The min-heap abstraction below is defined for any element type that is comparable. The following type signatures define our Heap interface:
type Heap a
empty : Heap comparable
isEmpty : Heap comparable -> Bool
findMin : Heap comparable -> Maybe comparable
deleteMin : Heap comparable -> Maybe (comparable, Heap comparable)
insert : comparable -> Heap comparable -> Heap comparable
merge : Heap comparable -> Heap comparable -> Heap comparableMax-heaps are defined similarly.
One way to represent a min-heap is as a complete binary tree that satisfies the min-heap property, which requires that every node be no bigger than its children. Complete binary trees can be implemented using arrays.
There is a scheme for the representing the nodes of a complete binary tree in breadth-first order…
Depth 0 0
Depth 1 1 2
Depth 2 3 4 5 6
. . . . . . . .… using an an array a:
i: 0 1 2 3 4 5 6 ...
----------------------------------
a[i]: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | ...
----------------------------------And then navigating the parent and child edges for a node stored in a[i] is easy:
a[(i-1)//2],a[(2*i)+1], anda[(2*i)+2].In imperative languages, retrieving an arbitrary element in an array takes O(1) time. In functional languages, purely functional arrays are often implemented in terms of balanced search trees, which typically provide O(log n) access to arbitrary elements.
Although we have not yet talked about how to implement balanced search trees, we will take as given an implementation of balanced search trees. In Elm, the Array library is implemented with a data structure called Relaxed Radix Trees, which provides O(log n) operations for getting and setting arbitrary indices.
We will make use of the Array library to implement the heap abstraction. If we choose to define and export
type alias Heap a = Array athen clients of our module will be able to see that we have implemented Heaps using Arrays and will have access to the Array values that we will use to represent Heaps. If we wish to hide the representation from clients, we can instead define a new type
type Heap a = Heap (Array a)and export only those functions that we want clients to use:
module Heaps exposing
(Heap, empty, isEmpty, findMin, deleteMin, insert merge)
...We can’t create a Heap value directly…
> import ArrayHeap exposing (..)
> import Array
> Heap Array.empty
-- NAMING ERROR ---------------------------------------------- repl-temp-000.elm
Cannot find variable `Heap`
6| Heap Array.empty
^^^^because the module declaration exposes only the Heap type and not the Heap data constructor. We could change the module declaration to expose the data constructors, too (notice the Heap(..)):
module Heaps exposing
(Heap(..), empty, isEmpty, findMin, deleteMin, insert merge)
...Then we would be able to use the Heap data constructor to directly create Heap values from outside the module:
> Heap Array.empty
Heap (Array.fromList []) : ArrayHeap.Heap aHowever, we choose not to expose the internal representation of the Heap type to the client. Furthermore, we will not provide any functions of type Heap comparable -> Array comparable that allow accessing the internal representation. So, clients will not be able to manipulate Heaps directly with Array operators.
Creating abstraction boundaries like this facilitate the software engineering process, by preventing clients from (intentionally or accidentally) violating invariants that the correctness of the module implementation depends on, as well as facilitating changes to the implementation by limiting and making explicit the boundaries between modules.
To implement the Heap abstraction, we will use the following Array operators with assumed bounds for their running times:
Array.empty : Array a -- O(1)
Array.isEmpty : Array a -> Bool -- O(1)
Array.length : Array a -> Int -- O(1)
Array.push : a -> Array a -> Array a -- O(log n)
Array.get : Int -> Array a -> Maybe a -- O(log n)
Array.set : Int -> a -> Array a -> Array a -- O(log n)
Array.slice : Int -> Int -> Array a -> Array a -- O(log n)The Array.push function creates a new array that is the same as the input array except that it contains an additional element at the end. Let’s “rename” it so that we remember which end it works with.
addToEnd : a -> Array a -> Array a
addToEnd = Array.pushFor convenience, we will define an analogous function called removeFromEnd that retrieves the last element and creates a new array without it.
removeFromEnd : Array a -> Maybe (a, Array a)
removeFromEnd array =
let n = Array.length array in
case Array.get (n-1) array of
Nothing -> Nothing
Just last -> Just (last, Array.slice 0 -1 array)We will start with the simple operators.
empty : Heap comparable
empty = Heap Array.empty
isEmpty : Heap comparable -> Bool
isEmpty (Heap array) =
Array.isEmpty arrayNotice that the definition of isEmpty uses pattern matching to, at once, deconstruct the argument value and bind the underlying array value to the variable array. This definition is equivalent to all of the following:
isEmpty h = case h of Heap array -> Array.isEmpty array
isEmpty = \h -> case h of Heap array -> Array.isEmpty array
isEmpty = \h -> let (Heap array) = h in Array.isEmpty array
isEmpty = \(Heap array) -> Array.isEmpty arrayBecause values of type Heap can only be constructed by the Heap data constructor, we can use patterns in bindings (rather than case expressions) and be sure that we have covered all patterns.
