SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98599
D. Huttenlocher, K. Kedem
{"title":"Computing the minimum Hausdorff distance for point sets under translation","authors":"D. Huttenlocher, K. Kedem","doi":"10.1145/98524.98599","DOIUrl":"https://doi.org/10.1145/98524.98599","url":null,"abstract":"We consider the problem of computing a translation that minimizes the Hausdorff distance between two sets of points. For points in @@@@<supscrpt>1</supscrpt> in the worst case there are ⊖(<italic>mn</italic>) translations at which the Hausdorff distance is a local minimum, where <italic>m</italic> is the number of points in one set and <italic>n</italic> is the number in the other. For points in @@@@<supscrpt>2</supscrpt> there are ⊖(<italic>mn</italic>(<italic>m</italic> + <italic>n</italic>)) such local minima. We show how to compute the minimal Hausdorff distance in time <italic>&Ogr;</italic>(<italic>mn</italic> log <italic>mn</italic>) for points in @@@@<supscrpt>1</supscrpt> and in time <italic>&Ogr;</italic>(<italic>m</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>2</supscrpt>α(<italic>mn</italic>)) for points in @@@@<supscrpt>2</supscrpt>. The results for the one-dimensional case are applied to the problem of comparing polygons under general affine transformations, where we extend the recent results of Arkin et al on polygon resemblance under rigid body motion. The two-dimensional case is closely related to the problem of finding an approximate congruence between two points sets under translation in the plane, as considered by Alt et al.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115507673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98544
P. Gritzmann, V. Klee, J. Westwater
{"title":"On the limited power of linear probes and other optimization oracles","authors":"P. Gritzmann, V. Klee, J. Westwater","doi":"10.1145/98524.98544","DOIUrl":"https://doi.org/10.1145/98524.98544","url":null,"abstract":"Throughout this note, X denotes a real vector space of finite dimension d _> 2. As the term is used here, a convex body in X is a compact convex subset C of X whose interior is \u0000nonempty. When the origin 0 is interior to C, the gauge functional of C is the positively homogenous function gc whose value is 1 at all points of C's boundary bd C. This function is \u0000subadditive, and is symmetric (gc(x) = gc(-x)) precisely when C itself is symmetric about 0 (i.e. C = -C). In this case, gc is a norm for the space X. Here we consider the problem of \u0000using norm-computations to find a convex polytope P (not necessarily in any sense the smallest one) that contains C. By finding P we mean to produce its vertex-set or, equivalently for our model, the set of hyperplanes determined by P's facets. It is assumed that the usual arithmetic operations in R and the usual vector operations in X are available without cost, so the problem's difficulty is measured solely in terms of the number of calls to the \"oracle\" that computes norms. Computing the norm of a point x E X {0} is equivalent to finding the point at which the ray [0, co)x intersects the boundary bd C. Thus we refer to C's ray-oracle (Pc, which, when presented with any ray R issuing from 0, returns the point R N (bd C). Such probing oracles are currently of widespread interest because of their use in robotics and tomography (see, e.g.,)","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127342492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98551
B. Chazelle, H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir
{"title":"Slimming down by adding; selecting heavily covered points","authors":"B. Chazelle, H. Edelsbrunner, L. Guibas, J. Hershberger, R. Seidel, M. Sharir","doi":"10.1145/98524.98551","DOIUrl":"https://doi.org/10.1145/98524.98551","url":null,"abstract":"<italic>We show that for any set &Pgr; of n points in three-dimensional space there is a set Q of &Ogr;(n<supscrpt>1/2</supscrpt> log<supscrpt>3</supscrpt> n) points so that the Delaunay triangulation of &Pgr; ∪ Q has at most &Ogr;(n<supscrpt>3/2</supscrpt> log<supscrpt>3</supscrpt> n) edges — even though the Delaunay triangulation of &Pgr; may have &OHgr;(n<supscrpt>2</supscrpt>) edges. The main tool of our construction is the following geometric covering result: For any set &Pgr; of n points in three-dimensional space and any set S of m spheres, where each sphere passes through a distinct point pair in &Pgr;, there exists a point x, not necessarily in &Pgr;, that is enclosed by</italic> &OHgr;(<italic>m</italic><supscrpt>2</supscrpt>/<italic>n</italic><supscrpt>2</supscrpt> log<supscrpt>3</supscrpt> <italic>n</italic><supscrpt>2</supscrpt>/<italic>m</italic>) <italic>of the spheres in S.</italic>\u0000par><italic>Our results generalize to arbitrary fixed dimensions, to geometric bodies other than spheres, and to geometric structures other than Delaunay triangulations.</italic>","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128546545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98546
