Posets, boolean representations and quick path searching
The aim of this paper is twofold: first, the concept of boolean dimension of posets is introduced and its relevance is discussed in the representation of posets and transitive closure of dags by means of labeling of nodes and boolean formulae defined on order predicates between pairs of labels.
Second, such concept is used in the framework of efficient searching for a path between two nodes in a given acyclic digraph G (with n nodes and m arcs), obtaining a query time 0(k·l·lgcn) and a corresponding space complexity 0(k(n+m lgn)lgcn), where c is a suitable constant, k≤n is a parameter related to the length of the formula and l is the length of the resulting path, which behaves favorably compared to the best known results in the case of k bounded by √n: such result is obtained via the use of a data structure derived from Segment trees and Cartesian trees.
Some interesting classes of dags are finally presented for which query time and space complexity 0(l·lgcn) and 0((n+m lgcn) respectively can be obtained.
Finally, as a side result, a simple algorithm for path searching for dags with indegree or outdegree bounded by a constant is presented which has time complexity 0(f·l) and space complexity 0(n·t+f), where f is the length of the boolean formula and t is the number of variables in the formula.
KeywordsTime Complexity Space Complexity Linear Extension Transitive Closure Priority Queue
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