Water collected between towers
- Task
In a two-dimensional world, we begin with any bar-chart (or row of close-packed 'towers', each of unit width), and then it rains, filling any convex enclosures in the chart with water.
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In the example above, a bar chart representing the values [5, 3, 7, 2, 6, 4, 5, 9, 1, 2] has filled, collecting 14 units of water.
Write a function, in your language, from a given array of heights, to the number of water units that would be collected in this way, by a corresponding bar chart.
Display test results for water collected by bar charts for the following 6 series of bar heights:
[[1, 5, 3, 7, 2],
[5, 3, 7, 2, 6, 4, 5, 9, 1, 2],
[5, 5, 5, 5],
[5, 6, 7, 8],
[8, 7, 7, 6],
[6, 7, 10, 7, 6]]
See, also:
- Four Solutions to a Trivial Problem – a Google Tech Talk by Guy Steele
- Water collected between towers on Stack Overflow, from which the example above is taken)
- An interesting Haskell solution, using the Tardis monad, by Phil Freeman in a Github gist.
AppleScript
<lang AppleScript>-- waterCollected :: [Int] -> Int on waterCollected(xs)
set leftWalls to tail(scanl(my max, 0, xs))
set rightWalls to init(scanr(my max, 0, xs))
set waterLevels to zipWith(my min, leftWalls, rightWalls)
-- positive :: Num a => a -> Bool
script positive
on lambda(x)
x > 0
end lambda
end script
-- minus :: Num a => a -> a -> a
script minus
on lambda(a, b)
a - b
end lambda
end script
sum(filter(positive, zipWith(minus, waterLevels, xs)))
end waterCollected
-- TEST ------------------------------------------------------------------ on run
map(waterCollected, ¬
[[1, 5, 3, 7, 2], ¬
[5, 3, 7, 2, 6, 4, 5, 9, 1, 2], ¬
[5, 5, 5, 5], ¬
[5, 6, 7, 8], ¬
[8, 7, 7, 6], ¬
[6, 7, 10, 7, 6]])
--> {2, 14, 0, 0, 0, 0}
end run
-- GENERIC FUNCTIONS ------------------------------------------------------
-- scanl :: (b -> a -> b) -> b -> [a] -> [b] on scanl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
set lst to {startValue}
repeat with i from 1 to lng
set v to lambda(v, item i of xs, i, xs)
set end of lst to v
end repeat
return lst
end tell
end scanl
-- scanr :: (b -> a -> b) -> b -> [a] -> [b] on scanr(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
set lst to {startValue}
repeat with i from lng to 1 by -1
set v to lambda(v, item i of xs, i, xs)
set end of lst to v
end repeat
return reverse of lst
end tell
end scanr
-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] on zipWith(f, xs, ys)
set nx to length of xs
set ny to length of ys
if nx < 1 or ny < 1 then
{}
else
set lng to cond(nx < ny, nx, ny)
set lst to {}
tell mReturn(f)
repeat with i from 1 to lng
set end of lst to lambda(item i of xs, item i of ys)
end repeat
return lst
end tell
end if
end zipWith
-- map :: (a -> b) -> [a] -> [b] on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to lambda(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- filter :: (a -> Bool) -> [a] -> [a] on filter(f, xs)
tell mReturn(f)
set lst to {}
set lng to length of xs
repeat with i from 1 to lng
set v to item i of xs
if lambda(v, i, xs) then set end of lst to v
end repeat
return lst
end tell
end filter
-- sum :: Num a => [a] -> a on sum(xs)
script add
on lambda(a, b)
a + b
end lambda
end script
foldl(add, 0, xs)
end sum
-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to lambda(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property lambda : f
end script
end if
end mReturn
-- init :: [a] -> [a] on init(xs)
if length of xs > 1 then
items 1 thru -2 of xs
else
{}
end if
end init
-- tail :: [a] -> [a] on tail(xs)
if length of xs > 1 then
items 2 thru -1 of xs
else
{}
end if
end tail
-- max :: Ord a => a -> a -> a on max(x, y)
if x > y then
x
else
y
end if
end max
-- min :: Ord a => a -> a -> a on min(x, y)
if y < x then
y
else
x
end if
end min
-- cond :: Bool -> a -> a -> a on cond(bool, f, g)
if bool then
f
else
g
end if
end cond</lang>
- Output:
<lang AppleScript>{2, 14, 0, 0, 0, 0}</lang>
Haskell
Following cdk's Haskell solution at Stack Overflow:
<lang haskell>waterCollected :: [Int] -> Int waterCollected xs =
-- water collects only in flanked concavities
-- (i.e. where there is a gap between the high tide mark
-- and the sea bed beneath)
sum (filter (> 0) (zipWith (-) levels xs))
where
-- high tide is the level of the lower of the
-- two flanking walls
levels = zipWith min
(init $ scanr max 0 xs)
(tail $ scanl max 0 xs)
main :: IO () main = mapM_ (putStrLn . show . waterCollected) [
[1, 5, 3, 7, 2], [5, 3, 7, 2, 6, 4, 5, 9, 1, 2], [5, 5, 5, 5], [5, 6, 7, 8], [8, 7, 7, 6], [6, 7, 10, 7, 6]]</lang>
- Output:
2 14 0 0 0 0
JavaScript
ES6
<lang JavaScript>(() => {
'use strict';
const waterCollected = xs => {
const maxToRight = scanr(max, 0, xs),
maxToLeft = scanl(max, 0, xs),
levels = zipWith(min, init(maxToLeft), tail(maxToRight));
return sum(zipWith(difference, levels, xs)
.filter(x => x > 0));
};
// GENERIC FUNCTIONS ----------------------------------------
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = (f, xs, ys) => {
const ny = ys.length;
return (xs.length <= ny ? xs : xs.slice(0, ny))
.map((x, i) => f(x, ys[i]));
}
// scanl :: (b -> a -> b) -> b -> [a] -> [b]
const scanl = (f, startValue, xs) => {
const lst = [startValue];
return (
xs.reduce((a, x) => {
const v = f(a, x);
return (lst.push(v), v);
}, startValue),
lst
);
};
// scanr :: (b -> a -> b) -> b -> [a] -> [b]
const scanr = (f, startValue, xs) => {
const lst = [startValue];
return (
xs.reduceRight((a, x) => {
const v = f(a, x);
return (lst.push(v), v);
}, startValue),
lst.reverse()
);
};
// difference :: (Num a) => a -> a -> a const difference = (a, b) => a - b;
// sum :: (Num a) => [a] -> a const sum = xs => xs.reduce((a, x) => a + x, 0);
// max :: Ord a => a -> a -> a const max = (a, b) => a > b ? a : b;
// min :: Ord a => a -> a -> a const min = (a, b) => b < a ? b : a;
// init :: [a] -> [a] const init = xs => xs.length ? xs.slice(0, -1) : undefined;
// tail :: [a] -> [a] const tail = xs => xs.length ? xs.slice(1) : undefined;
// TEST ---------------------------------------------------
return [
[1, 5, 3, 7, 2],
[5, 3, 7, 2, 6, 4, 5, 9, 1, 2],
[5, 5, 5, 5],
[5, 6, 7, 8],
[8, 7, 7, 6],
[6, 7, 10, 7, 6]
].map(waterCollected);
//--> [2, 14, 0, 0, 0, 0]
})();</lang>
- Output:
<lang>[2, 14, 0, 0, 0, 0]</lang>