Copyright	(C) 2011-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	portable
Safe Haskell	Safe
Language	Haskell98

Data.LCA.Online

Description

Provides online calculation of the the lowest common ancestor in O(log h) by compressing the spine of a Path using a skew-binary random access list.

This library implements the technique described in my talk

http://www.slideshare.net/ekmett/skewbinary-online-lowest-common-ancestor-search

to improve the known asymptotic bounds on both online lowest common ancestor search

http://en.wikipedia.org/wiki/Lowest_common_ancestor

and the online level ancestor problem:

http://en.wikipedia.org/wiki/Level_ancestor_problem

Algorithms used here assume that the key values chosen for k are globally unique.

Synopsis

data Path a
lca :: Path a -> Path b -> Path a
empty :: Path a
cons :: Int -> a -> Path a -> Path a
uncons :: Path a -> Maybe (Int, a, Path a)
view :: Path a -> View Path a
null :: Foldable t => t a -> Bool
length :: Foldable t => t a -> Int
isAncestorOf :: Path a -> Path b -> Bool
keep :: Int -> Path a -> Path a
drop :: Int -> Path a -> Path a
traverseWithKey :: Applicative f => (Int -> a -> f b) -> Path a -> f (Path b)
toList :: Path a -> [(Int, a)]
fromList :: [(Int, a)] -> Path a
(~=) :: Path a -> Path b -> Bool

Documentation

data Path a #

Compressed paths using skew binary random access lists

Instances

Functor Path #
Instance details Defined in Data.LCA.Online Methods fmap :: (a -> b) -> Path a -> Path b # (<$) :: a -> Path b -> Path a #
Foldable Path #
Instance details Defined in Data.LCA.Online Methods fold :: Monoid m => Path m -> m # foldMap :: Monoid m => (a -> m) -> Path a -> m # foldr :: (a -> b -> b) -> b -> Path a -> b # foldr' :: (a -> b -> b) -> b -> Path a -> b # foldl :: (b -> a -> b) -> b -> Path a -> b # foldl' :: (b -> a -> b) -> b -> Path a -> b # foldr1 :: (a -> a -> a) -> Path a -> a # foldl1 :: (a -> a -> a) -> Path a -> a # toList :: Path a -> [a] # null :: Path a -> Bool # length :: Path a -> Int # elem :: Eq a => a -> Path a -> Bool # maximum :: Ord a => Path a -> a # minimum :: Ord a => Path a -> a # sum :: Num a => Path a -> a # product :: Num a => Path a -> a #
Traversable Path #
Instance details Defined in Data.LCA.Online Methods traverse :: Applicative f => (a -> f b) -> Path a -> f (Path b) # sequenceA :: Applicative f => Path (f a) -> f (Path a) # mapM :: Monad m => (a -> m b) -> Path a -> m (Path b) # sequence :: Monad m => Path (m a) -> m (Path a) #
Read a => Read (Path a) #
Instance details Defined in Data.LCA.Online Methods readsPrec :: Int -> ReadS (Path a) # readList :: ReadS [Path a] # readPrec :: ReadPrec (Path a) # readListPrec :: ReadPrec [Path a] #
Show a => Show (Path a) #
Instance details Defined in Data.LCA.Online Methods showsPrec :: Int -> Path a -> ShowS # show :: Path a -> String # showList :: [Path a] -> ShowS #

lca :: Path a -> Path b -> Path a #

O(log h) Compute the lowest common ancestor of two paths.

empty :: Path a #

The empty Path

cons :: Int -> a -> Path a -> Path a #

O(1) Invariant: most operations assume that the keys k are globally unique

Extend the Path with a new node ID and value.

uncons :: Path a -> Maybe (Int, a, Path a) #

O(1) Extract the node ID and value from the newest node on the Path.

view :: Path a -> View Path a #

O(1) Extract the node ID and value from the newest node on the Path, slightly faster than uncons.

null :: Foldable t => t a -> Bool #

Test whether the structure is empty. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.

length :: Foldable t => t a -> Int #

Returns the size/length of a finite structure as an Int. The default implementation is optimized for structures that are similar to cons-lists, because there is no general way to do better.

isAncestorOf :: Path a -> Path b -> Bool #

O(log h) xs isAncestorOf ys holds when xs is a prefix starting at the root of Path ys.

keep :: Int -> Path a -> Path a #

O(log (h - k)) to keep k elements of Path of length h

This solves the online version of the "level ancestor problem" with no preprocessing in O(log h) time, improving known complexity bounds.

http://en.wikipedia.org/wiki/Level_ancestor_problem

drop :: Int -> Path a -> Path a #

O(log k) to drop k elements from a Path

traverseWithKey :: Applicative f => (Int -> a -> f b) -> Path a -> f (Path b) #

Traverse a Path with access to the node IDs.

toList :: Path a -> [(Int, a)] #

Convert a Path to a list of (ID, value) pairs.

fromList :: [(Int, a)] -> Path a #

Build a Path from a list of (ID, value) pairs.

(~=) :: Path a -> Path b -> Bool infix 4 #

O(1) Compare to see if two trees have the same leaf key