# Graph Reachability

In this example we cover:

• Implementing a recursive algorithm (graph reachability) via cyclic dataflow
• Operators to union data from multiple inputs (union), and send data to multiple outputs (tee)
• Indexing multi-output operators by appending a bracket expression
• An example of how a cyclic dataflow in one stratum executes to completion before starting the next stratum.

To expand from graph neighbors to graph reachability, we want to find vertices that are connected not just to origin, but also to vertices reachable transitively from origin. Said differently, a vertex is reachable from origin if it is one of two cases:

1. a neighbor of origin or
2. a neighbor of some other vertex that is itself reachable from origin.

It turns out this is a very small change to our Hydroflow program! Essentially we want to take all the reached vertices we found in our graph neighbors program, and treat them recursively just as we treated origin. To do this in a language like Hydroflow, we introduce a cycle in the flow: we take the join output and have it flow back into the join input. The modified intuitive graph looks like this:

Note that we added a Reached Vertices box to the diagram to union the two inbound edges corresponding to our two cases above. Similarly note that the join box V ⨝ E now has two outbound edges; the sketch omits the operator to copy ("tee") the output along two paths.

Now lets look at a modified version of our graph neighbor code that implements this full program, including the loop as well as the Hydroflow union and tee. Modify src/main.rs to look like this:

use hydroflow::hydroflow_syntax;pub fn main() {    // An edge in the input data = a pair of usize vertex IDs.    let (edges_send, edges_recv) = hydroflow::util::unbounded_channel::<(usize, usize)>();    let mut flow = hydroflow_syntax! {        // inputs: the origin vertex (vertex 0) and stream of input edges        origin = source_iter(vec!);        stream_of_edges = source_stream(edges_recv);        reached_vertices = union();        origin -> reached_vertices;        // the join        my_join_tee = join() -> flat_map(|(src, ((), dst))| [src, dst]) -> tee();        reached_vertices -> map(|v| (v, ())) -> my_join_tee;        stream_of_edges -> my_join_tee;        // the loop and the output        my_join_tee -> reached_vertices;        my_join_tee -> unique() -> for_each(|x| println!("Reached: {}", x));    };    println!(        "{}",        flow.meta_graph()            .expect("No graph found, maybe failed to parse.")            .to_mermaid()    );    edges_send.send((0, 1)).unwrap();    edges_send.send((2, 4)).unwrap();    edges_send.send((3, 4)).unwrap();    edges_send.send((1, 2)).unwrap();    edges_send.send((0, 3)).unwrap();    edges_send.send((0, 3)).unwrap();    flow.run_available();}

And now we get the full set of vertices reachable from 0:

cargo run<build output><graph output>Reached: 3Reached: 0Reached: 2Reached: 4Reached: 1

## Examining the Hydroflow Code​

Let's review the significant changes here. First, in setting up the inputs we have the addition of the reached_vertices variable, which uses the union() op to union the output of two operators into one. We route the origin vertex into it as one input right away:

    reached_vertices = union();    origin -> reached_vertices;

Note the square-bracket syntax for differentiating the multiple inputs to union() is the same as that of join() (except that union can have an unbounded number of inputs, whereas join() is defined to only have two.)

Now, join() is defined to only have one output. In our program, we want to copy the joined output output to two places: to the original for_each from above to print output, and also back to the union operator we called reached_vertices. We feed the join() output through a flat_map() as before, and then we feed the result into a tee() operator, which is the mirror image of union(): instead of merging many inputs to one output, it copies one input to many different outputs. Each input element is cloned, in Rust terms, and given to each of the outputs. The syntax for the outputs of tee() mirrors that of union: we append an output index in square brackets to the tee or variable. In this example we have my_join_tee -> and my_join_tee ->.

Finally, we process the output of the join as passed through the tee. One branch pushes reached vertices back up into the reached_vertices variable (which begins with a union), while the other prints out all the reached vertices as in the simple program.

        my_join_tee -> reached_vertices;        my_join_tee -> for_each(|x| println!("Reached: {}", x));

Note the syntax for differentiating the outputs of a tee() is symmetric to that of union(), showing up to the right of the variable rather than the left.

Below is the diagram rendered by mermaid showing the structure of the full flow:

This is similar to the flow for graph neighbors, but has a few more operators that make it look more complex. In particular, it includes the union and tee operators, and a cycle-forming back-edge that passes through an auto-generated handoff operator. This handoff is not a stratum boundary (after all, it connects stratum 0 to itself!) It simply enforces the rule that a push producer and a pull consumer must be separated by a handoff.

Meanwhile, note that there is once again a stratum boundary between the stratum 0 with its recursive loop, and stratum 1 that computes unique, with the blocking input. This means that Hydroflow will first run the loop of stratum 0 repeatedly until all the transitive reached vertices are found, before moving on to compute the unique reached vertices.