We consider the problems of deciding whether a given collection of polygons (polyhedra resp.) forms (i) a partition or (ii) a cell complex decomposition of a given polygon (polyhedron resp.). We describe simple O(n log n)-time
and O(n)-space algorithms for these problems, where n is the total description size of the input. If, in the input, vertices are referenced by means of indices to an array of distinct vertices, then our cell complex decomposition verification algorithms run in O(n) time

Palios, Leonidas

English

idUS. Depósito de Investigación Universidad de Sevilla

Verification of Partitions of 2d and 3d ObjectsLeonidas PaliosDepartment of Computer Science, University of Ioannina, 45110 Ioannina, GreeceAbstractWe consider the problems of deciding whether a given collection of polygons (polyhedra resp.) forms (i) a partitionor (ii) a cell complex decomposition of a given polygon (polyhedron resp.). We describe simple O(n log n)-timeand O(n)-space algorithms for these problems, where n is the total description size of the input. If, in the input,vertices are referenced by means of indices to an array of distinct vertices, then our cell complex decompositionverification algorithms run in O(n) time.1. IntroductionIn recent years, there has been growing interestin algorithms that enable us to check the outputof a program. This issue has been considered fromdifferent viewpoints, and the research yielded re-sults ranging from characterizations of problemsthat are checkable [2] to special-purpose checkers.In particular, in computational geometry, this is-sue proves to be crucial as the newer and more ef-ficient algorithms are more and more complicated.Usually, the authors of an algorithm for computingsome geometric object describe properties of theobject which can be used to verify the correctnessof the computation (see e.g. [1] and [4] for 2d and3d triangulations of point sets). Often, however,more machinery is needed.The first verification algorithms (“checkers”)were provided by Mehlhorn et al. who defined theproperties that a checker should have (correct-ness, simplicity, efficiency) and described check-ers for convex polyhedra and convex hulls [5];they have also studied checkers for trapezoidaldecompositions, planar point location structuresfor trapezoidal decompositions, and Voronoi andDelaunay diagrams. Devillers et al. extended thenotion of checkers and provided checkers for con-vex polytopes in two and higher dimensions, andfor various types of planar subdivisions [3]. Theiralgorithms run in linear time, assuming that theinput is a 2d or a 3d ordered geometric graph.Email address: palios@cs.uoi.gr (Leonidas Palios).In this paper, we present simple and efficientverification algorithms for partitions and cellcomplex decompositions of simple polygons andpolyhedra. The algorithms rely on appropriatelymatching the edges of the 2d decompositions andthe facets of the 3d decompositions, and run inO(n log n) time using O(n) space, where n is thetotal size of the input. If, in the input, vertices arereferenced by means of indices to an array of dis-tinct vertices, then our cell complex decompositionverification algorithms run in optimal O(n) time.Note: In the following, a polygon or a polyhedronis understood to be a simple one.2. The Two-dimensional CaseWe assume that each polygon in the input isdescribed by the sequence of its vertices along itsboundary, where each vertex is given by its coor-dinates.2.1. Verification of a partition of a polygonLemma 1 Let P and C be a polygon and a collec-tion of polygons respectively. Additionally, let EI ={e− ∂P | e is an edge of a polygon in C} and EB ={e∩∂P | e is an edge of a polygon in C}. Then, thecollection C forms a partition of P if and only if(i) the set EI of (parts of) edges of the polygons inC can be partitioned into sets S and S′ such that•⋃s∈Sclosure(s) =⋃s∈S′closure(s),20th EWCG Seville, Spain (2004)20th European Workshop on Computational Geometry• the closures of the segments in S and S′ definea partition of⋃s∈Sclosure(s), and• for each (part of) edge