Difference between revisions of "Graham Scan Implementation"

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<haskell>
 
<haskell>
--Graham Scan exercise
 
  +
import Data.List (sortBy)
  +
import Data.Function (on)
   
--Direction type
 
  +
data Direction = LEFT | RIGHT | STRAIGHT
data Direction = LeftTurn
 
  +
deriving (Show, Eq)
| RightTurn
 
| Straight
 
deriving (Show, Eq)
 
   
--Point type
 
  +
data Pt = Pt (Double, Double)
data Point = Point (Double, Double)
 
  +
deriving (Show, Eq, Ord)
deriving (Show)
 
   
--some points
 
  +
isTurned :: Pt -> Pt -> Pt -> Direction
p0 = Point (2.1,2.0)
 
  +
isTurned (Pt (ax, ay)) (Pt (bx, by)) (Pt (cx, cy)) = case sign of
p1 = Point (4.2,2.0)
 
  +
EQ -> STRAIGHT
p2 = Point (0.5,2.5)
 
  +
LT -> RIGHT
p3 = Point (3.2,3.5)
 
  +
GT -> LEFT
p4 = Point (1.2,4.0)
 
  +
where sign = compare ((bx - ax) * (cy - ay)) ((cx - ax) * (by - ay))
p5 = Point (0.7,4.7)
 
p6 = Point (1.0,1.0)
 
p7 = Point (3.0,5.2)
 
p8 = Point (4.0,4.0)
 
p9 = Point (3.5,1.5)
 
pA = Point (0.5,1.0)
 
points = [p0,p1,p2,p3,p4,p5,p6,p7,p8,p9,pA]
 
   
-- Actually, I'd leave it as EQ, GT, LT. Then, actually,
 
  +
gScan :: [Pt] -> [Pt]
-- if you wanted to sort points rotationally around a single point,
 
  +
gScan pts
-- sortBy (dir x) would actually work. --wasserman.louis@gmail.com
 
  +
| length pts >= 3 = scan [pt0] rests
--Get direction of single set of line segments
 
  +
| otherwise = pts
dir :: Point -> Point -> Point -> Direction
 
  +
where
dir (Point (ax, ay)) (Point (bx, by)) (Point (cx, cy)) = case sign of
 
  +
-- Find the most bottom-left point pt0
EQ -> Straight
 
  +
pt0 = foldr bottomLeft (Pt (1/0, 1/0)) pts where
GT -> LeftTurn
+
bottomLeft pa pb = case ord of
LT -> RightTurn
+
LT -> pa
where sign = compare ((bx - ax) * (cy - ay) - (by - ay) * (cx - ax)) 0
+
GT -> pb
  +
EQ -> pa
  +
where ord = (compare `on` (\ (Pt (x, y)) -> (y, x))) pa pb
   
--Get a list of Directions from a list of Points
 
  +
-- Sort other points based on angle
dirlist :: [Point] -> [Direction]
 
  +
rests = tail (sortBy (compare `on` compkey pt0) pts) where
dirlist (x:y:z:xs) = dir x y z : dirlist (y:z:xs)
 
  +
compkey (Pt (x0, y0)) (Pt (x, y)) = (atan2 (y - y0) (x - x0),
dirlist _ = []
 
  +
abs (x - x0))
   
--Compare Y axes
 
  +
-- Scan the points to find out convex
sortByY :: [Point] -> [Point]
 
  +
-- -- handle the case that all points are collinear
sortByY xs = sortBy lowestY xs
 
  +
scan [p0] (p1:ps)
where lowestY (Point(x1,y1)) (Point (x2,y2)) = if y1 == y2
+
| isTurned pz p0 p1 == STRAIGHT = [pz, p0]
then compare x1 x2
+
where pz = last ps
else compare y1 y2
 
--get COT of line defined by two points and the x-axis
 
pointAngle :: Point -> Point -> Double
 
pointAngle (Point (x1, y1)) (Point (x2, y2)) = (x2 - x1) / (y2 - y1)
 
   
--compare based on point angle
 
  +
scan (x:xs) (y:z:rsts) = case isTurned x y z of
pointOrdering :: Point -> Point -> Ordering
 
  +
RIGHT -> scan xs (x:z:rsts)
pointOrdering a b = compare (pointAngle a b) 0.0
 
  +
STRAIGHT -> scan (x:xs) (z:rsts) -- skip collinear points
  +
LEFT -> scan (y:x:xs) (z:rsts)
   
