diff --git a/django/db/migrations/graph.py b/django/db/migrations/graph.py
index 27b4105dcc..1daa902162 100644
--- a/django/db/migrations/graph.py
+++ b/django/db/migrations/graph.py
@@ -94,31 +94,26 @@ class MigrationGraph(object):
         """
         Dynamic programming based depth first search, for finding dependencies.
         """
-        cache = {}
+        visited = []
+        visited.append(start)
+        path = [start]
+        stack = sorted(get_children(start))
+        while stack:
+            node = stack.pop(0)
 
-        def _dfs(start, get_children, path):
-            # If we already computed this, use that (dynamic programming)
-            if (start, get_children) in cache:
-                return cache[(start, get_children)]
-            # If we've traversed here before, that's a circular dep
-            if start in path:
-                raise CircularDependencyError(path[path.index(start):] + [start])
-            # Build our own results list, starting with us
-            results = []
-            results.append(start)
-            # We need to add to results all the migrations this one depends on
-            children = sorted(get_children(start))
-            path.append(start)
-            for n in children:
-                results = _dfs(n, get_children, path) + results
-            path.pop()
-            # Use OrderedSet to ensure only one instance of each result
-            results = list(OrderedSet(results))
-            # Populate DP cache
-            cache[(start, get_children)] = results
-            # Done!
-            return results
-        return _dfs(start, get_children, [])
+            if node in path:
+                raise CircularDependencyError()
+            path.append(node)
+
+            visited.insert(0, node)
+            children = sorted(get_children(node))
+
+            if not children:
+                path = []
+
+            stack = children + stack
+
+        return list(OrderedSet(visited))
 
     def __str__(self):
         return "Graph: %s nodes, %s edges" % (len(self.nodes), sum(len(x) for x in self.dependencies.values()))
diff --git a/tests/migrations/test_graph.py b/tests/migrations/test_graph.py
index ada8a5a1e5..23ec450246 100644
--- a/tests/migrations/test_graph.py
+++ b/tests/migrations/test_graph.py
@@ -134,6 +134,21 @@ class GraphTests(TestCase):
             graph.forwards_plan, ("app_a", "0003"),
         )
 
+    def test_dfs(self):
+        graph = MigrationGraph()
+        root = ("app_a", "1")
+        graph.add_node(root, None)
+        expected = [root]
+        for i in xrange(2, 1000):
+            parent = ("app_a", str(i - 1))
+            child = ("app_a", str(i))
+            graph.add_node(child, None)
+            graph.add_dependency(str(i), child, parent)
+            expected.append(child)
+
+        actual = graph.dfs(root, lambda x: graph.dependents.get(x, set()))
+        self.assertEqual(expected[::-1], actual)
+
     def test_plan_invalid_node(self):
         """
         Tests for forwards/backwards_plan of nonexistent node.