Preprocessing Ambiguous Imprecise Points

Ivor Van Der Hoog, Irina Kostitsyna, Maarten Löffler, Bettina Speckmann, Michael Wagner
2019 International Symposium on Computational Geometry  
Let R = {R1, R2, . . . , Rn} be a set of regions and let X = {x1, x2, . . . , xn} be an (unknown) point set with xi ∈ Ri. Region Ri represents the uncertainty region of xi. 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.
more » ... 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 fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n log n) time such that we can sort X (if d = 1) and reconstruct a quadtree on X (if d ≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, 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. In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P . We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of Ω(A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight. Finally, our results imply that one can approximate the entropy of interval graphs in O(n log n) time, improving the O(n 2.5 ) bound by Cardinal et al. ACM Subject Classification Theory of computation → Design and analysis of algorithms the regions: for each region we count only the overlap with regions that appear earlier in the permutation. A proper technical definition of ambiguity can be found in Section 2. We also show how to compute a 3-approximation of the ambiguity in O(n log n) time. Ambiguity and entropy. In one dimension, R is a set of intervals and the ambiguity is linked to interval (and graph) entropy (refer to the full version for a definition), which in turn relates to the well-studied problem of sorting under partial information. Fredman [9] shows that if the only information we are given about a set of values is a partial order P , and e(P ) is the number of linear extensions (total orders compatible with) of P , then we need at least Ω(log e(P )) comparisons to sort the values. Brightwell and Winkler prove that computing the number of linear extensions e(P ) is #P -complete [1]. Hence efforts have concentrated on computing approximations, most notably via the concept of graph entropy as introduced by Körner [14] . Specifically, Khan and Kim [13] prove that log e(P ) = Θ(n · H(G)) where H(G) denotes the entropy of the incomparability graph G of the poset P . To the best of our knowledge there is currently no exact algorithm to compute H(G). Cardinal et al. [4] describe the fastest known algorithm to approximate H(G), which runs in O(n 2.5 ) time. Refer to the full version for a more in-depth discussion of sorting and its relation to graph entropy. We consider the special case where the partial order is induced by uncertainty intervals. We define the entropy H(R) of a set of intervals as the entropy of their intersection graph (which is also an incomparability graph) using the definition of graph entropy given by Körner. In this setting we prove that the ambiguity A(R) provides a constant-factor approximation of the interval entropy (see Section 2). Since we can compute a constant-factor approximation of the ambiguity in O(n log n) time, we can hence also compute a constant-factor approximation of the entropy of interval graphs in O(n log n) time, thereby improving the result by Cardinal et al. [4] for this special case.
doi:10.4230/lipics.socg.2019.42 dblp:conf/compgeom/HoogKLS19 fatcat:wgcoyvfydncrvhykoufl6baie4