FP Cvičení 5

Generátory seznamů

Implementujte následující funkce s využitím genrátoru seznamu:

• Vytvořte funkci, která vygeneruje seznam všech lichých čísel v daném intervalu.

oddList :: Int -> Int -> [Int]

*Main> oddList 1 10   
[1,3,5,7,9]

oddList :: Int -> Int -> [Int]
oddList a b = [ x |x<-[a..b], odd x]

• Vztvořte funkci která odstraní všechna velká písmena z řetězce.

removeAllUpper :: String -> String

*Main> removeAllUpper "ABCabcABC"
"abc"

import Data.Char

removeAllUpper :: String -> String
removeAllUpper xs = [ x |x<-xs, not (isUpper x)]

• Vytvořte funkci která spočítá sjednocení a průnik dvou množin.

union :: Eq a => [a] -> [a] -> [a]
intersection :: Eq a => [a] -> [a] -> [a]

*Main> union [1..5] [3..10]
[1,2,3,4,5,6,7,8,9,10]
*Main> intersection [1..5] [3..10]
[3,4,5]

union :: Eq a => [a] -> [a] -> [a]
union xs ys = xs ++ [y| y<-ys, not (elem y xs)]

intersection ::  Eq a =>  [a] -> [a] -> [a]
intersection xs ys = [y| y<-ys, elem y xs]

Komplexnější funkce

• Vytvořte funkci která spočítá počet výskytů všech znaků v řetězci.

countThem :: String -> [(Char, Int)]

*Main>countThem "hello hello hello"
[('h',3),('e',3),('l',6),('o',3),(' ',2)]

unique :: String -> String
unique n = reverse(tmp n "") where
  tmp [] store = store
  tmp (x:xs) store | x `elem` store = tmp xs store
                   | otherwise = tmp xs (x:store)

unique' :: String -> String                   
unique' [] = []
unique' (x:xs) = x: unique' (filter (/=x)xs)

countThem :: String -> [(Char, Int)]
countThem xs = let u = unique xs
               in [(x, length (filter (==x) xs)) |x<-u]

• Goldbachova hypotéza říká, že každé kladné sudé číslo větší než 2 lze vyjádřit jako součet dvou prvočísel. Příklad: 28 = 5 + 23. Je to jeden z nejslavnějších hypotéz v teorii čísel, která ještě nebyla dokázána. Vytvořte funkci která pro dané celé číslo spočítá seznam dvojic prvočísel, které splňují Goldbachovu hypotézu.

goldbach :: Int-> [(Int, Int)]

*Main>goldbach 28
[(5, 23),(11,17)]

isPrime :: Int -> Bool
isPrime n = null [x |x<-[2..ceiling (sqrt (fromIntegral n)::Double)], n `mod` x == 0]

goldbach :: Int-> [(Int, Int)]
goldbach n = let primes = [x |x<-[2..(n `div` 2)], isPrime x]
             in [(x,n-x) |x<-primes, isPrime (n-x)]

• In most cases, if an even number is written as the sum of two prime numbers, one of them is very small. We will be searching for cases that violates this rule. Create a function, that has three parameters. First two defines an interval, where we will be searching for Goldbach numbers. The last parameter is the limit. For each number in this interval, find Goldbach's pair with smallest prime number. If this smallest number is bigger than given limit, the corresponding pair will be in the result.

goldbachList :: Int -> Int-> Int -> [(Int, Int)]

*Main>goldbachList 4 2000 50
[(73,919),(61,1321),(67,1789),(61,1867)]

isPrime :: Int -> Bool
isPrime n = null [x |x<-[2..ceiling (sqrt (fromIntegral n)::Double)], n `mod` x == 0]

goldbach :: Int-> [(Int, Int)]
goldbach n = let primes = [x |x<-[2..(n `div` 2)+1], isPrime x]
             in [(x,n-x) |x<-primes, isPrime (n-x)]

goldbachList :: Int -> Int-> Int -> [(Int, Int)]
goldbachList a b limit = filter (\(x,_)-> x>limit) [head (goldbach x) | x<-[a..b], even x]

• Create a function that generates all combinations of given length from the characters from given string. You can assume, that all character are unique and the given length is not bigger then the length of this string.

combinations :: Int -> String -> [String]

*Main> combinations 3 "abcdef"
["abc","abd","abe",...]

combinations :: Int -> String -> [String]
combinations 1 xs = [[x]| x<-xs]
combinations n (x:xs) | n == length (x:xs) = [(x:xs)]
                      |otherwise = [[x] ++ y |y<-combinations (n-1) xs ] 
                                    ++ (combinations n xs)