Merge
General Information
Algorithmic problem: Merging two sorted sequences
Type of algorithm: loop
Auxiliary data: There are two current positions: [math]\displaystyle{ i_1 \in \{0,\dots,|S_1|\} }[/math] and [math]\displaystyle{ i_2 \in \{0,\dots,|S_2|\} }[/math] .
Abstract View
Invariant: Before and after each iteration:
- [math]\displaystyle{ S }[/math] consists exactly of all elements [math]\displaystyle{ S_1[1],\dots,S_1[i_1] }[/math] and [math]\displaystyle{ S_2[1],\dots,S_2[i_2] }[/math].
- [math]\displaystyle{ S }[/math] is sorted according to the comparison on [math]\displaystyle{ S_1 }[/math] and [math]\displaystyle{ S_2 }[/math].
Variant: [math]\displaystyle{ i_1 + i_2 }[/math] increases by [math]\displaystyle{ 1 }[/math]; neither [math]\displaystyle{ i_1 }[/math] nor [math]\displaystyle{ i_2 }[/math] decreases.
Break condition: [math]\displaystyle{ i_1 = |S_1| }[/math] and [math]\displaystyle{ i_2 = |S_2| }[/math].
Induction Basis
Abstract view: [math]\displaystyle{ i_1 := 0 }[/math] and [math]\displaystyle{ i_2 := 0 }[/math].
Implementation: Obvious.
Proof: Nothing to show.
Induction Step
Abstract view:
- If [math]\displaystyle{ i_1 = |S_1| }[/math] and [math]\displaystyle{ i_2 = |S_2| }[/math], terminate the algorithm.
- Otherwise, if [math]\displaystyle{ i_1 = |S_1| }[/math], append [math]\displaystyle{ S_2[i_2 + 1] }[/math] at the end of [math]\displaystyle{ S }[/math] and increase [math]\displaystyle{ i_2 }[/math] by [math]\displaystyle{ 1 }[/math].
- Otherwise, if [math]\displaystyle{ i_2 = |S_2| }[/math], append [math]\displaystyle{ S_1[i_1 + 1] }[/math] at the end of [math]\displaystyle{ S }[/math] and increase [math]\displaystyle{ i_1 }[/math] by [math]\displaystyle{ 1 }[/math].
- Otherwise, if [math]\displaystyle{ S_1[i_1 +1] \lt S_2[i_2 +1] }[/math], append [math]\displaystyle{ S_1[i_1 +1] }[/math] at the of [math]\displaystyle{ S }[/math] and increase [math]\displaystyle{ i_1 }[/math] by [math]\displaystyle{ 1 }[/math].
- Otherwise, append [math]\displaystyle{ S_2[i_2 +1] }[/math] at the of [math]\displaystyle{ S }[/math] and increase [math]\displaystyle{ i_2 }[/math] by [math]\displaystyle{ 1 }[/math].
Implementation: Obvious.
Correctness: Obvious.