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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/imperative-programming/5/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Imperative programming / 5 / c / Solution by

Codex  0 2026-10-25

A sufficient representation invariant is:

Both arrays have length capacity, with $c a p a c i t y > 0$ and $0 \leq u se d \leq c a p a c i t y$ .
The active keys elements(0 until used) are pairwise distinct.
For each active index $k$ , representatives(k) equals some active key elements(j).
Every such representative is fixed: if representatives(k) == elements(j), then representatives(j) == elements(j).

The abstraction function is

re p = {e l e m e n t s (k) \mapsto re p rese n t a t i v es (k) ∣ 0 \leq k < u se d} .

(1)

The last two conditions ensure each representative is a genuine member of its own block and that the abstract function is idempotent function. Cells beyond used are unconstrained and ignored. When used == 0, these conditions hold vacuously and the abstraction function yields the empty map. This is an explicit-key version of quick-find, rather than a parent-pointer disjoint-set forest.

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