An efficient implicit data structure for relation testing and searching in partially ordered sets

In this paper, we introduce an implicit data structure which represents a forest-structured partial order to efficiently perform, with respect to time and space, the following operations: 1) testing the relation among two elements (checking whether the two elements are related) and 2) given an element u, searching for its father. The first operation can be performed in constant time, while the second one requires polylog time (logarithmic in the case of bounded degree). The data structure represents the order relation by referring only to internal nodes of the forest, thus achieving in many cases a significant saving in space occupation. Finally, the algorithm is shown to be optimal in a restricted computation model by deriving a lower bound on the space complexity within such a model