Kotiinorder

Kotiinorder describes a class of constrained permutations of a finite set relative to a relation on the set. It is a notion used primarily in theoretical discussions to explore locality constraints in ordered arrangements.

It imposes locality: related elements must lie within a bounded distance in the resulting sequence.

Let X be finite and R a symmetric relation on X. For a positive integer k, a kotiinorder of (X,R,k) is a permutation π of X such that, for all a,b with aRb, the positions satisfy |posπ(a) - posπ(b)| ≤ k, where posπ(x) is the index of x in π.

This concept is closely related to the graph bandwidth problem. If one views R as the edge set of a graph on vertex set X, then a kotiinorder is a linear layout whose bandwidth does not exceed k.

Existence is not guaranteed; deciding whether a kotiinorder with bandwidth at most k exists for a given X, R, and k is the bandwidth decision problem, which is NP-complete for general graphs.

Origin and usage: The term is not standard in published literature and is mainly used in theoretical discussions or illustrative contexts. It serves as a convenient way to talk about locality-preserving orderings without committing to a specific formal framework.

Applications and relevance include data layout and cache-aware algorithms, where arranging related items close together can improve performance, and in graph layout optimization where locality constraints influence visualization and processing.

See also: graph bandwidth, bandwidth minimization, linear extension, graph layout.