The findMin implementation is also straightforward.
findMin : Heap comparable -> Maybe comparable
findMin (Heap array) =
Array.get 0 arrayLet’s now look at the first non-trivial operator, inserting an element into a Heap. The idea is to add the element to the next position in the complete binary tree (conveniently represented as the last element in the array) and then “bubble” or “percolate” the element up the tree until it is no longer bigger than its parent.
insert : comparable -> Heap comparable -> Heap comparable
insert x (Heap array) =
array
|> addToEnd x
|> bubbleUp (Array.length array)
|> HeapAs we manipulate the underlying Array representations, we will make sure to always access elements within bounds. So we define an “unsafe” version of get:
justGet : Int -> Array a -> a
justGet i array =
case Array.get i array of
Nothing -> Debug.crash ("justGet: " ++ toString i)
Just x -> xWe also define a helper function to swap two elements in an Array:
swap : Int -> Int -> Array a -> Array a
swap i j array =
let ai = justGet i array in
let aj = justGet j array in
array |> Array.set i aj |> Array.set j aiThe bubbleUp function is defined to swap node i with its parent node (i-1)//2 if the parent is larger and, if so, recursively traverses up the tree. Notice how bubbleUp is a function that works directly with the internal representation and is not exposed to clients.
parentIdx i = (i-1) // 2
bubbleUp : Int -> Array comparable -> Array comparable
bubbleUp i array =
let
child = justGet i array
parent = justGet (parentIdx i) array
in
if parent <= child
then array
else array |> swap i (parentIdx i) |> bubbleUp (parentIdx i)Let’s now consider the worst-case time cost of the insertion algorithm. The insert function computes the length of the Array (which runs in O(1) time, by looking at the library implementation with native code), pushes an element on the end of it (which runs in worst-case O(log n) time) and calls bubbleUp.
The bubbleUp function makes use of justGet and swap. Because Array.get runs in O(log n) time, so does the wrapper function justGet. The swap function makes several calls to Array.set and justGet, each of which takes O(log n) time. Thus, swap takes O(log n) time. The bubbleUp function visits at most O(log n) elements because the index i is divided in half before each recursive call. Therefore, there are O(log n) calls to bubbleUp, each of which performs O(log n) work. So the running time of bubbleUp, and hence insert, is O(log2n). In an imperative language, where array operations take worst-case O(1) time, the insertion algorithm runs in worst-case O(log n) time.
To delete the minimum element, which is stored at index 0, we first overwrite the root with the value currently stored in the last position (conveniently stored in the last element of the Array). We pop the last element because its value is now stored in the root and then “bubble” or “percolate” this value down as long as necessary. When recursively bubbling down, we choose the child tree whose root is smaller in order to maintain the heap order property.
deleteMin : Heap comparable -> Maybe (comparable, Heap comparable)
deleteMin (Heap array) =
case removeFromEnd array of
Nothing -> Nothing
Just (lastElement, choppedArray) ->
let minElement = justGet 0 array in
let newArray =
choppedArray
|> Array.set 0 lastElement
|> bubbleDown 0
in
Just (minElement, Heap newArray)If choppedArray has no elements, then the minimum element is the last element, so there is nothing left to bubble down. Rather than checking for this case explicitly, however, we rely on the fact that Array.set and bubbleDown leave the array unchanged if the index argument (i.e. 0) is out of bounds.