e in S (resp. S′), if S′[e](resp. S[e]) is the set of segments of S′ (resp. S)that are collinear with and intersect e, then,locally around e, the interior of the polygon inC with edge e and the interiors of the polygonsin C whose edges contributed the elements ofS′[e] (resp. S[e]) lie on opposite sides of theline supporting e;(ii) the closures of the segments in EB form a parti-tion of the boundary ∂P of P , and for each (partof) edge e in EB, locally around e, the interior ofthe polygon in C with e as an edge and the inte-rior of P lie on the same side with respect to theline supporting e.The partition verification algorithm appliesLemma 1. It uses an (initially empty) array Aof size equal to the total number of edges of thepolygons in C and of the polygon P .2d Partition Verification Algorithm1. Orient the boundary of polygon P and theboundaries of the polygons in C in a compat-ible fashion (e.g., the boundary is traversedin a ccw fashion).2. For each polygon Q in C dofor each edge−→uv of Q doif u is lex-smaller than vthen add the entry (u, v, +1) in A;else add the entry (v, u,−1) in A;3. For each edge−→uv of P doif u is lex-smaller than vthen add the entry (u, v,−1) in A;else add the entry (v, u, +1) in A;4. Sort the entries (ui, vi,±1) of the array A byslope of the line uivi and, in case of ties, lex-icographically taking into account the coor-dinates of the vertices ui and vi.5. For each slope separately, traverse the relatedentries of A verifying that those with “+1”and those with “−1” partition the same setof segments. If this terminates successfully,then the collection C of polygons defines apartition of the polygon P , otherwise it doesnot.Time complexity. Steps 1, 2, and 3 can be com-pleted in time linear in the total description size nof the input, while Step 4 takes O(n log n) time.Step 5 takes O(n) time: for each slope, the sort-ing of the entries implies that each new entry with“+1” either is the extension of the latest entry with“+1” or starts a new segment in the union of the“+1”-entries (if not, C does not form a partition ofP ), and similarly for the “−1”-entries. The algo-rithm uses O(n) space.2.2. Verification of a cell complex decompositionof a polygonLemma 2 Let P and C be a polygon and a collec-tion of polygons respectively. Then, the collection Cforms a cell complex decomposition of P if and onlyif the set of edges of the polygons in C can be parti-tioned into two sets EI and EB such that(i) for each edge e in EI there exists exactly oneother edge, say d, in EI such that e = d and,locally around e, the interiors of the polygons inC with edges e and d lie on opposite sides of theline supporting e;(ii) the edges in EB form a partition of the bound-ary ∂P of P and for each edge e in EB, locallyaround e, the interior of the polygon in C withedge e and the interior of P lie on the same sidewith respect to the line supporting e.The cell complex verification algorithm appliesLemma 2; it too uses an (initially empty) array Aof size equal to the total number of edges of thepolygons in C.2d Cell Complex Verification Algorithm1. Collect all the vertices and assign to each oneof them a distinct positive integer id( ).2. Orient the boundary of polygon P and theboundaries of the polygons in C in a compa-tible fashion.3. For each polygon Q in C dofor each edge−→uv of Q doif u is lex-smaller than vthen add (id(u), id(v), +1) in A;else add (id(v), id(u),−1) in A;4. Sort the array A lexicographically.5. Traverse the sorted array A deleting match-ing entries (i.e., (k, k′,−1) and (k, k′, +1)),which now appear next to each other.6. Collect the unmatched entries and by ap-plying a depth-first traversal construct thegraph that these entries form. If this graphis a closed path identical to the boundary ofthe polygon P , then the collection C of poly-March 25-26, 2004 Seville (Spain)gons defines a cell complex decomposition ofP , otherwise it does not.Time complexity. If n is the total description sizeof the input, Step 1 can be executed in O(n log n)time by means of a balanced binary search treeto