Stephan Guenther wrote:

Is it possible to change a particular node of the doubly linked list? That is to say, that would like to have a function: update :: DList a -> a -> DList a where update node newValue returns a list where only the value at the node which is passed in is set to the new Value and all other values are the same. All this of course in a pure way, that is without using (M/T/TM)Vars or IORefs.

It is possible to do all of this, and more: - no rebuilding of the whole list on updates to the list - the update operation takes constant time (for lists longer than 32 elements on 32-bit platform) - both cyclic and terminated lists can be handled, uniformly - no monads used or mentioned - let alone no IORef, STRef, TVars, etc. The algorithm is essentially imperative (and so permits identity checking and in-place `updates') but implemented purely functionally. No destructive updates are ever used. Therefore, all the changes can be undone and re-done, and the code is MT-safe. The code is easily generalizable to 2D. Here are the tests

testl = fromList [1..5] testl_s = takeDL 11 testl

*FL> testl_s [5,1,2,3,4,5,1,2,3,4,5]

testl1 = update (-1) testl testl1_s = takeDL 11 testl1 *FL> testl1_s [-1,1,2,3,4,-1,1,2,3,4,-1]

testl2 = update (-2) . move_right' . move_right' $ testl1 testl2_s = takeDL 11 testl2 *FL> testl2_s [-2,3,4,-1,1,-2,3,4,-1,1,-2]

-- Old testl is still available testl3 = update (-2) . move_right' . move_right' $ testl testl3_s = takeDL 11 testl3 *FL> testl3_s [-2,3,4,5,1,-2,3,4,5,1,-2]

It is not for nothing Haskell is called the best imperative language. One can implement imperative algorithms just as they are -- purely functionally, without any monads or other categorical notions. module FL where import qualified Data.IntMap as IM -- Representation of the double-linked list type Ref = Int -- positive, we shall treat 0 specially data Node a = Node{node_val :: a, node_left :: Ref, node_right :: Ref} data DList a = DList{dl_counter :: Ref, -- to generate new Refs dl_current :: Ref, -- current node dl_mem :: IM.IntMap (Node a)} -- main `memory' -- Operations on the DList a empty :: DList a empty = DList{dl_counter = 1, dl_current = 0, dl_mem = IM.empty} -- In a well-formed list, dl_current must point to a valid node -- All operations below preserve well-formedness well_formed :: DList a -> Bool well_formed dl | IM.null (dl_mem dl) = dl_current dl == 0 well_formed dl = IM.member (dl_current dl) (dl_mem dl) is_empty :: DList a -> Bool is_empty dl = IM.null (dl_mem dl) -- auxiliary function get_curr_node :: DList a -> Node a get_curr_node DList{dl_current=curr,dl_mem=mem} = maybe (error "not well-formed") id $ IM.lookup curr mem -- The insert operation below makes a cyclic list -- The other operations don't care -- Insert to the right of the current element, if any -- Return the DL where the inserted node is the current one insert_right :: a -> DList a -> DList a insert_right x dl | is_empty dl = let ref = dl_counter dl -- the following makes the list cyclic node = Node{node_val = x, node_left = ref, node_right = ref} in DList{dl_counter = succ ref, dl_current = ref, dl_mem = IM.insert ref node (dl_mem dl)} insert_right x dl@DList{dl_counter = ref, dl_current = curr, dl_mem = mem} = DList{dl_counter = succ ref, dl_current = ref, dl_mem = IM.insert ref new_node $ IM.insert curr curr_node' mem} where curr_node = get_curr_node dl curr_node'= curr_node{node_right = ref} new_node = Node{node_val = x, node_left = curr, node_right = node_right curr_node} get_curr :: DList a -> a get_curr = node_val . get_curr_node move_right :: DList a -> Maybe (DList a) move_right dl = if next == 0 then Nothing else Just (dl{dl_current=next}) where next = node_right $ get_curr_node dl -- If no right, just stay inplace move_right' :: DList a -> DList a move_right' dl = maybe dl id $ move_right dl fromList :: [a] -> DList a fromList = foldl (flip insert_right) FL.empty takeDL :: Int -> DList a -> [a] takeDL 0 _ = [] takeDL n dl | is_empty dl = [] takeDL n dl = get_curr dl : (maybe [] (takeDL (pred n)) $ move_right dl) -- Update the current node update :: a -> DList a -> DList a update x dl@(DList{dl_current = curr, dl_mem = mem}) = dl{dl_mem = IM.insert curr (curr_node{node_val = x}) mem} where curr_node = get_curr_node dl testl = fromList [1..5] testl_s = takeDL 11 testl testl1 = update (-1) testl testl1_s = takeDL 11 testl1 testl2 = update (-2) . move_right' . move_right' $ testl1 testl2_s = takeDL 11 testl2 -- Old testl is still available testl3 = update (-2) . move_right' . move_right' $ testl testl3_s = takeDL 11 testl3