nearadjacent

Nearadjacent is a term used to describe elements that are close to each other within a specified notion of proximity, but are not directly adjacent. It is used in discussions of locality and neighborhood in sequences, graphs, and spatial data. The term is not tied to a single formal discipline, but rather to a family of proximity relations defined by a parameter that bounds how far apart two elements can be to count as nearadjacent.

In a linear sequence, two elements a_i and a_j (i ≠ j) are nearadjacent if the index distance satisfies 1 ≤ |i − j| ≤ d, where d is a chosen positive integer. The value of d determines the scope of “nearness.” For example, with d = 2 in the sequence [A, B, C, D, E], the pairs (A, C), (B, D), and (C, E) are nearadjacent.

In a graph, nearadjacency can be defined via the shortest-path distance. Two vertices u and v are nearadjacent if the distance between them is at least 2 and at most d, excluding direct neighbors (distance 1) and leaving room for a defined neighborhood size. The concept extends to metric spaces by using a distance function and a radius or diameter parameter, again omitting immediate adjacency when desired.

Applications of nearadjacent relations include analyzing locality in algorithms, clustering nearby events in time series, cache-aware data processing, and defining local neighborhoods in graphs. The precise interpretation depends on the chosen proximity parameter, making nearadjacency a flexible concept across disciplines.

See also: adjacency, neighborhood, locality, proximity, k-nearest neighbor, graph distance.