TY - JOUR

T1 - Computing depth orders for fat objects and related problems

AU - Agarwal, Pankaj K.

AU - Katz, Matthew J.

AU - Sharir, Micha

N1 - Funding Information:
Work on this paperb y the first authorh as been supportedb y NationalS cienceF oundationG rant CCR-93-0125a9n da n NYI award.W orkon this paperb y the thirda uthorh asb eens upportebd y NSF Grant CCR-91-22103b, y a Max-PlanckR esearchA ward,and by grantsf rom the U.S.-IsraeliB inationaSl cience Foundationt,h e IsraelS cienceF und administerebdy the Israeli Academyo f Sciencesa, nd the G.I.F., the German-IsraeFloi undatiofno r ScientificR esearchan dD evelopment. * Correspondinagu thor.

PY - 1995/11

Y1 - 1995/11

N2 - Let K be a set of n non-intersecting objects in 3-space. A depth order of K, if it exists, is a linear order < of the objects in K such that if K, L ε{lunate} 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 log5n). (ii) If K is a set of n convex and simply-shaped objects whose xy-projections are all 'fat' and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(nλs 1 2(n) log4n), where s is the maximum number of intersections between the boundaries of the xy-projections of any pair of objects in K, and λs(n) is the maximum length of (n,s) Davenport-Schinzel sequences.

AB - Let K be a set of n non-intersecting objects in 3-space. A depth order of K, if it exists, is a linear order < of the objects in K such that if K, L ε{lunate} 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 log5n). (ii) If K is a set of n convex and simply-shaped objects whose xy-projections are all 'fat' and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(nλs 1 2(n) log4n), where s is the maximum number of intersections between the boundaries of the xy-projections of any pair of objects in K, and λs(n) is the maximum length of (n,s) Davenport-Schinzel sequences.

UR - http://www.scopus.com/inward/record.url?scp=34547949769&partnerID=8YFLogxK

U2 - 10.1016/0925-7721(95)00005-8

DO - 10.1016/0925-7721(95)00005-8

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AN - SCOPUS:34547949769

VL - 5

SP - 187

EP - 206

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 4

ER -