I/O-Efficient Algorithms for Computing Contours on a Terrain

A terrain M is the graph of a bivariate function. We assume that M is represented as a triangulated surface with N vertices. A contour (or isoline) of M is a connected component of a level set of M. Generically, each contour is a closed polygonal curve; at “critical” levels these curves may touch each other or collapse to a point. We present I/Oefficient algorithms for the following two problems related to computing contours of M: (i) Given a sequence `1 < · · · < `s of real numbers, we present an I/O-optimal algorithm that reports all contours of M at heights `1, . . . , `s using O(sort(N) + T/B) I/Os, where T is the total number edges in the output contours, B is the “block size,” and sort(N) is the number of I/Os needed to sort N elements. The algorithm uses O(N/B) disk blocks. Each contour is generated individually with its composing segments sorted in clockwise or counterclockwise order. Moreover, our algorithm generates information on how the contours are nested. (ii) We can preprocess M, using O(sort(N)) I/Os, into a linear-size data structure so that all contours at a given height can be reported using O(logB N + T/B) I/Os, where T is the output size. Each contour is generated individually with its composing segments sorted in clockwise or counterclockwise order. ∗The first and fourth authors are supported by NSF under grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, by an NIH grant 1P50-GM-08183-01, by a DOE grant OEGP200A070505, and by a grant from the U.S.–Israel Binational Science Foundation. The second and third authors are supported in part by the US Army Research Office through grant W911NF-04-01-0278, by an Ole Roemer Scholarship from the Danish National Science Research Council, a NABIIT grant from the Danish Strategic Research Council, and by the Danish National Research Foundation, and in part by MADALGO: Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SCG’08, June 9–11, 2008, College Park, Maryland, USA. Copyright 2008 ACM 978-1-60558-071-5/08/04 ...$5.00.