4
    [6] => 6
)*/
$g->topologicalSort();

Application

As I already mentioned above this algorithm is practically used to sort the execution of different tasks that depend on each other. However this isn’t its only use. Actually any kind of objects that depend on each other can be modeled with a graph. Indeed sometimes these graphs may be a trees, but most of the cases that isn’t true.