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**1 - 2**of**2**### 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 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...

### The Subset Partial Order: . . .

"... 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 ∑ ..."

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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 ∑