Rectangles are Better than Chains for Encoding Partially Ordered Sets
نویسندگان
چکیده
Partial orders have applications in various areas in computer science such as distributed systems, object oriented languages, knowledge representation systems and databases. In this paper we present a new technique to encode partially ordered sets or posets. Specifically, we use the family of two-dimensional posets, which we refer to as rectangles, to encode a given poset; an element x is less than another element y in the partial order if and only if x is less than y in at least one of the rectangular orders and y is not less than x in any of the rectangular orders. The least number of rectangular orders needed to represent a poset is referred to as its rectangular dimension. We establish upper bounds on rectangular dimension of posets in general and special families of posets in particular. We also provide examples of families of posets that can be encoded optimally using our technique but require much larger number of bits per element using other techniques such as adjacency matrix, adjacency list and dimension theory.
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