Iterateddistinct

Iterateddistinct is a concept encountered in discrete mathematics and programming that describes repeatedly applying a distinct-operation to the elements of a sequence. The goal is to obtain a stabilized sequence in which all elements are pairwise distinct.

Formal idea: Let a sequence be given, possibly with duplicates. The distinct operator D extracts from the sequence the subsequence consisting of the first occurrence of each value, preserving the original order. Iterating: a(0) is the initial sequence, and a(n+1) = D(a(n)). The process continues until a(n+1) equals a(n), which occurs when the sequence contains only distinct elements.

Convergence and fixed points: For finite initial sequences, the iteration reaches a fixed point after at most as many steps as there are distinct values in the initial sequence. The fixed point is the first-occurrence deduplicated version of the original sequence. For infinite sequences, convergence depends on the limiting behavior of the values, so practical use often relies on a stopping criterion or a bounded iteration.

Algorithms and variants: In practice, D can be implemented with a single pass using a hash set to track seen values, producing the sequence of first occurrences in linear time. Iterating D on finite sequences typically yields the same result after the first application, making additional iterations mostly of theoretical interest. Variants may define D to keep the last occurrence or apply alternative tie-breaking rules, leading to different fixed points or cycles.

Applications: Iterateddistinct informs data cleaning, sequence canonicalization, and preprocessing steps where deduplication with preserved order is desired. It also appears in theoretical discussions of fixed-point operators and iterative simplification processes.

See also: Deduplication, stable ordering, fixed point iteration.