Functions for iterating trees.
A `List`

whose underlying monad is also a `List`

is a tree.

It's nodes are accessible, in contrast to the list monad, which can also be seen as a tree, except only its leafs are accessible and only in dfs order.

import Control.Monad.Trans.List.Funcs (repeatM) import Data.List.Class (genericTake, scanl, takeWhile, toList, lastL) import Prelude hiding (scanl, takeWhile) appendToEnd xs x = xs ++ [x] bits = scanl appendToEnd [] (repeatM "01") > take 3 (bfsLayers bits) [[""],["0","1"],["00","01","10","11"]] > take 10 (bfs bits) ["","0","1","00","01","10","11","000","001","010"] > dfs (genericTake 4 bits) ["","0","00","000","001","01","010","011","1","10","100","101","11","110","111"] > toList $ genericTake 3 bits [["","0","00"],["","0","01"],["","1","10"],["","1","11"]]

Examples of pruning with `prune`

and `takeWhile`

:

> dfs . takeWhile (not . isSuffixOf "11") $ genericTake 4 bits ["","0","00","000","001","01","010","1","10","100","101"] > lastL . takeWhile (not . isSuffixOf "11") $ genericTake 4 bits ["000","001","010","01","100","101","1"] > lastL . prune (not . isSuffixOf "11") $ genericTake 4 bits ["000","001","010","100","101"]

- class (List t, List (ItemM t)) => Tree t
- type TreeT m a = ListT (ListT m) a
- type TreeItemM t = ItemM (ItemM t)
- dfs :: Tree t => t a -> ItemM t a
- bfs :: Tree t => t a -> ItemM t a
- bfsLayers :: Tree t => t a -> ItemM t (ItemM t a)
- bestFirstSearchOn :: (Ord b, Tree t) => (a -> b) -> t a -> ItemM t a
- bestFirstSearchSortedChildrenOn :: (Ord b, Tree t) => (a -> b) -> t a -> ItemM t a
- sortChildrenOn :: (Ord b, Tree t) => (a -> b) -> t a -> ListT (ItemM t) a
- prune :: MonadPlus m => (a -> Bool) -> ListT m a -> ListT m a
- pruneM :: MonadPlus m => (a -> m Bool) -> ListT m a -> ListT m a
- branchAndBound :: (Ord b, Tree t) => (a -> (Maybe b, Maybe b)) -> t a -> TreeT (StateT (Maybe b) (TreeItemM t)) a

# Documentation

Search algorithms

bfsLayers :: Tree t => t a -> ItemM t (ItemM t a)Source

Transform a tree into lists of the items in its different layers

bestFirstSearchOn :: (Ord b, Tree t) => (a -> b) -> t a -> ItemM t aSource

Best First Search given a scoring function.

bestFirstSearchSortedChildrenOn :: (Ord b, Tree t) => (a -> b) -> t a -> ItemM t aSource

Best-First-Search given that a node's children are in sorted order (best first) and given a scoring function.
Especially useful for trees where nodes have an infinite amount of children, where `bestFirstSearchOn`

will get stuck.

Example: Find smallest Pythagorian Triplets

import Control.Monad import Control.Monad.Generator import Control.Monad.Trans.Class import Data.List.Tree import Data.Maybe pythagorianTriplets = catMaybes . fmap fst . bestFirstSearchSortedChildrenOn snd . generate $ do x <- lift [1..] yield (Nothing, x) y <- lift [1..] yield (Nothing, x + y) z <- lift [1..] yield (Nothing, x + y + z) lift . guard $ x^2 + y^2 == z^2 yield (Just (x, y, z), 0) > print $ take 10 pythagorianTriplets [(3,4,5),(4,3,5),(6,8,10),(8,6,10),(5,12,13),(12,5,13),(9,12,15),(12,9,15),(15,8,17),(8,15,17)]

Pruning methods

branchAndBound :: (Ord b, Tree t) => (a -> (Maybe b, Maybe b)) -> t a -> TreeT (StateT (Maybe b) (TreeItemM t)) aSource

Generalized Branch and Bound. A method for pruning.

The result of this function
would usually be given to another search algorithm,
such as `dfs`

, in order to find the node with lowest value.

This augments the regular search by pruning the tree. Given a function to calculate a lower and upper bound for a subtree, we keep the lowest upper bound (hence the State monad) encountered so far, and we prune any subtree whose lower bound is over the known upper bound.