Abstract
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 ϵ R i. Region Ri 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 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 (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.
Original language | English |
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Title of host publication | 35th International Symposium on Computational Geometry (SoCG 2019) |
Editors | Gill Barequet, Yusu Wang |
Place of Publication | Dagstuhl, Germany |
Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Pages | 42:1-42:16 |
Number of pages | 16 |
Volume | 129 |
ISBN (Electronic) | 9783959771047 |
ISBN (Print) | 978-3-95977-104-7 |
DOIs | |
Publication status | Published - 1 Jun 2019 |
Event | 35th International Symposium on Computational Geometry, (SoCG2019) - Portland, United States Duration: 18 Jun 2019 → 21 Jun 2019 http://www.wikicfp.com/cfp/servlet/event.showcfp?eventid=80745©ownerid=35838 |
Publication series
Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 129 |
ISSN (Print) | 1868-8969 |
Conference
Conference | 35th International Symposium on Computational Geometry, (SoCG2019) |
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Abbreviated title | SoCG2019 |
Country/Territory | United States |
City | Portland |
Period | 18/06/19 → 21/06/19 |
Internet address |
Keywords
- And phrases preprocessing
- Entropy
- Imprecise points
- Proximity structures
- Sorting