I/O-Efficient Algorithms for Computing Contour Maps on Terrains

A terrain M is the graph of a continuous 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 points. We present I/O-efficient 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 of 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. ∗Supported by NSF under grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-041-0278 and W911NF-07-1-0376, by an NIH grant 1P50-GM-08183-01, by a DOE grant OEG-P200A070505, and by a grant from the U.S.–Israel Binational Science Foundation. Supported by NSF under grants EIA-01-31905, CCR02-04118, and DEB-04-25465, by an ARO grant W911NF-04-1-0278, and by a grant from the U.S.–Israel Binational Science Foundation. †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. ‡Center for Massive Data Algorithmics — a Center of the Danish National Research Foundation. §Supported in part 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. ¶Work performed primarily when this author was at Duke University and supported by NSF under grants CNS05-40347, 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 OEG-P200A070505, and by a grant from the U.S.–Israel Binational Science Foundation. Supported by NSF under grants EIA-01-31905, CCR-02-04118, and DEB-04-25465, by an ARO grant W911NF-04-1-0278, and by a grant from the U.S.–Israel Binational Science Foundation.