determine identical vertices. Steps 2 and 3 takeO(n) time; so does Step 4 (by using radix sorting),and Steps 5 and 6. Thus, the algorithm takes atotal of O(n log n) time. The space required by thealgorithm is O(n).Observe that all the steps of the above algorithmbut Step 1 take O(n) time. Moreover, if the inputconsists of the list V of distinct vertices of the poly-gon P and of the polygons in the collection C, fol-lowed by the vertices of each polygon, where eachvertex is referenced by its index to the list V , thenwe do not need to execute Step 1. Such a represen-tation is commonly used, and is easily produced bypartitioning programs. Then, it can be determinedwhether the collection C forms a cell complex de-composition of P in O(n) time using O(n) space.3. The Three-dimensional CaseWe assume that each polyhedron in the input isdescribed by a list of its facets, each representedby the sequence of its vertices along its boundary,where each vertex is given by its coordinates.Our verification algorithms rely on two lemmatasimilar to Lemma 1 and Lemma 2.3.1. Verification of a partition of a polyhedronLemma 3 Let P and C be a polyhedron and a col-lection of polyhedra respectively. Additionally, letFI = {f − ∂P | f is a facet of a polyhedron in C}and FB = {f ∩ ∂P | f is a facet of a polyhedron inC}. Then, the collection C forms a partition of P ifand only if(i) the set FI of (parts of) facets of the polyhedra inC can be partitioned into two sets S and S′ suchthat the closures of the polygons in each of thesesets defines a partition of⋃s∈Sclosure(s), andfor each (part of) facet f in S (S′ resp.), theinterior of the polyhedron of C with facet f andthe interiors of the polyhedra of C with facets thepolygons of S′ (S resp.) which intersect with f lie(locally around f) on opposite sides of the planesupporting f ;(ii) the closures of the polygons in FB form a parti-tion of the boundary ∂P of P , and for each (partof) facet f in FB, locally around f , the interiorof the polyhedron of C with facet f and the inte-rior of P lie on the same side with respect to theplane supporting f .In light of Lemma 3, we have:3d Partition Verification Algorithm1. Orient the facets of the polyhedron P and ofthe polyhedra in C in a compatible fashion(i.e., in counterclockwise order as seen fromoutside the polyhedron).2. For each polyhedron Q in C dofor each facet f of Q dou← the lex-smallest vertex of f ;if f ’s boundary forms a left turn at uthen add the entry (f, +1) in A;else add the entry (f,−1) in A;3. For each facet f of P dou← the lex-smallest vertex of f ;if f ’s boundary forms a left turn at uthen add the entry (f,−1) in A;else add the entry (f, +1) in A;4. Sort the entries (fi,±1) of the array A byslope of the plane supporting fi and, in caseof ties, lexicographically on the second fieldof the entry.5. For each slope separately, verify that the re-lated facet records with “+1” and those with“−1” partition the same polygonal regions.If this matching terminates successfully, thenthe collection C defines a partition of thepolyhedron P , otherwise it does not.Time complexity. Steps 1, 2, and 3 can be com-pleted in time linear in the total description size nof the input. Step 4 takes O(n log n) time. Step 5can also be completed in O(n log n) time: for eachdifferent plane slope, collect the entries with “+1”as second field, and apply on the associated facetsSteps 2 and 4 of the 2d partition verification algo-rithm (see Section 2.1); Step 5 of that algorithmis applied next, except that the difference of theunion of the “+1” and the “−1” entries is com-puted and used to form a graph; the same pro-cedure is applied to the entries of the array Awith “−1” as second field; finally, the two resultinggraphs are checked to see whether they are identi-20th European Workshop on Computational Geometrycal. Thus, the algorithm takes a total of O(n log n)time. The algorithm uses O(n) space.3.2. Verification of a cell complex decompositionof a polyhedronLemma 4 Let P and C be a polyhedron and a col-lection of polyhedra respectively. Then, the