• Both arrays have length capacity, with and .
  • The active keys elements(0 until used) are pairwise distinct.
  • For each active index , 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
(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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