Order Dependency

Synonyms

OD

Definition

Order dependencies (ODs) describe the relationship among lexicographical orderings of sets of tuples. An order specification is a list of attributes marked as asc (ascending) or desc (descending). This is the notion of order used in SQL and within query optimization, as by the order-by operator (nested sort). Let X = [A |T] be a list of marked attributes, the marked attribute A is the head of the list, and the marked list T is the tail. For two tuples s and t, s ≼_Xt iff X = [ ]; or X = [A asc|T] and s_A < t_A; or X = [A desc|T] and s_A > t_A; or X = [A asc|T] or X = [A desc|T], s_A = t_A, and s ≼_Tt. Let r be a table over a relation R that contains the attributes that appear in X and Y. Given two order specifications X and Y a table r satisfies an OD X ↦ Y, read as X orders Y, iff for all s, t ∈r, s ≼_Xt implies s ≼_Yt [8]. An OD is unidirectional when attributes within it are all marked as asc or desc [4, 6, 9, 7]. Unidirectional ODs are subsumed by ODs by definition.

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