diff --git a/lib/src/dag_walk.rs b/lib/src/dag_walk.rs
index ff698f964..a5707fbcb 100644
--- a/lib/src/dag_walk.rs
+++ b/lib/src/dag_walk.rs
@@ -488,14 +488,28 @@ where
 {
     let start: Vec<T> = start.into_iter().try_collect()?;
     let mut heads: HashSet<T> = start.iter().cloned().collect();
-    dfs_ok(start.into_iter().map(Ok), id_fn, |node| {
-        let neighbors: Vec<Result<T, E>> = neighbors_fn(node).into_iter().collect();
-        for neighbor in neighbors.iter().filter_map(|x| x.as_ref().ok()) {
-            heads.remove(neighbor);
+    // Do a BFS until we have only one item left in the frontier. That frontier must
+    // have originated from one of the heads, and since there can't be cycles,
+    // it won't be able to eliminate any other heads.
+    let mut frontier: Vec<T> = heads.iter().cloned().collect();
+    let mut visited: HashSet<ID> = heads.iter().map(&id_fn).collect();
+    let mut root_reached = false;
+    while frontier.len() > 1 || (!frontier.is_empty() && root_reached) {
+        frontier = frontier
+            .iter()
+            .flat_map(|node| {
+                let neighbors = neighbors_fn(node).into_iter().collect_vec();
+                if neighbors.is_empty() {
+                    root_reached = true;
+                }
+                neighbors
+            })
+            .try_collect()?;
+        for node in &frontier {
+            heads.remove(node);
         }
-        neighbors
-    })
-    .try_for_each(|r| r.map(|_| ()))?;
+        frontier.retain(|node| visited.insert(id_fn(node)));
+    }
     Ok(heads)
 }
 
@@ -1211,10 +1225,14 @@ mod tests {
         let neighbors_fn = |node: &char| neighbors[node].clone();
 
         let result = heads_ok([Ok('C')], id_fn, neighbors_fn);
-        assert_eq!(result, Err('X'));
+        assert_eq!(result, Ok(hashset! {'C'}));
         let result = heads_ok([Ok('B')], id_fn, neighbors_fn);
-        assert_eq!(result, Err('X'));
+        assert_eq!(result, Ok(hashset! {'B'}));
         let result = heads_ok([Ok('A')], id_fn, neighbors_fn);
         assert_eq!(result, Ok(hashset! {'A'}));
+        let result = heads_ok([Ok('C'), Ok('B')], id_fn, neighbors_fn);
+        assert_eq!(result, Err('X'));
+        let result = heads_ok([Ok('C'), Ok('A')], id_fn, neighbors_fn);
+        assert_eq!(result, Err('X'));
     }
 }