Efficient heaps in purely functional languages

Question

As an exercise in Haskell, I'm trying to implement heapsort. The heap is usually implemented as an array in imperative languages, but this would be hugely inefficient in purely functional languages. So I've looked at binary heaps, but everything I found so far describes them from an imperative viewpoint and the algorithms presented are hard to translate to a functional setting. How to efficiently implement a heap in a purely functional language such as Haskell?

Edit: By efficient I mean it should still be in O(n*log n), but it doesn't have to beat a C program. Also, I'd like to use purely functional programming. What else would be the point of doing it in Haskell?

Answer 1

I tried to port standard binary heap into functional settings. There is an article with described idea: A Functional Approach to Standard Binary Heaps. All the source code listings in the article are in Scala. But it might be ported very easy into any other functional language.

Answer 2

Arrays in Haskell aren't as hugely inefficient as you might think, but typical practice in Haskell would probably be to implement this using ordinary data types, like this:

data Heap a = Empty | Heap a (Heap a) (Heap a)
fromList :: Ord a => [a] -> Heap a
toSortedList :: Ord a => Heap a -> [a]
heapSort = toSortedList . fromList

If I were solving this problem, I might start by stuffing the list elements into an array, making it easier to index them for heap creation.

import Data.Array
fromList xs = heapify 0 where
  size = length xs
  elems = listArray (0, size - 1) xs :: Array Int a
  heapify n = ...

If you're using a binary max heap, you might want to keep track of the size of the heap as you remove elements so you can find the bottom right element in O(log N) time. You could also take a look at other types of heaps that aren't typically implemented using arrays, like binomial heaps and fibonacci heaps.

A final note on array performance: in Haskell there's a tradeoff between using static arrays and using mutable arrays. With static arrays, you have to create new copies of the arrays when you change the elements. With mutable arrays, the garbage collector has a hard time keeping different generations of objects separated. Try implementing the heapsort using an STArray and see how you like it.

Answer 3

As an exercise in Haskell, I implemented an imperative heapsort with the ST Monad.

{-# LANGUAGE ScopedTypeVariables #-}

import Control.Monad (forM, forM_)
import Control.Monad.ST (ST, runST)
import Data.Array.MArray (newListArray, readArray, writeArray)
import Data.Array.ST (STArray)
import Data.STRef (newSTRef, readSTRef, writeSTRef)

heapSort :: forall a. Ord a => [a] -> [a]
heapSort list = runST $ do
  let n = length list
  heap <- newListArray (1, n) list :: ST s (STArray s Int a)
  heapSizeRef <- newSTRef n
  let
    heapifyDown pos = do
      val <- readArray heap pos
      heapSize <- readSTRef heapSizeRef
      let children = filter (<= heapSize) [pos*2, pos*2+1]      
      childrenVals <- forM children $ \i -> do
        childVal <- readArray heap i
        return (childVal, i)
      let (minChildVal, minChildIdx) = minimum childrenVals
      if null children || val < minChildVal
        then return ()
        else do
          writeArray heap pos minChildVal
          writeArray heap minChildIdx val
          heapifyDown minChildIdx
    lastParent = n `div` 2
  forM_ [lastParent,lastParent-1..1] heapifyDown
  forM [n,n-1..1] $ \i -> do
    top <- readArray heap 1
    val <- readArray heap i
    writeArray heap 1 val
    writeSTRef heapSizeRef (i-1)
    heapifyDown 1
    return top

btw I contest that if it's not purely functional then there is no point in doing so in Haskell. I think my toy implementation is much nicer than what one would achieve in C++ with templates, passing around stuff to the inner functions.

Answer 4

And here is a Fibonacci Heap in Haskell:

https://github.com/liuxinyu95/AlgoXY/blob/algoxy/datastruct/heap/other-heaps/src/FibonacciHeap.hs

Here are the pdf file for some other k-ary heaps based on Okasaki's work.

https://github.com/downloads/liuxinyu95/AlgoXY/kheap-en.pdf

Answer 5

Here is a page containing an ML version of HeapSort. It's quite detailed and should provide a good starting point.

http://flint.cs.yale.edu/cs428/coq/doc/Reference-Manual021.html

Answer 6

There are a number of Haskell heap implementations in an appendix to Okasaki's Purely Functional Data Structures. (The source code can be downloaded at the link. The book itself is well worth reading.) None of them are binary heaps, per se, but the "leftist" heap is very similar. It has O(log n) insertion, removal, and merge operations. There are also more complicated data structures like skew heaps, binomial heaps, and splay heaps which have better performance.