cga/Algorithms/ConvexHull/GrahamScan.hs

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{-# OPTIONS_HADDOCK ignore-exports #-}
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module Algorithms.ConvexHull.GrahamScan where
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import Algebra.Vector
import Algebra.VectorTypes
import MyPrelude
-- |Get all points on a convex hull by using the graham scan
-- algorithm.
{--
========== FUNCTIONAL PSEUDO CODE ======================
input: unsorted list us'
output: sorted convex hull list
variables:
(lowerHull, restl) = splitAt3IntoTuple (sort us')
(upperHull, restu) = reverse (splitAt3IntoTuple (sort us'))
main scope:
return (scanHalf upperHull restu) ++
(stripFirstAndLastElem(scanHalf lowerHull restl))
=== begin scanHalf function ===
scanHalf (min 3 elem => lowerHull) (min 1 elem => rest)
| isNotClockWise (last3Elements lowerHull) == True
= scanHalf (lowerHull + head rest) (tail rest)
| otherwise
= scanHalf (deleteSndToLastElem lowerHull + head rest)
(rest)
scanHalf (min 3 elem => lowerHull ) []
| isNotClockWise (last3Elements lowerHull) == True
= return lowerHull
| otherwise
= scanHalf (deleteSndToLastElem lowerHull) []
scanHalf lowerHull (min 1 elem => rest) = scanHalf (lowerHull + head rest)
(tail rest)
scanHalf lowerHull _ = lowerHull
=== end scanHalf function ===
============= SIMULATION ===================================
xs = [(100, 100), (200, 450), (250, 250)]
ys = [(300, 400), (400, 200)]
notcw (100, 100) (200, 450) (250, 250) => false, pop snd2last of xs
===
move first of ys to end of xs
xs = [(100, 100), (250, 250), (300, 400)]
ys = [(400, 200)]
notcw (100, 100), (250, 250) (300, 400) => true
===
move first of ys to end of xs
xs = [(100, 100), (250, 250), (300, 400), (400, 200)]
ys = []
notcw (250, 250) (300, 400) (400, 200) => false, pop snd2last of xs
===
xs = [(100, 100), (250, 250), (400, 200)]
ys = []
notcw (100, 100) (250, 250) (400, 200) => false, pop snd2last of xs
===
xs = [(100, 100), (400, 200)]
ys = []
===
return [(100, 100), (400, 200)]
=========================================================
--}
grahamCH :: [PT] -> [PT]
grahamCH vs = grahamUCH vs ++ (tailInit . grahamLCH $ vs)
-- |Get the lower part of the convex hull.
grahamLCH :: [PT] -> [PT]
grahamLCH vs = uncurry (\x y -> last . scanH x $ y)
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(first reverse . splitAt 3 . sortedXY $ vs)
-- |Get the upper part of the convex hull.
grahamUCH :: [PT] -> [PT]
grahamUCH vs = uncurry (\x y -> last . scanH x $ y)
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(first reverse . splitAt 3 . reverse . sortedXY $ vs)
-- |This scans only a half of the convex hull, but all steps (the last
-- is the end result).
-- If it's the upper or lower half depends on the input.
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-- Also, the first list is expected to be reversed since we only care
-- about the last 3 elements and want to stay efficient.
scanH :: [PT] -- ^ the first 3 starting points in reversed order
-> [PT] -- ^ the rest of the points
-> [[PT]] -- ^ all convex hull points iterations for the half
scanH hs@(x:y:z:xs) (r':rs')
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| notcw z y x = hs : scanH (r':hs) rs'
| otherwise = hs : scanH (x:z:xs) (r':rs')
scanH hs@(x:y:z:xs) []
| notcw z y x = [hs]
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| otherwise = hs : scanH (x:z:xs) []
scanH hs (r':rs') = hs : scanH (r':hs) rs'
scanH hs _ = [hs]
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-- |Compute all steps of the graham scan algorithm to allow
-- visualizing it.
-- Whether the upper or lower hull is computed depends on the input.
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grahamCHSteps :: Int -> [PT] -> [PT] -> [[PT]]
grahamCHSteps c xs' ys' = take c . scanH xs' $ ys'
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-- |Get all iterations of the upper hull of the graham scan algorithm.
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grahamUHSteps :: [PT] -> [[PT]]
grahamUHSteps vs =
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(++) [getLastX 2 . sortedXY $ vs]
. rmdups
. grahamCHSteps ((* 2) . length $ vs) uH
$ uHRest
where
(uH, uHRest) = first reverse . splitAt 3 . reverse . sortedXY $ vs
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-- |Get all iterations of the lower hull of the graham scan algorithm.
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grahamLHSteps :: [PT] -> [[PT]]
grahamLHSteps vs =
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(++) [take 2 . sortedXY $ vs]
. rmdups
. grahamCHSteps ((* 2) . length $ vs) lH
$ lHRest
where
(lH, lHRest) = first reverse . splitAt 3 . sortedXY $ vs