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By the Central Element Theorem of Linial and Saks, it follows that for the problem of (generalised) searching in posets, the information-theoretic lower bound of log N comparisons (where N is the number of order-ideals in the poset) is tight asymptotically. We observe that this implies that the problem of (generalised) sorting in posets has complexity (n · log N) (where n is the number of elements in the poset). We present schemes for (efficiently) transforming a randomised generation procedure for central elements (which often exists for some classes of posets) into randomised procedures for approximately counting ideals in the poset and for testing if an arbitrary element is central.

Searching, sorting and randomised algorithms for Central Elements and ideal counting in posets