# Computing depth orders and related problems

Pankaj K. Agarwal, Matthew J. Katz, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

## Abstract

Let K: be a set of n non-intersecting objects in 3-space. A depth order of K, if exists, is a linear order < of the objects in K such that if K, L ε K: and K lies vertically below L then K < L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If K is a set of n triangles whose xy-projections are all ‘fat’, then a depth order for K: can be computed in time O(n log6 n). (ii) If K: is a set of n convex and simply-shaped objects whose xy-projections are all ‘fat’ and their sizes axe within a constant ratio from one another, then a depth order for K: can be computed in time O(nλs1/2 12 (n)log4 n), where s is the maximum number of intersections between the xy-projections of the boundaries of any pair of objects in/C.

Original language English Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings Erik M. Schmidt, Sven Skyum Springer Verlag 1-12 12 9783540582182 https://doi.org/10.1007/3-540-58218-5_1 Published - 1994 4th Scandinavian Workshop on Algorithm Theory, SWAT 1994 - Aarhus, DenmarkDuration: 6 Jul 1994 → 8 Jul 1994

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 824 LNCS 0302-9743 1611-3349

### Conference

Conference 4th Scandinavian Workshop on Algorithm Theory, SWAT 1994 Denmark Aarhus 6/07/94 → 8/07/94

### Funding

FundersFunder number
Israeli Academy of Sciences
U.S.-Israeli Binationa| Science Foundation
National Science FoundationCCR-93-01259, CCR-91-22103
German-Israeli Foundation for Scientific Research and Development

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