XQuery/Topological Sort

Motivation

You have a Directed Acyclic Graph (DAG) to track things such as a dependency graph. You want to sort in input DAG of nodes so that in the output reflects the dependency structure.

The Topological Sort of a Directed Acyclic Graph puts nodes in a sequence such that every node references only preceding nodes.

This ordering is needed for example in scheduling processes in a Pipeline.

For example, given a DAG defined as

```<node id="a">
<ref id="b"/>
<ref id="c"/>
</node>
<node id="b">
<ref  id="c"/>
</node>
<node id="c"/>
```

the topological order would be:

```<node id="c"/>
<node id="b">
<ref  id="c"/>
</node>
<node id="a">
<ref id="b"/>
<ref id="c"/>
</node>
```

The definition of topological order can be simply expressed in XQuery:

```declare function local:topological-sorted(\$nodes) as xs:boolean {
every \$n in \$nodes satisfies
every \$id in \$n/ref/@id
satisfies \$id = \$n/preceding::node/@id
};
```

A recursive algorithm is also straightforward:

```declare function local:topological-sort(\$unordered, \$ordered )   {
if (empty(\$unordered))
then \$ordered
else
let \$nodes := \$unordered [ every \$id in ref/@id satisfies \$id = \$ordered/@id]
return
if (\$nodes)
then local:topological-sort( \$unordered except \$nodes, (\$ordered, \$nodes ))
else ()  (: cycles so no order possible :)
};
```

which is invoked as

```let \$graph :=
<graph>
<node id="a">
<ref id="b"/>
<ref id="c"/>
</node>
<node id="b">
<ref  id="c"/>
</node>
<node id="c"/>
</graph>
let \$sortedNodes := <graph>{local:topological-sort(\$graph/node,())}</graph>
return local:topological-sorted(\$sortedNodes/node)
```

Explanation

\$ordered is initially the original sequence, \$ordered is empty. At each iteration, the set of nodes which are dependent only on the ordered nodes are calculated and these are removed from the unordered nodes and added to the ordered nodes.