For a given index i, the index of the left child is j = 2*i + 1 and of the right child as k = 2*i + 2.
leftIdx i = 2*i + 1
rightIdx i = 2*i + 2If there is no left child (in which case, neither is there a right one), then there is nothing left to bubble down:
bubbleDown i array =
if leftIdx i >= Array.length array then
arrayIf there is a left child but not left child, we need to compare the value of the left child to the current to decide whether or not to bubble down to the left:
else if rightIdx i >= Array.length array then
let
this = justGet i array
left = justGet (leftIdx i) array
in
if this <= left
then array
else bubbleDownLeft ()If both children are defined, there are several cases:
let
this = justGet i array
left = justGet (leftIdx i) array
right = justGet (rightIdx i) array
in
if this <= left && this <= right then array
else if left < this && this <= right then bubbleDownLeft ()
else if right < this && this <= left then bubbleDownRight ()
else {- left <= this && right <= this -}
if left <= right
then bubbleDownLeft ()
else bubbleDownRight ()In the above, we defined a few helpers to reduce some of the commonalities in bubbling down to the left and right in various cases:
let swapAndRecurse j = array |> swap i j |> bubbleDown j in
let bubbleDownLeft () = swapAndRecurse (leftIdx i) in
let bubbleDownRight () = swapAndRecurse (rightIdx i) in(Note: how we use functions with dummy arguments so that we can give names to expressions that we may want to evaluate in one or more subsequent expressions. Functions with dummy arguments, called thunks, are often useful in functional programming to “delay” the evaluation of an expression until later.)
Nevertheless, this has become overly verbose. Instead, we can define a single operation (called smaller below) to check if each child index is in bounds and, if so, whether its value is less than the others.
bubbleDown : Int -> Array comparable -> Array comparable
bubbleDown i array =
let n = Array.length array in
let smaller j acc =
if j < n && justGet j array < justGet acc array
then j
else acc
in
let smallest = i |> smaller (leftIdx i) |> smaller (rightIdx i) in
if i == smallest
then array
else array |> swap i smallest |> bubbleDown smallestIf the value at index i is smaller than both children, the output array is unchanged. Otherwise, the value at index i is swapped with the smaller value among the two subtrees and bubbleDown recurses on that subtree.
The analysis of bubbleDown and deleteMin is similar to the insertion algorithm, resulting in a O(log2n) worst-case time cost.
We will not go through the algorithm for merging heaps that are as arrays in this course. But you may be curious to go read about it if you have not seen it before.
In our Elm implementation, we will pretend that we implement the merge function faithfully, but instead we will always trigger a run-time error.
merge _ _ = Debug.crash "merge not implemented"The ArrayHeap.elm file contains the implementation above.
The ArrayHeapTest.elm file provides a tiny bit of testing.
> import ArrayHeapTest exposing (..)
> simpleHeapSort [100,123,235,235,1,999,998]
[1,100,123,235,235,998,999] : List numberIn general, it would be a good idea to design your own testing infrastructure to help sanity check your implementations of the various data structures we will study.
Recall that Elm provides the special polymorphic type variable comparable. We may want to define interfaces in terms of types that come with operators besides those specified in the comparable “type class.” Elm does not provide a way for the programmer to define an interface that describes a set of types.
In Haskell, user-defined type classes can be used to specify the heap abstraction as follows.
class Ord a where
(<) :: a -> a -> Bool
(>=) :: a -> a -> Bool
(>) :: a -> a -> Bool
(<=) :: a -> a -> Bool
max :: a -> a -> a
min :: a -> a -> a
class Ord a => Heap a where
empty : Heap a
isEmpty : Heap a -> Bool
findMin : Heap a -> Maybe a
insert : a -> Heap a -> Heap a
deleteMin : Heap a -> Maybe (a, Heap a)
merge : Heap a -> Heap a -> Heap aThe second class definition says that a type a “is a” Heap if (1) a “is a” Ord and (2) the six functions specified can be defined for that type. Particular implementations are then defined to “implement” the Heap “interface”.
type ArrayHeap a = Heap (Array a)
instance ArrayHeap (Heap a) where
insert = ...
...For simplicity, the definition above does not mention the Haskell Eq type class.
In Standard ML and OCaml, signatures are used to describe modules, which are structures that contain values and types.
signature ORDERED =
sig
type T
val eq : T -> T -> bool
val lt : T -> T -> bool
val leq : T -> T -> bool
end
signature HEAP =
sig
structure Elem : ORDERED
type H
val empty : H
val isEmpty : H -> bool
val insert : Elem.T -> H -> H
val merge : H -> H -> H
val findMin : H -> Elem.t option
val deleteMin : H -> (Elem.T * H) option
endAn implementation satisfies the Heap signature by defining a functor that takes a module as an argument and returns another module. Note that these ML functors are the not same thing as Haskell Functors.
functor MyHeap (Element : ORDERED) : HEAP =
struct
structure Elem = Element
datatype H = ...
fun insert = ...
...
end