Array list: number

Algorithmic problem: Ordered sequence: number

Prerequisites:

Type of algorithm: loop

Auxiliary data:

1. A pointer [math]\displaystyle{ p }[/math] of type "pointer to array list item of type [math]\displaystyle{ K }[/math]".
2. A counter [math]\displaystyle{ c \in \mathbb{N}_0 }[/math].

Abstract view

Invariant: After [math]\displaystyle{ i \ge 0 }[/math] iterations:

1. The pointer [math]\displaystyle{ p }[/math] points to the array list item at position [math]\displaystyle{ i+1 }[/math] (or is void if there is no such item).
2. The value of [math]\displaystyle{ c }[/math] is the sum of the values n of all array list items at positions [math]\displaystyle{ 1,...,i }[/math].

Variant: [math]\displaystyle{ p }[/math] is moved one step forward so as to point to the next array list item.

Break condition: It is [math]\displaystyle{ p= }[/math] void.

Induction basis

Abstract view: Initialize [math]\displaystyle{ p }[/math] so as to point to the first array list item.

Implementation:

1. Set [math]\displaystyle{ p:= }[/math]first.
2. Set [math]\displaystyle{ c:=0 }[/math].

Proof: Nothing to show.

Induction step

Abstract view: Add the number of elements in the current array and then go forward.

Implementation:

1. If [math]\displaystyle{ p= }[/math]void, terminate the algorithm and return the value of [math]\displaystyle{ c }[/math].
2. Otherwise:
   1. Set [math]\displaystyle{ c:=c+p }[/math].n.
   2. Set [math]\displaystyle{ p:=p }[/math].next.

Correctness: Nothing to show.

Complexity

Statement: Linear in the length of the sequence in the worst case (if the arrays of all array list items have identical lengths).

Proof: Obvious.