Heap as array: Difference between revisions

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# There is an internal type '''heap item''' with two components:
# There is an internal type '''heap item''' with two components:
## a component named <math>key</math> of type <math>\mathcal{K}</math>
## a component named <math>key</math> of type <math>\mathcal{K}</math>
## a component named <math>ID</math>, which is a [[natural numbers|natural number]] and ranges from <math>1,...,N_\text{max}</math>.
## a component named <math>ID</math>, which is a [[Numbers#natural numbers|natural number]] and ranges from <math>1,...,N_\text{max}</math>.
# There is an array <math>TheHeap</math> with index range <math>1,...,N_\text{max}</math> and component type heap item. The '''currently stored items''' are <math>TheHeap[1],...,TheHeap[N]</math>.
# There is an array <math>TheHeap</math> with index range <math>1,...,N_\text{max}</math> and component type heap item. The '''currently stored items''' are <math>TheHeap[1],...,TheHeap[N]</math>.
# There is an array <math>Positions</math> with index range <math>1,...,N_\text{max}</math> and integral components from <math>\{1,...,N_\text{max}\}</math>.
# There is an array <math>Positions</math> with index range <math>1,...,N_\text{max}</math> and integral components from <math>\{1,...,N_\text{max}\}</math>.
# There is an [[ordered sequence]] <math>Unused</math> of [[natural numbers]], which has length <math>N_\text{max}-N</math> and stores pairwise different numbers from <math>\{1,...,N_\text{max}\}</math>.
# There is an [[ordered sequence]] <math>Unused</math> of [[Numbers#natural numbers|natural numbers]], which has length <math>N_\text{max}-N</math> and stores pairwise different numbers from <math>\{1,...,N_\text{max}\}</math>.
# For each <math>i\in\{1,...,N_\text{max}\}</math> not in <math>Unused</math>:
# For each <math>i\in\{1,...,N_\text{max}\}</math> not in <math>Unused</math>:
## <math>Positions[i]</math> is the position of one of the <math>N</math> heap items in <math>TheHeap</math>. As long as this heap item is stored, <math>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>i</math> is one-to-one.
## <math>Positions[i]</math> is the position of one of the <math>N</math> heap items in <math>TheHeap</math>. As long as this heap item is stored, <math>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>i</math> is one-to-one.

Revision as of 17:44, 30 September 2014

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.

Remark

The implementations of the methods Bounded priority queue: number and Bounded priority queue: find minimum are trivial and, hence, left out here.

References