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content/blog/immutable-bfs-unfold.md

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+---
+title: "Immutable Traversals with Unfold"
+date: 2022-11-12T21:27:42-05:00
+draft: false
+tags: ["Functional Programming", "Scala"]
+math: false
+---
+
+Let's consider the following binary tree:
+
+```goat
+                   a                                                                       
+                  / \                          
+                 /   \         
+                b     d                    
+                 \        
+                  \       
+                   c       
+```
+
+
+
+
+
+We can encode this with the following Scala code:
+
+```scala
+final case class BinNode(
+    val label: String,
+    val left: Option[BinNode],
+    val right: Option[BinNode]
+)
+
+// Leaf Nodes
+val c_node = BinNode("c", None, None)
+val d_node = BinNode("d", None, None)
+// Rest of nodes
+val b_node = BinNode("b", None, Some(c_node))
+val a_node = BinNode("a", Some(b_node), Some(d_node))
+```
+
+For depth first search, an intuitive immutable implementation would be a recursive function.
+
+```scala
+// Using Preorder traversal
+def DFS(node: BinNode): Iterator[BinNode] = 
+    lazy val left_side = node.left.fold(Iterator.empty[BinNode])(DFS)
+    lazy val right_side = node.right.fold(Iterator.empty[BinNode])(DFS)
+    Iterator(node) ++ left_side ++ right_side
+```
+
+Let's evaluate this using our example above:
+
+```scala
+DFS(a_node).toList.map(_.label)
+// List(a, b, c, d)
+```
+
+The recursive implementation inherently uses the system stack to keep track of the nodes.  This means that the last element gets evaluated in each step. Otherwise called last-in-first-out (LIFO). Breadth first search, however, uses a queue based approach where the first one added to the data structure is the first one considered (FIFO). 
+
+To preserve immutability in our code, we can use `unfold`. Here our state is the queue of nodes.
+
+```scala
+def BFS(node: BinNode): Iterator[BinNode] =
+    Iterator.unfold(List(node))(q =>
+        if q.isEmpty then
+            None
+        else
+            val crnt_node = q.head
+            val next_q = q.tail ++ crnt_node.left ++ crnt_node.right
+            Some(crnt_node, next_q)
+    )
+```
+
+Evaluating on our example:
+
+```scala
+BFS(a_node).toList.map(_.label)
+// List(a, b, d, c)
+```
+
+We can also use `unfold` for the depth first search approach as well. We can replace the list used with a stack.
+
+```scala
+import scala.collection.mutable.Stack
+def DFS2(node: BinNode): Iterator[BinNode] = 
+    Iterator.unfold(Stack(node))(s =>
+        if s.isEmpty then
+            None
+        else
+            val crnt_node = s.pop()
+            s.pushAll(crnt_node.right)
+            s.pushAll(crnt_node.left)
+            Some(crnt_node, s)
+    )
+```
+
+Using a stack introduces some mutability. We can use the immutable list data structure instead, as long as we satisfy the LIFO ordering.
+
+```scala
+def DFS3(node: BinNode): Iterator[BinNode] = 
+    Iterator.unfold(List(node))(s =>
+        if s.isEmpty then
+            None
+        else
+            val crnt_node = s.last
+            val next_s = s.init ++ crnt_node.right ++ crnt_node.left
+            Some(crnt_node, next_s)
+    )
+```
+
+
+