Recursion Patterns and Time-analysis - Departamento de ...

foldr :: (a -> b -> b) -> b -> [a] -> bfoldr f z [] = zfoldr f z (x:xs) = f x (foldr f z xs)For instance, the function that sums the elements in a list is written as sum = foldr (+) 0.This idea can be generalized for any regular type, the difference being that the programmer will have to writethe appropriate parameterized function that encapsulates the recursion pattern. We exemplify this for two differenttypes of binary trees, respectively leaf-labeled and node-labeled, defined as follows.data LTree a = Leaf a | Fork (LTree a) (LTree a)data BTree a = Nil | Node a (BTree a) (BTree a)Folds over these trees can be captured by the functions cataLTree and cataBTree. In order to define these functions,one just has to bear in mind that when processing a node the results of performing recursion on the sub-trees of thenode are combined with the contents of the node (if any) to give the final result.cataLTree :: (b -> b -> b) -> (a->b) -> LTree a -> bcataLTree f g (Leaf x) = g xcataLTree f g (Fork l r) = f l’ r’ where l’ = cataLTree f g lr’ = cataLTree f g rcataBTree :: (a -> b -> b -> b) -> b -> BTree a -> bcataBTree f z Nil = zcataBTeee f z (Node x l r) = f x l’ r’ where l’ = cataBTeee f z lr’ = cataBTeee f z rOften the arguments of these functions are called the genes of the folds. In the case of binary trees, the genes are pairsof functions, the first of which is used to process nodes, and the second to process leaves (in fact, for node-labeledtrees, this second component is a constant since there is no information to process in an empty leaf).As examples of folds over trees, let us write the height functions for both types of trees.heightLTree = cataLTree (\ l r -> 1 + (max l r)) (\_ -> 0)heightBTree = cataBTree (\x l r -> 1 + (max l r)) 0Unfolds. In the context of a programming language with non-strict semantics such as Haskell, recursive and corecursivetypes coincide. Unfolds stand to corecursive programs as folds stand to recursive programs (they are dualconcepts); they are however less widely used by programmers [4]. Consider the Haskell function that constructs (wheninvoked with 0) a binary tree where each node is labeled with its depth in the tree, stopping at level 100.inflevels100 100 = Nilinflevels100 n = Node n (inflevels100 (n+1)) (inflevels100 (n+1))This function has the property that when a node is returned, both its sub-trees are recursively constructed by thesame function being defined. This clearly indicates that a corecursion pattern is being used, which can be writtenonce and for all. The following functions capture these patterns for both types of trees we have been using:anaLTree :: (b -> (Either a (b,b))) -> b -> LTree aanaLTree h x = case (h x) of Left y -> Leaf yRight (a,b) -> Fork (anaLTree h a) (anaLTree h b)anaBTree :: (a -> Maybe (b,a,a)) -> a -> BTree banaBTree h x = case (h x) of Nothing -> NilJust (y,a,b) -> Node y (anaBTree h a) (anaBTree h b)