# Is there a route between these two nodes in this directed graph?

Question 4.2 of *Cracking the Coding Interview*:

Given a directed graph, design an algorithm to find out whether there is a route between two nodes.

Given any node in the graph,
we can generate a full list of nodes reachable from that node,
by exploring the graph from that node outwards.
We can then determine whether a route exists from node `n1`

to node `n2`

by generating the full list of nodes reachable from `n1`

,
and asking whether `n2`

is in that list.

To generate the “reachable set” from a node,
we partition the graph into three sets:
`explored`

nodes, `boundary`

nodes, and the rest.
We repeatedly look for new nodes
by looking at outgoing edges from nodes in the `boundary`

set.
When we’ve looked at all the outgoing edges of a node,
we move that node to `explored`

so we don’t look at its edges again.
Eventually, our `boundary`

set becomes empty,
there are no new nodes to explore,
and the `explored`

set is all the reachable nodes.

Here’s an implementation in Haskell:

```
module DirectedGraphRoute where
import qualified Data.Set as Set
import Data.Set (Set)
type Node = Int
type Edge = (Node,Node)
type Graph = Set Edge
isRoute :: Graph -> Node -> Node -> Bool
isRoute g n1 n2 = Set.member n2 $ reachableSet g n1
reachableSet :: Graph -> Node -> Set Node
reachableSet g n = go Set.empty (Set.singleton n) where
go explored boundary
| Set.null boundary = explored
| otherwise = go newExplored $ Set.fromList [ t |
(f,t) <- Set.toList g,
Set.member f boundary,
not (Set.member t newExplored)
]
where newExplored = Set.union explored boundary
```

One optimization could be to stop as soon as we find `n2`

while exploring.
Another optimization could be to expand from `n1`

and `n2`

concurrently
(exploring edges in the reverse direction from `n2`

),
and stopping as soon as the explored sets overlap.

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