Elementmost

Elementmost is a term used to describe the element of a collection that is maximal with respect to a specified criterion. The concept is commonly used in optimization and data analysis, where each element is assigned a score by a function f.

Formal definition: Let S be a finite set and f: S → R a real-valued scoring function. An element x in S is called the elementmost of S with respect to f if f(x) ≥ f(y) for all y in S. When multiple elements share the maximal score, a predetermined tie-breaking rule determines the elementmost (for example, the smallest index).

Relation to existing terms: The elementmost is equivalent to the argmax of f on S; in practice, "elementmost" is a casual label for the top-scoring element.

Computation: Computing the elementmost involves evaluating f on each element and keeping track of the current maximum. This linear scan runs in O(n) time and uses O(1) extra space beyond storage of the set and the scores. For streaming data, a running maximum can be maintained, updating the elementmost as new elements arrive.

Variants and limitations: The concept assumes a well-defined maximum. If f has no maximum or S is infinite without additional constraints, an elementmost may not exist. In finite sets with a real-valued f, a maximum always exists, and an elementmost can be determined given a tie-breaking rule.

Examples: In a list of items with scores A: 3.2, B: 4.1, C: 4.1, the elementmost is either B or C depending on the tie-breaker.

Etymology and usage: The term is a neologism and not standard in formal literature; it appears mainly in informal discussions of selection by a score function. See also argmax, maximum, top element.