G. Vegter, C. Yap
{"title":"Computational complexity of combinatorial surfaces","authors":"G. Vegter, C. Yap","doi":"10.1145/98524.98546","DOIUrl":"https://doi.org/10.1145/98524.98546","url":null,"abstract":"We investigate the computational problems associated with combinatorial surfaces. Specifically, we present an algorithm (based on the Brahana-Dehn-Heegaard approach) for transforming the polygonal schema of a closed triangulated surface into its canonical form in <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>) time, where <italic>n</italic> is the total number of vertices, edges and faces. We also give an <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic> + <italic>gn</italic>) algorithm for constructing canonical generators of the fundamental group of a surface of genus <italic>g</italic>. This is useful in constructing homeomorphisms between combinatorial surfaces.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128172667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98559
B. Natarajan
{"title":"On computing the intersection of B-splines (extended abstract)","authors":"B. Natarajan","doi":"10.1145/98524.98559","DOIUrl":"https://doi.org/10.1145/98524.98559","url":null,"abstract":"We consider the problem of computing a piecewise linear approximation to the intersection of a pair of tensor product B-spline surfaces in 3-space. The problem is rather central in solid modeling. We present a fast and robust divide-and-conquer algorithm for the problem, that is a generalization of the bisection algorithm for computing the roots of non-linear equations. The algorithm is guaranteed to solve a “nearby” problem, and our analysis proves that its expected run-time is linear in the worst-case size of the output. To our knowledge, this is the first such analysis, resulting in the first provably efficient algorithm for the problem.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"255 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132494178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98539
M. Goodrich, Steven B. Shauck, S. Guha
{"title":"Parallel methods for visibility and shortest path problems in simple polygons (preliminary version)","authors":"M. Goodrich, Steven B. Shauck, S. Guha","doi":"10.1145/98524.98539","DOIUrl":"https://doi.org/10.1145/98524.98539","url":null,"abstract":"In this paper we give efficient parallel algorithms for solving a number of visibility and shortest path problems for simple polygons. Our algorithms all run in <italic>&Ogr;</italic>(log <italic>n</italic>) time and are based on the use of a new data structure for implicitly representing all shortest paths in a simple polygon <italic>P</italic>, which we call the <italic>stratified decomposition tree</italic>. We use this approach to derive efficient parallel methods for computing the visibility of <italic>P</italic> from an edge, constructing the visibility graph of the vertices of <italic>P</italic> (using an output-sensitive number of processors), constructing the shortest path tree from a vertex of <italic>P</italic>, and determining all-farthest neighbors for the vertices in <italic>P</italic>. The computational model we use is the CREW PRAM.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133512031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98564
J. Bentley
