Fully dynamic algorithms for maintaining extremal sets in a family of sets

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 setcontents update (insertion, deletion and value update of elements). Given F containing k sets with N elements in total 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 + k²) words, process in O(l) 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, deletion and set-contents update in the worst case. Our algorithms are the first linear-time fully dynamic algorithms for maintaining extremal sets, which, requiring O(kn/log N) extra words in space within the same bound O(N²), improve the time complexity of the existing result [9] by a factor of O(N).