Heap as array

General information

Abstract Data Structure: Bounded priority queue

Implementation Invariant:

1. For each object of "Heap as array", there is
   1. a specific maximum number of items [math]\displaystyle{ N_\text{max}\in\mathbb{N} }[/math], which is constant throughout the object's lifetime.
   2. a current number of items [math]\displaystyle{ N\in\mathbb{N}_{0} }[/math], which is dynamically changing, but it is [math]\displaystyle{ N\le N_\text{max} }[/math] at any time.
2. There is an internal type heap item with two components:
   1. a component named [math]\displaystyle{ key }[/math] of type [math]\displaystyle{ \mathcal{K} }[/math]
   2. a component named [math]\displaystyle{ ID }[/math], which is a natural number and ranges from [math]\displaystyle{ 1,...,N_\text{max} }[/math].
3. There is an array [math]\displaystyle{ TheHeap }[/math] with index range [math]\displaystyle{ 1,...,N_\text{max} }[/math] and component type heap item. The currently stored items are [math]\displaystyle{ TheHeap[1],...,TheHeap[N] }[/math].
4. There is an array [math]\displaystyle{ Positions }[/math] with index range [math]\displaystyle{ 1,...,N_\text{max} }[/math] and integral components from [math]\displaystyle{ \{1,...,N_\text{max}\} }[/math].
5. There is an ordered sequence [math]\displaystyle{ Unused }[/math] of natural numbers, which has length [math]\displaystyle{ N_\text{max}-N }[/math] and stores pairwise different numbers from [math]\displaystyle{ \{1,...,N_\text{max}\} }[/math].
6. For each [math]\displaystyle{ i\in\{1,...,N_\text{max}\} }[/math] not in [math]\displaystyle{ Unused }[/math]:
   1. [math]\displaystyle{ Positions[i] }[/math] is the position of one of the [math]\displaystyle{ N }[/math] heap items in [math]\displaystyle{ TheHeap }[/math]. As long as this heap item is stored, [math]\displaystyle{ i }[/math] is permanently associated with this heap item (and can hence be used to locate and access this heap item at any time). The correspondence between the heap items and these numbers [math]\displaystyle{ i }[/math] is one-to-one.
   2. [math]\displaystyle{ TheHeap[Positions[i]].ID=i }[/math].
7. Heap property: For each [math]\displaystyle{ i\in\{2,...,N\} }[/math], it is [math]\displaystyle{ TheHeap[i].key\ge TheHeap[\lfloor i/2 \rfloor ].key. }[/math].

In summary, [math]\displaystyle{ Positions }[/math] stores the positions of all heap items, [math]\displaystyle{ Unused }[/math] stores all currently unused indexes of [math]\displaystyle{ Positions }[/math], and the ID of each heap item refers back to its corresponding index in [math]\displaystyle{ Positions }[/math].

The heap property may be viewed as follows: Imagine a binary tree of [math]\displaystyle{ N }[/math] nodes such that each height level except for the last one is full, and the last one is filled from left to right. The nodes have pairwise different indexes from [math]\displaystyle{ \{ 1,...,N \} }[/math]:

1. The root is indexed [math]\displaystyle{ 1 }[/math].
2. For a node with [math]\displaystyle{ i }[/math], the left child (if existing) has index [math]\displaystyle{ 2*i }[/math], and the right child (if existing) has index [math]\displaystyle{ 2*i+1 }[/math].

Now associate the node with index [math]\displaystyle{ i }[/math] with the key [math]\displaystyle{ TheHeap[i].key }[/math]. Then the heap property says that the key of a node is not larger than the keys of its children. In this case, we say that i fulfills the heap property.

In particular, the minimum key is associated with the root.