An Efficient Data Structure for Lattice Operations

Abstract

In this paper, we consider the representation and management of an element set on which a lattice partial order relation is defined. In particular, let n be the element set size. We present an \nradn-space implicit data structure for performing the following set of basic operations: 1. Test the presence of an order relation between two given elements, in constant time. 2. Find a path between two elements whenever one exists, in O(l) steps, where l is the path length. 3. Compute the successors and/or predecessors set of a given element, in O(h) steps, where h is the size of the returned set. 4. Given two elements, find all elements between them, in time O(k log d), where k is the size of the returned set and d is the maximum in-degree or out-degree in the transitive reduction of the order relation. 5. Given two elements, find the least common ancestor and/or the greatest common successor in $O(\sqrt{n})$-time. 6. Given k elements, find the least common ancestor and/or the greatest common successor in $O(\sqrt{n}+k \log n)$time. (Unless stated otherwise, all logarithms are to the base 2.) The preprocessing time is O(n2 ). Focusing on the first operation, representing the building-box for all the others, we derive an overall \nradn-space\,$\times$\,time bound which beats the order n2 bottleneck representing the present complexity for this problem. Moreover, we will show that the complexity bounds for the first three operations are optimal with respect to the worst case. Additionally, a stronger result can be derived. In particular, it is possible to represent a lattice in space $O(n\sqrt{t})$, where t is the minimum number of disjoint chains which partition the element set.