Preprocessing Ambiguous Imprecise Points Let R  R_1, R_2, ..., R_n be a set of regions and let  X  x_1, x_2, ..., x_n be an (unknown) point set with x_i in R_i. Region R_i represents the uncertainty region of x_i. We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more finegrained measure of the degree of overlap in R. We show how to preprocess a set of ddimensional disks in O(n log n) time such that we can sort X (if d1) and reconstruct a quadtree on X (if dgeq 1 but constant) in O(A(R)) time. If A(R) is sublinear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time.  :200,one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the wellstudied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lowerbounded by the graph entropy of P. We show that if P is an interval order, then the ambiguity provides a constantfactor approximation of the graph entropy. This gives a lower bound of Omega(A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight.

A Fully Polynomial Time Approximation Scheme for the Smallest Diameter of Imprecise Points Abstract   Given a set    D        d    1    ,  xe2x80xa6  ,    d    n        of imprecise points modeled as disks, the minimum diameter problem is to locate a set    P        p    1    ,  xe2x80xa6  ,    p    n        of fixed points, where      p    i    xe2x88x88    d    i     , such that the furthest distance between any pair of points in P is as small as possible. This introduces a tight lower bound on the size of the diameter of any instance P. In this paper, we present a fully polynomial time approximation scheme (FPTAS) for computing the minimum diameter of a set of disjoint disks that runs in    O  (    n    2      xcfxb5    xe2x88x92  1    )    time. Then we relax the disjointness assumption and provide a    O  (    n    2      xcfxb5    xe2x88x92  2    )    randomize algorithm. We also show that our results can be generalized in      R    d      when the dimension  d  is an arbitrary fixed constant.