Pub Date : 2011-09-21DOI: 10.1007/978-3-642-25878-7_7
V. Dujmovic, W. Evans, S. Lazard, William J. Lenhart, G. Liotta, D. Rappaport, S. Wismath
{"title":"On point-sets that support planar graphs","authors":"V. Dujmovic, W. Evans, S. Lazard, William J. Lenhart, G. Liotta, D. Rappaport, S. Wismath","doi":"10.1007/978-3-642-25878-7_7","DOIUrl":"https://doi.org/10.1007/978-3-642-25878-7_7","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"49 1","pages":"29-50"},"PeriodicalIF":0.0,"publicationDate":"2011-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90648893","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}
J. D. Loera, Brandon E. Dutra, M. Köppe, Stanislav Moreinis, Gregory Pinto, Jianqiu Wu
We are interested in quickly computing the exact value of integrals of polynomial functions over domains that are decomposable into convex polyhedra (e.g., a tetrahedral or cubical mesh decomposition of space). We describe a software implementation, part of the software LattE, and provide benchmark computations.
{"title":"Software for exact integration of polynomials over polyhedra","authors":"J. D. Loera, Brandon E. Dutra, M. Köppe, Stanislav Moreinis, Gregory Pinto, Jianqiu Wu","doi":"10.1145/2110170.2110175","DOIUrl":"https://doi.org/10.1145/2110170.2110175","url":null,"abstract":"We are interested in quickly computing the exact value of integrals of polynomial functions over domains that are decomposable into convex polyhedra (e.g., a tetrahedral or cubical mesh decomposition of space). We describe a software implementation, part of the software LattE, and provide benchmark computations.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"6 1","pages":"232-252"},"PeriodicalIF":0.0,"publicationDate":"2011-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79833953","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}
A partially embedded graph (or PEG) is a triple (G,H,EH), where G is a graph, H is a subgraph of G, and EH is a planar embedding of H. We say that a PEG (G,H,EH) is planar if the graph G has a planar embedding that extends the embedding EH. We introduce a containment relation of PEGs analogous to graph minor containment, and characterize the minimal non-planar PEGs with respect to this relation. We show that all the minimal non-planar PEGs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar PEGs. Furthermore, by extending an existing planarity test for PEGs, we obtain a polynomial-time algorithm which, for a given PEG, either produces a planar embedding or identifies a minimal obstruction.
{"title":"A kuratowski-type theorem for planarity of partially embedded graphs","authors":"Vít Jelínek, Jan Kratochvíl, Ignaz Rutter","doi":"10.1145/1998196.1998214","DOIUrl":"https://doi.org/10.1145/1998196.1998214","url":null,"abstract":"A partially embedded graph (or PEG) is a triple (G,H,EH), where G is a graph, H is a subgraph of G, and EH is a planar embedding of H. We say that a PEG (G,H,EH) is planar if the graph G has a planar embedding that extends the embedding EH.\u0000 We introduce a containment relation of PEGs analogous to graph minor containment, and characterize the minimal non-planar PEGs with respect to this relation. We show that all the minimal non-planar PEGs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar PEGs.\u0000 Furthermore, by extending an existing planarity test for PEGs, we obtain a polynomial-time algorithm which, for a given PEG, either produces a planar embedding or identifies a minimal obstruction.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"56 1","pages":"466-492"},"PeriodicalIF":0.0,"publicationDate":"2011-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78729595","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}
Snap rounding is a popular method for rounding the vertices of a planar arrangement of line segments to the integer grid. It has many advantages, including minimum perturbation of the segments, preservation of the arrangement topology, and ease of implementation. However, snap rounding has one significant weakness: it is not stable (i.e., not idempotent). That is, applying snap rounding to a snap-rounded arrangement of n segments may cause additional segment perturbation, and the number of iterations of snap rounding needed to reach stability may be as large as Θ(n2). This paper introduces stable snap rounding, a variant of snap rounding that has all of snap rounding's advantages and is also idempotent. In particular, stable snap rounding does not change any arrangement whose vertices are already grid points (such as those produced by stable snap rounding or standard snap rounding).
{"title":"Stable snap rounding","authors":"J. Hershberger","doi":"10.1145/1998196.1998226","DOIUrl":"https://doi.org/10.1145/1998196.1998226","url":null,"abstract":"Snap rounding is a popular method for rounding the vertices of a planar arrangement of line segments to the integer grid. It has many advantages, including minimum perturbation of the segments, preservation of the arrangement topology, and ease of implementation. However, snap rounding has one significant weakness: it is not stable (i.e., not idempotent). That is, applying snap rounding to a snap-rounded arrangement of n segments may cause additional segment perturbation, and the number of iterations of snap rounding needed to reach stability may be as large as Θ(n2).\u0000 This paper introduces stable snap rounding, a variant of snap rounding that has all of snap rounding's advantages and is also idempotent. In particular, stable snap rounding does not change any arrangement whose vertices are already grid points (such as those produced by stable snap rounding or standard snap rounding).","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"32 1","pages":"403-416"},"PeriodicalIF":0.0,"publicationDate":"2011-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89975992","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}
Pub Date : 2010-07-21DOI: 10.1016/j.comgeo.2011.09.001
O. Aichholzer, F. Aurenhammer, E. Demaine, F. Hurtado, P. Ramos, J. Urrutia
{"title":"On k-convex polygons","authors":"O. Aichholzer, F. Aurenhammer, E. Demaine, F. Hurtado, P. Ramos, J. Urrutia","doi":"10.1016/j.comgeo.2011.09.001","DOIUrl":"https://doi.org/10.1016/j.comgeo.2011.09.001","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"90 1","pages":"73-87"},"PeriodicalIF":0.0,"publicationDate":"2010-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79583026","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}
Pub Date : 2010-07-01DOI: 10.1016/j.comgeo.2009.12.003
N. Chernov, Y. Stoyan, T. Romanova
{"title":"Mathematical model and efficient algorithms for object packing problem","authors":"N. Chernov, Y. Stoyan, T. Romanova","doi":"10.1016/j.comgeo.2009.12.003","DOIUrl":"https://doi.org/10.1016/j.comgeo.2009.12.003","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"53 1","pages":"535-553"},"PeriodicalIF":0.0,"publicationDate":"2010-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90756445","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}
Pub Date : 2010-07-01DOI: 10.1016/j.comgeo.2009.11.001
M. D. Berg, H. Haverkort, Shripad Thite, Laura Toma
{"title":"Star-quadtrees and guard-quadtrees: I/O-efficient indexes for fat triangulations and low-density planar subdivisions","authors":"M. D. Berg, H. Haverkort, Shripad Thite, Laura Toma","doi":"10.1016/j.comgeo.2009.11.001","DOIUrl":"https://doi.org/10.1016/j.comgeo.2009.11.001","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"12 1","pages":"493-513"},"PeriodicalIF":0.0,"publicationDate":"2010-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75339203","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}
Pub Date : 2010-07-01DOI: 10.1016/j.comgeo.2009.10.001
Juan José Jiménez-Delgado, R. Segura, F. Feito-Higueruela
{"title":"A robust segment/triangle intersection algorithm for interference tests. Efficiency study","authors":"Juan José Jiménez-Delgado, R. Segura, F. Feito-Higueruela","doi":"10.1016/j.comgeo.2009.10.001","DOIUrl":"https://doi.org/10.1016/j.comgeo.2009.10.001","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"56 1","pages":"474-492"},"PeriodicalIF":0.0,"publicationDate":"2010-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83915900","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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