Computing depth orders and related problems

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

doi:10.1007/3-540-58218-5_1

Computing depth orders and related problems

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

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

9 Scopus citations

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(nλ_s^1/2 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
Title of host publication	Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings
Editors	Erik M. Schmidt, Sven Skyum
Publisher	Springer Verlag
Pages	1-12
Number of pages	12
ISBN (Print)	9783540582182
DOIs	https://doi.org/10.1007/3-540-58218-5_1
State	Published - 1994
Event	4th Scandinavian Workshop on Algorithm Theory, SWAT 1994 - Aarhus, Denmark Duration: 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)
Volume	824 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

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

Funding

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

Access to Document

10.1007/3-540-58218-5_1

Cite this

Agarwal, P. K., Katz, M. J., & Sharir, M. (1994). Computing depth orders and related problems. In E. M. Schmidt, & S. Skyum (Eds.), Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings (pp. 1-12). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 824 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-58218-5_1

Agarwal, Pankaj K. ; Katz, Matthew J. ; Sharir, Micha. / Computing depth orders and related problems. Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings. editor / Erik M. Schmidt ; Sven Skyum. Springer Verlag, 1994. pp. 1-12 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Computing depth orders and related problems",

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 {\textquoteleft}fat{\textquoteright}, 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 {\textquoteleft}fat{\textquoteright} 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.",

author = "Agarwal, {Pankaj K.} and Katz, {Matthew J.} and Micha Sharir",

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Agarwal, PK, Katz, MJ & Sharir, M 1994, Computing depth orders and related problems. in EM Schmidt & S Skyum (eds), Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 824 LNCS, Springer Verlag, pp. 1-12, 4th Scandinavian Workshop on Algorithm Theory, SWAT 1994, Aarhus, Denmark, 6/07/94. https://doi.org/10.1007/3-540-58218-5_1

Computing depth orders and related problems. / Agarwal, Pankaj K.; Katz, Matthew J.; Sharir, Micha.
Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings. ed. / Erik M. Schmidt; Sven Skyum. Springer Verlag, 1994. p. 1-12 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 824 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - 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.

AB - 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.

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Agarwal PK, Katz MJ, Sharir M. Computing depth orders and related problems. In Schmidt EM, Skyum S, editors, Algorithm Theory – SWAT 1994 - 4th Scandinavian Workshop on Algorithm Theory, Proceedings. Springer Verlag. 1994. p. 1-12. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-58218-5_1

Computing depth orders and related problems

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