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
by generating the full list of nodes reachable from
and asking whether
n2 is in that list.
To generate the “reachable set” from a node,
we partition the graph into three sets:
boundary nodes, and the rest.
We repeatedly look for new nodes
by looking at outgoing edges from nodes in the
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.
boundary set becomes empty,
there are no new nodes to explore,
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
(exploring edges in the reverse direction from
and stopping as soon as the explored sets overlap.