A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal sets, i.e., extremal sets) in worst-case time O(N 2 = log N ). This paper develops a simple algorithm that uses only simple data structures, and gives a simple analysis that establishes the above worst-case bound on its running time. The algorithm exploits a variation on lexicographic order that may be of independent interest. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. Pritchard [4] presented algorithms for finding those sets in F that have no subset in F . Starting from a naive O(N 2 ) algorithm 1 , an algorithm was obtained that had worst-case complexity O...

A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worst-case time O(N 2 = log N ). This paper gives a variant implementation of a previously proposed algorithm which is shown to have a worst-case complexity of O(N 2 (log log N) 2 = log 2 N) operations on a RAM with \Theta(log N) bit words. This is the first known o(N 2 = log N) worst-case running time. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. In [5] we presented algorithms for finding those sets in F that have no subset in F , and obtained a fast algorithm for the important special case when all sets in F are small. A particular implementation was later shown [6] to have worst-cas...

Functional models are frequently used in computer vision and photogrammetry, as they enable the mathematical formulation of several problems such as pose computation and more generally the parameter estimation problem. However, the structural properties of such models have only seldom been studied. This contribution is dedicated to the analysis of such properties. We propose a new formalism that enables the analysis and design of functional models.

Given a family F of k sets with cardinalities s1, s2,..., sk and N = ∑k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O ( ∑ si≤B 2s ∑

Abstract. We propose an extension of the join calculus with pattern matching on algebraic data types. Our initial motivation is twofold: to provide an intuitive semantics of the interaction between concurrency and pattern matching; to define a practical compilation scheme from extended join definitions into ordinary ones plus ML pattern matching. To assess the correctness of our compilation scheme, we develop a theory of the applied join calculus, a calculus with value passing and value matching. We implement this calculus as an extension of the current JoCaml system. 1.