--Sort by angle
 
  +
scan xs [z] = z : xs
sortByAngle :: [Point] -> [Point]
 
  +
</haskell>
sortByAngle ps = bottomLeft : sortBy (compareAngles bottomLeft) (tail (sortedPs))
 
where sortedPs = sortByY ps
 
bottomLeft = head (sortedPs)
 
   
 
   
--Compare angles
 
  +
== Test ==
compareAngles :: Point -> Point -> Point -> Ordering
 
  +
<haskell>
compareAngles base a b = compare (pointAngle base b) (pointAngle base a)
 
  +
import Test.QuickCheck
  +
import Control.Monad
  +
import Data.List
   
--Graham Scan
 
  +
gscan :: [Point] -> [Point]
 
  +
convex' = map (\(Pt x) -> x) . convex . map Pt
gscan ps = scan (sortByAngle ps)
 
  +
where scan (x:y:z:xs) = if dir x y z == RightTurn
 
  +
prop_convex n = forAll (replicateM n arbitrary) (\xs -> sort (convex' xs) == sort (convex' (convex' xs)))
then x: scan (z:xs)
 
  +
else x: scan (y:z:xs)
 
  +
</haskell>
scan [x,y] = [x,y] -- there's no shame in a pattern match
 
  +
-- of this type!
 
  +
The results are:
scan _ = []
 
  +
  +
<haskell>
  +
> quickCheck (prop_convex 0)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 1)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 2)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 3)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 4)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 5)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 10)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 100)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 1000)
  +
+++ OK, passed 100 tests.
  +
> quickCheck (prop_convex 10000)
  +
+++ OK, passed 100 tests.
 
</haskell>
 
</haskell>

Latest revision as of 14:36, 5 January 2015

Descriptions of this problem can be found in Real World Haskell, Chapter 3

import Data.List (sortBy)
import Data.Function (on)

data Direction = LEFT | RIGHT | STRAIGHT
               deriving (Show, Eq)

data Pt = Pt (Double, Double)
        deriving (Show, Eq, Ord)

isTurned :: Pt -> Pt -> Pt -> Direction
isTurned (Pt (ax, ay)) (Pt (bx, by)) (Pt (cx, cy)) = case sign of
    EQ -> STRAIGHT
    LT -> RIGHT
    GT -> LEFT
    where sign = compare ((bx - ax) * (cy - ay)) ((cx - ax) * (by - ay))

gScan :: [Pt] -> [Pt]
gScan pts 
    | length pts >= 3 = scan [pt0] rests
    | otherwise       = pts
    where 
        -- Find the most bottom-left point pt0
        pt0 = foldr bottomLeft (Pt (1/0, 1/0)) pts where
            bottomLeft pa pb = case ord of
                               LT -> pa
                               GT -> pb
                               EQ -> pa
                       where ord = (compare `on` (\ (Pt (x, y)) -> (y, x))) pa pb

        -- Sort other points based on angle
        rests = tail (sortBy (compare `on` compkey pt0) pts) where
            compkey (Pt (x0, y0)) (Pt (x, y)) = (atan2 (y - y0) (x - x0),
                                       abs (x - x0))

        -- Scan the points to find out convex
        -- -- handle the case that all points are collinear
        scan [p0] (p1:ps)
            | isTurned pz p0 p1 == STRAIGHT = [pz, p0]
            where pz = last ps

        scan (x:xs) (y:z:rsts) = case isTurned x y z of
            RIGHT    -> scan xs (x:z:rsts)
            STRAIGHT -> scan (x:xs) (z:rsts) -- skip collinear points
            LEFT     -> scan (y:x:xs) (z:rsts)

        scan xs [z] = z : xs


Test

import Test.QuickCheck
import Control.Monad
import Data.List


convex' = map (\(Pt x) -> x) . convex . map Pt

prop_convex n = forAll (replicateM n arbitrary) (\xs -> sort (convex' xs) == sort (convex' (convex' xs)))

The results are:

> quickCheck (prop_convex 0)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 1)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 2)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 3)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 4)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 5)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 10)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 100)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 1000)
+++ OK, passed 100 tests.
> quickCheck (prop_convex 10000)
+++ OK, passed 100 tests.