Shortest Path in AQL

With the shortest path algorithm, you can find one shortest path between two vertices using AQL

General query idea

This type of query is supposed to find the shortest path between two given documents (startVertex and targetVertex) in your graph. For all vertices on this shortest path you will get a result in form of a set with two items:

  1. The vertex on this path.
  2. The edge pointing to it.

Example execution

Let’s take a look at a simple example to explain how it works. This is the graph that you are going to find a shortest path on:

traversal graph

You can use the following parameters for the query:

  1. You start at the vertex A.
  2. You finish with the vertex D.

So, obviously, you have the vertices A, B, C and D on the shortest path in exactly this order. Then, the shortest path statement returns the following pairs:

VertexEdge
Anull
BA → B
CB → C
DC → D

Note that the first edge is always null because there is no edge pointing to the startVertex.

Syntax

The next step is to see how you can write a shortest path query. You have two options here, you can either use a named graph or a set of edge collections (anonymous graph).

Working with named graphs 

FOR vertex[, edge]
  IN OUTBOUND|INBOUND|ANY SHORTEST_PATH
  startVertex TO targetVertex
  GRAPH graphName
  [OPTIONS options]
  • FOR: emits up to two variables:
    • vertex (object): the current vertex on the shortest path
    • edge (object, optional): the edge pointing to the vertex
  • IN OUTBOUND|INBOUND|ANY: defines in which direction edges are followed (outgoing, incoming, or both)
  • startVertex TO targetVertex (both string|object): the two vertices between which the shortest path will be computed. This can be specified in the form of an ID string or in the form of a document with the attribute _id. All other values will lead to a warning and an empty result. If one of the specified documents does not exist, the result is empty as well and there is no warning.
  • GRAPH graphName (string): the name identifying the named graph. Its vertex and edge collections will be looked up.
  • OPTIONS options (object, optional): used to modify the execution of the traversal. Only the following attributes have an effect, all others are ignored:
    • weightAttribute (string): a top-level edge attribute that should be used to read the edge weight. If the attribute is not existent or not numeric, the defaultWeight will be used instead. The attribute value must not be negative.
    • defaultWeight (number): this value will be used as fallback if there is no weightAttribute in the edge document, or if it is not a number. The value must not be negative. The default is 1.
Shortest Path traversals do not support negative weights. If a document attribute (as specified by weightAttribute) with a negative value is encountered during traversal, or if defaultWeight is set to a negative number, then the query is aborted with an error.

Working with collection sets 

FOR vertex[, edge]
  IN OUTBOUND|INBOUND|ANY SHORTEST_PATH
  startVertex TO targetVertex
  edgeCollection1, ..., edgeCollectionN
  [OPTIONS options]

Instead of GRAPH graphName you may specify a list of edge collections (anonymous graph). The involved vertex collections are determined by the edges of the given edge collections. The rest of the behavior is similar to the named version.

Traversing in mixed directions

For shortest path with a list of edge collections you can optionally specify the direction for some of the edge collections. Say for example you have three edge collections edges1, edges2 and edges3, where in edges2 the direction has no relevance, but in edges1 and edges3 the direction should be taken into account. In this case you can use OUTBOUND as general search direction and ANY specifically for edges2 as follows:

FOR vertex IN OUTBOUND SHORTEST_PATH
  startVertex TO targetVertex
  edges1, ANY edges2, edges3

All collections in the list that do not specify their own direction will use the direction defined after IN (here: OUTBOUND). This allows to use a different direction for each collection in your path search.

Conditional shortest path

The SHORTEST_PATH computation only finds an unconditioned shortest path. With this construct it is not possible to define a condition like: “Find the shortest path where all edges are of type X”. If you want to do this, use a normal Traversal instead with the option {order: "bfs"} in combination with LIMIT 1.

Please also consider using WITH to specify the collections you expect to be involved.

Examples

Creating a simple symmetric traversal demonstration graph:

traversal graph

var examples = require("@arangodb/graph-examples/example-graph");
var graph = examples.loadGraph("traversalGraph");
db.circles.toArray();
db.edges.toArray();
Show output

Start with the shortest path from A to D as above:

db._query(`
  FOR v, e IN OUTBOUND SHORTEST_PATH 'circles/A' TO 'circles/D' GRAPH 'traversalGraph'
    RETURN [v._key, e._key]
`);

db._query(`
  FOR v, e IN OUTBOUND SHORTEST_PATH 'circles/A' TO 'circles/D' edges
    RETURN [v._key, e._key]
`);
Show output

You can see that expectations are fulfilled. You find the vertices in the correct ordering and the first edge is null, because no edge is pointing to the start vertex on this path.

You can also compute shortest paths based on documents found in collections:

db._query(`
  FOR a IN circles
    FILTER a._key == 'A'
    FOR d IN circles
      FILTER d._key == 'D'
      FOR v, e IN OUTBOUND SHORTEST_PATH a TO d GRAPH 'traversalGraph'
        RETURN [v._key, e._key]
`);

db._query(`
  FOR a IN circles
    FILTER a._key == 'A'
    FOR d IN circles
      FILTER d._key == 'D'
      FOR v, e IN OUTBOUND SHORTEST_PATH a TO d edges
        RETURN [v._key, e._key]
`);
Show output

And finally clean it up again:

var examples = require("@arangodb/graph-examples/example-graph");
examples.dropGraph("traversalGraph");
Show output

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