{-# LANGUAGE UndecidableInstances #-}
--------------------------------------------------------------------------------
-- |
-- Module : Algorithms.BinarySearch
-- Copyright : (C) Frank Staals
-- License : see the LICENSE file
-- Maintainer : Frank Staals
--------------------------------------------------------------------------------
module Algorithms.BinarySearch where
import Control.Applicative ((<|>))
import Data.Sequence (Seq, ViewL(..),ViewR(..))
import qualified Data.Sequence as Seq
import Data.Set (Set)
import qualified Data.Set.Internal as Set
import qualified Data.Vector.Generic as V
--------------------------------------------------------------------------------
-- | Given a monotonic predicate p, a lower bound l, and an upper bound u, with:
-- p l = False
-- p u = True
-- l < u.
--
-- Get the index h such that everything strictly smaller than h has: p i =
-- False, and all i >= h, we have p h = True
--
-- running time: \(O(\log(u - l))\)
{-# SPECIALIZE binarySearch :: (Int -> Bool) -> Int -> Int -> Int #-}
{-# SPECIALIZE binarySearch :: (Word -> Bool) -> Word -> Word -> Word #-}
binarySearch :: Integral a => (a -> Bool) -> a -> a -> a
binarySearch p = go
where
go l u = let d = u - l
m = l + (d `div` 2)
in if d == 1 then u else if p m then go l m
else go m u
-- | Given a value \(\varepsilon\), a monotone predicate \(p\), and two values \(l\) and
-- \(u\) with:
--
-- - \(p l\) = False
-- - \(p u\) = True
-- - \(l < u\)
--
-- we find a value \(h\) such that:
--
-- - \(p h\) = True
-- - \(p (h - \varepsilon)\) = False
--
-- >>> binarySearchUntil (0.1) (>= 0.5) 0 (1 :: Double)
-- 0.5
-- >>> binarySearchUntil (0.1) (>= 0.51) 0 (1 :: Double)
-- 0.5625
-- >>> binarySearchUntil (0.01) (>= 0.51) 0 (1 :: Double)
-- 0.515625
binarySearchUntil :: (Fractional r, Ord r)
=> r
-> (r -> Bool) -> r -> r -> r
binarySearchUntil eps p = go
where
go l u | u - l < eps = u
| otherwise = let m = (l + u) / 2
in if p m then go l m else go m u
--------------------------------------------------------------------------------
-- * Binary Searching in some data structure
class BinarySearch v where
type Index v :: *
type Elem v :: *
-- | Given a monotonic predicate p and a data structure v, find the
-- element v[h] such that that
--
-- for every index i < h we have p v[i] = False, and
-- for every inedx i >= h we have p v[i] = True
--
-- returns Nothing if no element satisfies p
--
-- running time: \(O(T*\log n)\), where \(T\) is the time to execute the
-- predicate.
binarySearchIn :: (Elem v -> Bool) -> v -> Maybe (Elem v)
-- | Given a monotonic predicate p and a data structure v, find the
-- index h such that that
--
-- for every index i < h we have p v[i] = False, and
-- for every inedx i >= h we have p v[i] = True
--
-- returns Nothing if no element satisfies p
--
-- running time: \(O(T*\log n)\), where \(T\) is the time to execute the
-- predicate.
binarySearchIdxIn :: (Elem v -> Bool) -> v -> Maybe (Index v)
--------------------------------------------------------------------------------
-- * Searching on a Sequence
instance BinarySearch (Seq a) where
type Index (Seq a) = Int
type Elem (Seq a) = a
-- ^ runs in \(O(T*\log^2 n)\) time.
binarySearchIn p s = Seq.index s <$> binarySearchIdxIn p s
-- ^ runs in \(O(T*\log^2 n)\) time.
binarySearchIdxIn p s = case Seq.viewr s of
EmptyR -> Nothing
(_ :> x) | p x -> Just $ case Seq.viewl s of
(y :< _) | p y -> 0
_ -> binarySearch p' 0 u
| otherwise -> Nothing
where
p' = p . Seq.index s
u = Seq.length s - 1
instance {-# OVERLAPPABLE #-} V.Vector v a => BinarySearch (v a) where
type Index (v a) = Int
type Elem (v a) = a
binarySearchIdxIn p' v | V.null v = Nothing
| not $ p n' = Nothing
| otherwise = Just $ if p 0 then 0 else binarySearch p 0 n'
where
n' = V.length v - 1
p = p' . (v V.!)
binarySearchIn p v = (v V.!) <$> binarySearchIdxIn p v
instance BinarySearch (Set a) where
type Index (Set a) = Int
type Elem (Set a) = a
binarySearchIn p = go
where
go = \case
Set.Tip -> Nothing
Set.Bin _ k l r | p k -> go l <|> Just k
| otherwise -> go r
binarySearchIdxIn p = go
where
go = \case
Set.Tip -> Nothing
Set.Bin _ k l r | p k -> go l <|> Just (Set.size l)
| otherwise -> (+ (1 + Set.size l)) <$> go r