Heap as array: decrease key

Algorithmic problem: Bounded priority queue: decrease key

Prerequisites:

Type of algorithm: loop

Auxiliary data:

1. A current index [math]\displaystyle{ k \in \mathbb{N} }[/math].
2. An auxiliary index [math]\displaystyle{ k' \in \mathbb{N} }[/math].

Abstract view

Invariant: Before and after each iteration:

1. The heap property is fulfilled for all heap items except for the one at position [math]\displaystyle{ \lfloor k/2 \rfloor }[/math] (for that one, the heap property may or may not be fulfilled).

Variant: [math]\displaystyle{ k }[/math] is at least halved.

Break condition: One of the following two conditions is fulfilled:

1. either it is [math]\displaystyle{ k = 1 }[/math];
2. or, otherwise, the heap property is fulfilled by the heap item at position [math]\displaystyle{ \lfloor k/2 \rfloor }[/math].

Induction basis

Abstract view: The heap item is identified and its key is replaced by [math]\displaystyle{ x }[/math].

Implementation:

1. Set [math]\displaystyle{ k := Positions[ID] }[/math].
2. Set [math]\displaystyle{ TheHeap[k].key := x }[/math].

Proof: Nothing to show.