collec-tion C forms a cell complex decomposition of P ifand only if the set of facets of the polyhedra in C canbe partitioned into two sets FI and FB such that(i) for each facet f in FI there exists exactly oneother facet, say f ′, in FI such that f = f′ and,locally around f , the interiors of the polyhedra inC with edges f and f ′ lie on opposite sides of theplane supporting f ;(ii) the edges in FB form a partition of the bound-ary ∂P of P and for each facet f in FB, locallyaround f , the interior of the polyhedron of C withfacet f and the interior of P lie on the same sidewith respect to the plane supporting f .The cell complex verification algorithm appliesLemma 4. It uses two auxiliary arrays A and B,which are initially empty; it works as follows:3d Cell Complex Verification Algorithm1. Collect all the vertices and assign to each oneof them a distinct positive integer id( ).2. Orient the facets of the polyhedron P and ofthe polyhedra in C in a compatible fashion.3. For each polyhedron Q in C dofor each facet f of Q doa← the lex-smallest vertex of f ;b← the lex-largest vertex of f ;c← f ’s vertex farthest away from theline ab (and lex-smallest, if ties);if f ’s boundary is directed a c bthen add ((id(a), id(b), id(c)), +1) in A;else add ((id(a), id(b), id(c)),−1) in A;4. Sort the array A lexicographically.5. Traverse the sorted array A, verify thatmatched entries (which now appear next toeach other) correspond to matching facetsand delete them.6. For each facet f whose entry in A has notbeen matched dofor each edge−→uv of f doif u is lex-smaller than vthen add (id(u), id(v), +1, f) in B;else add (id(v), id(u),−1, f) in B;7. Sort the array B lexicographically.8. If the entries in B do not appear in matchingpairs (i.e., (k, k′,−1, f) and (k, k′, +1, f ′)),then the collection C of polyhedra does notdefine a cell complex decomposition of thegiven polyhedron P .9. For each matching pair (id(u), id(v),−1, f)and (id(u), id(v), +1, f ′) in B doif the two facets f, f ′ are not coplanarthen make u and v adjacent;10. Apply a depth-first traversal and check thatthe graph formed matches the edge skeletonof the polyhedron P . If it does, then the col-lection C of polyhedra defines a cell complexdecomposition of P , otherwise it does not.Time complexity. Step 1 takes O(n log n) time,as described in Section 2.2. Steps 2, 3, 5, and 6take O(n) time. Steps 4 and 7 can be completed inlinear time as well by using radix sorting. Finally,Steps 8, 9, and 10 also take linear time. Thus, thealgorithm requires a total of O(n log n) time. Thespace required by the algorithm is O(n).Similarly to the two-dimensional case, if thepolyhedron P and the polyhedra in the collection Care represented by a list of facets, each describedby a sequence of indices to an array of distinctvertices which indicates the order of the verticesaround the boundary of the facet, then it can bedetermined whether the collection C forms a cellcomplex decomposition of P in time and spacelinear in the total description size of P and C.References[1] D. Avis and H. El Gindy, Triangulating point sets inspace, Discrete and Computational Geometry 2, 99–111, 1987.[2] M. Blum and S. Kannan, Designing programs thatcheck their work, Journal ACM 42(1), 269–291, 1995.[3] O. Devillers, G. Liotta, F.P. Preparata, andR. Tamassia, Checking the convexity of polytopesand the planarity of subdivisions, ComputationalGeometry: Theory and Applications 11(3-4), 187–208,1998.[4] H. Edelsbrunner, F. Preparata, and D. West,Tetrahedralizing point sets in 3 dimensions, Journalof Symbolic Computation, 10, 335–347, 1990.[5] K. Mehlhorn, S. Näher, T. Schilz, S. Schirra, M. Seel,R. Seidel, and C. Uhrig, Checking geometric programsor Verification of geometric structures, Proc. 12thSymp. on Computational Geometry, 159–165, 1996.

Verification of partitions of 2d and 3d objects

https://idus.us.es/bitstream/11441/55360/1/Verification%20of%20partitions%20of%202D%20and%203D%20objets.pdf

Verification of partitions of 2d and 3d objects

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla