Some previously proposed algorithms are re-examined. They were designed to find all sets in a collection that have no subset in the collection, but are easily modified to find all sets that have no supersets. One is shown to have a worst-case running-time of O(N 2 = log N ), where N is the sum of the sizes of all the sets. This is lower than the only previously known sub-quadratic worst-case upper bound for this problem. Key words: Analysis of algorithms, set-theoretic algorithms, extremal sets. 1 Introduction Yellin and Jutla [3] tackled the following fundamental problem, for some applications of which see [2]. Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain. A set is a minimal (resp. maximal) set of F iff it has no strict subset (resp. superset) in F . Find the extremal sets of F , i.e., those that are minimal or maximal. With the problem size chosen as N = P i S i , Yellin and Jutla presented an abstract algorithm that requires O(N ...

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...

. An algorithm due to Bengalloun that continuously enumerates the primes is adapted to give the first prime number sieve that is simultaneously sublinear, additive, and smoothly incremental: -- it employs only \Theta(n= log log n) additions of numbers of size O(n) to enumerate the primes up to n, equalling the performance of the fastest known algorithms for fixed n; -- the transition from n to n + 1 takes only O(1) additions of numbers of size O(n). (On average, of course, O(1) such additions increase the limit up to which all primes are known from n to n + \Theta(log log n)). 1 Introduction A so-called "formula" for the i'th prime has been a long-lived concern, if not quite the Holy Grail, of Elementary Number Theory. This concern seems poorly motivated, as evidenced by the extraordinary freak-show of solutions proffered over the ages. The natural setting is Algorithmic Number Theory, and what is desired is much better cast as an algorithm to compute the i'th prime. Given that app...

The extremal sets of a family F of sets consist of all minimal and maximal sets of F that have no subset and superset in F respectively. We consider the problem of efficiently maintaining all extremal sets in F when it undergoes dynamic updates including set insertion, deletion and set-contents update (insertion, deletion and value update of elements). Given F containing k sets with N elements and domain (the union of these sets) size n, where clearly k; n N for any F , we present a set of algorithms that, requiring a space of O(N + kn log N + k 2 ) words, process in O(1) time a query on whether a set of F is minimal and/or maximal, and maintain all extremal sets of F in O(N ) time per set insertion in the worst case, deletion and set-contents update. Both time bounds are tight. Our algorithms are the first known fully dynamic algorithms that answer an extremal set query in constant time and maintain extremal sets in linear time for any set insertion and deletion. Keywords: Dy...

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...

Let F be a family of sets containing N elements. The extremal sets of F are those that have no subset or superset in F and are hence minimal or maximal respectively. We consider the problem of maintaining all extremal sets in F in parallel when F undergoes dynamic updates including set insertion, deletion and set-contents update (insertion, deletion and value update of elements). We present a set of parallel algorithms that, using O( N log N ) processors on a CREW PRAM, maintain all extremal sets of F in O(logN ) time per set insertion, deletion and set-contents update in the worst case. We also show that a batch of q queries on whether a set of F is minimal and/or maximal can be answered in O(1) time using q CREW processors. With a cost matching the time complexity of the optimal sequential algorithm [7], our algorithms are the first known NC algorithms that use a sub-linear number of processors for fully dynamic maintenance of extremal sets of F . Keywords: CREW PRAM, dynamic a...

Yellin and Jutla [7] proposed an algorithm for the problem of finding the extremal sets in a family of sets containing N elements that can be implemented in O( N 2 log N ) time and O( N 2 log N ) space due to Pritchard [3] who also showed that an earlier algorithm can be adapted to solve the problem in O( N 2 log N ) time and O( N 2 log 2 N ) space. We show that this problem can be solved in O( N 2 log 2 N ) time and O( N 2 log 3 N ) space in the worst case when F is normal, thus present the first algorithm that reaches the lower bound both in time and space complexity for this case. Keywords: Complexity analysis, extremal set, partial order, set inclusion. 1 Introduction In a given family of sets F = fS 1 ; S 2 ; : : : ; S k g, where elements of S i are drawn from some finite domain, a set S i is said minimal (resp. maximal) if S j 6ae S i (resp. S i 6ae S j ) for all 1 j k [5]. The extremal sets of F consist of all the minimal and maximal sets of F . The proble...

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 ∑