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Fully Dynamic Algorithms for Maintaining Extremal Sets in A Family of Sets
, 1995
"... 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 upda ..."
Abstract

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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 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 setcontents 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...
Parallel Algorithms for Fully Dynamic Maintenance of Extremal Sets in
"... 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, d ..."
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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 setcontents 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 setcontents 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 sublinear number of processors for fully dynamic maintenance of extremal sets of F . Keywords: CREW PRAM, dynamic a...