{"title":"K-d trees for semidynamic point sets","authors":"J. Bentley","doi":"10.1145/98524.98564","DOIUrl":"https://doi.org/10.1145/98524.98564","url":null,"abstract":"A <italic>K</italic>-d tree represents a set of <italic>N</italic> points in <italic>K</italic>-dimensional space. Operations on a <italic>semidynamic</italic> tree may delete and undelete points, but may not insert new points. This paper shows that several operations that require <italic>&Ogr;</italic>(log <italic>N</italic>) expected time in general <italic>K</italic>-d trees may be performed in constant expected time in semidynamic trees. These operations include deletion, undeletion, nearest neighbor searching, and fixed-radius near neighbor searching (the running times of the first two are proved, while the last two are supported by experiments and heuristic arguments). Other new techniques can also be applied to general <italic>K</italic>-d trees: simple sampling reduces the time to build a tree from <italic>&Ogr;</italic>(<italic>KN</italic> log <italic>N</italic>) to <italic>&Ogr;</italic>(<italic>KN</italic> + <italic>N</italic> log <italic>N</italic>), and more advanced sampling builds a robust tree in the same time. The methods are straightforward to implement, and lead to a data structure that is significantly faster and less vulnerable to pathological inputs than ordinary <italic>K</italic>-d trees.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127112925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98529
J. Pach, G. Woeginger
{"title":"Some new bounds for Epsilon-nets","authors":"J. Pach, G. Woeginger","doi":"10.1145/98524.98529","DOIUrl":"https://doi.org/10.1145/98524.98529","url":null,"abstract":"Given any natural number <italic>d</italic>, 0 < <italic>ε</italic> < 1, let ƒ<italic><subscrpt>d</subscrpt></italic>(<italic>ε</italic>) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension <italic>d</italic> has an <italic>ε</italic>-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if <italic>d</italic> ≥ 2, then ƒ<italic><subscrpt>d</subscrpt></italic>(<italic>ε</italic>) > 1/48 <italic>d</italic>/<italic>ε</italic> log 1/ <italic>ε</italic> which is not far from being optimal, if <italic>d</italic> is fixed and <italic>ε</italic> → 0. Further, we prove that ƒ<subscrpt>1</subscrpt>(<italic>ε</italic>) = max(2,⌈1/<italic>ε</italic>⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132596397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98596
T. Snyder
{"title":"On minimal rectilinear Steiner trees in all dimensions","authors":"T. Snyder","doi":"10.1145/98524.98596","DOIUrl":"https://doi.org/10.1145/98524.98596","url":null,"abstract":"It is proved that the length of the longest possible minimum rectilinear Steiner tree of <italic>n</italic> points in the unit <italic>d</italic>-cube is asymptotic to Β<subscrpt><italic>d</subscrpt>n</italic> d-1/d, where Β<subscrpt><italic>d</italic></subscrpt>. In addition to replicating Chung and Graham's exact determination of Β<subscrpt>2</subscrpt> = 1, this generalization yields tight new bounds such as 1 ≤ Β<subscrpt>3</subscrpt> < 1.191 and 1 < Β<subscrpt>4</subscrpt> < √<italic>2</italic>.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130405013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SCG '90Pub Date : 1990-05-01DOI: 10.1145/98524.98600
Elefterios A. Melissaratos, D. Souvaine
{"title":"On solving geometric optimization problems using shortest paths","authors":"Elefterios A. Melissaratos, D. Souvaine","doi":"10.1145/98524.98600","DOIUrl":"https://doi.org/10.1145/98524.98600","url":null,"abstract":"We have developed techniques which contribute to efficient algorithms for certain geometric optimization problems involving simple polygons: computing minimum separators, maximum inscribed triangles, a minimum circumscribed concave quadrilateral, or a maximum contained triangle. The structure for our algorithms is as follows: a) decompose the initial problem into a low-degree polynomial number of easy optimization problems; b) solve each individual subproblem in constant time using the methods of cealcuIns, standard methods of numerical analysis, or linear programming. The decomposition step uses shorteat path trees inside simple polygons (Guibas et. al.~ 1987) and, in the case of inscribed triangles, produces a new class of polygons, the fan-shaped polygon. By extending the shortest-path algorithm to splinegons, we also generate splinegon-versions of the algorithms for some of the optimization problems. The problems we discuss fall into four subgroups: Separa tors : If two points z and y lie on the boundary of simple polygon P and define a directed line segment zy C_ P that separates P into two sets PL and Pa, then zy is called a separator.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115729220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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