How do I find all paths through a set of given nodes in a DAG?

Question

I have a list of items (blue nodes below) which are categorized by the users of my application. The categories themselves can be grouped and categorized themselves.

The resulting structure can be represented as a Directed Acyclic Graph (DAG) where the items are sinks at the bottom of the graph's topology and the top categories are sources. Note that while some of the categories might be well defined, a lot is going to be user defined and might be very messy.

Example:

_{(source: theuprightape.net)}

On that structure, I want to perform the following operations:

• find all items (sinks) below a particular node (all items in Europe)
• find all paths (if any) that pass through all of a set of n nodes (all items sent via SMTP from example.com)
• find all nodes that lie below all of a set of nodes (intersection: goyish brown foods)

The first seems quite straightforward: start at the node, follow all possible paths to the bottom and collect the items there. However, is there a faster approach? Remembering the nodes I already passed through probably helps avoiding unnecessary repetition, but are there more optimizations?

How do I go about the second one? It seems that the first step would be to determine the height of each node in the set, as to determine at which one(s) to start and then find all paths below that which include the rest of the set. But is this the best (or even a good) approach?

The graph traversal algorithms listed at Wikipedia all seem to be concerned with either finding a particular node or the shortest or otherwise most effective route between two nodes. I think both is not what I want, or did I just fail to see how this applies to my problem? Where else should I read?

Answer 1

It seems to me that its essentially the same operation for all 3 questions. You're always asking "Find all X below node(s) Y, where X is of type Z". All you need is a generic mechanism for 'locate all nodes below node', (solves Q3) and then you can filter the results for 'nodetype=sink' (solves Q1). For Q2, you have the starting-point (your node set) and your ending point (any sink below the starting point) so your solution set is all paths from starting node specified to the sink. So I would suggest that what you basically have a is a tree, and basic tree-traversal algorithms would be the way to go.

Answer 2

Despite the fact that your graph is acyclic, the operations you cite remind me of similar aspects of control flow graph analysis. There is a rich set of algorithms based on dominance that may be applicable. For example, your third operation reminds me od computing dominance frontiers; I believe that algorithm would work directly if you temporarily introduce "entry" and "exit" nodes. The entry node connects the "given set of nodes" and the exit nodes connects the sinks.

Also see Robert Tarjan's basic algorithms.