Step by Step solution about How to resolve the algorithm Partition function P step by step in the Haskell programming language

Memoization Utilities

The code defines a data structure called Memo for memoization. Memoization is a technique used to improve the efficiency of functions by storing their results for future use. The Memo data structure represents a node in a tree, where each node has a value, a left child, and a right child.

The memo function retrieves the value stored in the Memo data structure. If the value for the given input is already stored in the tree, it returns the stored value. Otherwise, it recursively traverses the tree to find the value. The function uses bit manipulation to determine which child to traverse based on the input.

Calculating Partitions

The code calculates the number of partitions of a given number using a recursive approach. A partition of a number is a way of dividing the number into smaller positive integers.

The partitions memo represents the number of partitions for each input. The partitionP function recursively calculates the number of partitions for a given input. It uses the memo function to access the stored results.

The partitionP function uses a formula to calculate the number of partitions. It sums up the product of signs and terms. The signs and terms are calculated using the signs and terms functions, respectively.

The signs function generates a cycle of alternating 1s and -1s. The terms function calls the memo function to retrieve the number of partitions for smaller inputs.

Main Function

The main function calls the partitionP function with an input of 6666 and prints the result.

{-# LANGUAGE DeriveFunctor #-}

------------------------------------------------------------
-- memoization utilities

data Memo a = Node a (Memo a) (Memo a)
  deriving (Functor)

memo :: Integral a => Memo p -> a -> p
memo (Node a l r) n
  | n == 0 = a
  | odd n = memo l (n `div` 2)
  | otherwise = memo r (n `div` 2 - 1)

nats :: Memo Int
nats =
  Node
    0
    ((+ 1) . (* 2) <$> nats)
    ((* 2) . (+ 1) <$> nats)

------------------------------------------------------------
-- calculating partitions

partitions :: Memo Integer
partitions = partitionP <$> nats

partitionP :: Int -> Integer
partitionP n
  | n < 2 = 1
  | otherwise = sum $ zipWith (*) signs terms
  where
    terms =
      [ memo partitions (n - i)
        | i <- takeWhile (<= n) ofsets
      ]
    signs = cycle [1, 1, -1, -1]

ofsets :: [Int]
ofsets = scanl1 (+) $ mix [1, 3 ..] [1, 2 ..]
  where
    mix a b = concat $ zipWith (\x y -> [x, y]) a b

main :: IO ()
main = print $ partitionP 6666