We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of higher genus. This is done in the language of angle labelings. Generalizing results of De Fraysseix and Ossona de Mendez, and Felsner, we establish a correspondence between these labelings and orientations and characterize the set of orientations of a map that correspond to such a Schnyder labeling. Furthermore, we study the set of these orientations of a given map and provide a natural partition into distributive lattices depending on the surface homology. This generalizes earlier results of Felsner and Ossona de Mendez. In the toroidal case, a new proof for the existence of Schnyder woods is derived from this approach.
我们提出了一种简单的将施耐德木从平面推广到高属可定向曲面上的映射。这是在角标记语言中完成的。推广De Fraysseix和Ossona De Mendez以及Felsner的结果，我们建立了这些标记和方向之间的对应关系，并表征了与此类施耐德标记对应的地图的方向集。此外，我们研究了给定映射的这些取向的集合，并根据表面同调提供了分配格的自然划分。这概括了Felsner和Ossona de Mendez早期的结果。在环面情况下，用这种方法得到了施耐德森林存在的一个新的证明。
{"title":"On the structure of Schnyder woods on orientable surfaces","authors":"K. Knauer, D. Gonçalves, Benjamin Lévêque","doi":"10.20382/JOCG.V10I1A5","DOIUrl":"https://doi.org/10.20382/JOCG.V10I1A5","url":null,"abstract":"We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of higher genus. This is done in the language of angle labelings. Generalizing results of De Fraysseix and Ossona de Mendez, and Felsner, we establish a correspondence between these labelings and orientations and characterize the set of orientations of a map that correspond to such a Schnyder labeling. Furthermore, we study the set of these orientations of a given map and provide a natural partition into distributive lattices depending on the surface homology. This generalizes earlier results of Felsner and Ossona de Mendez. In the toroidal case, a new proof for the existence of Schnyder woods is derived from this approach.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"37 1","pages":"127-163"},"PeriodicalIF":0.0,"publicationDate":"2015-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82043098","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}
We study point-sphere and point-plane incidences in the three-dimensional space. In particular, for $1 0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ spheres is [ O(m^{3/4+varepsilon}n^{3/4}k^{1/4}+n+mk).] Similarly, we prove that, for every $varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ planes is [ O(m^{4/5+varepsilon}n^{3/5}k^{2/5} + n + mk). ] These bounds are obtained by using the polynomial partitioning technique, recently introduced by Guth and Katz. More specifically, in our proofs we use a pair of constant-degree partitioning polynomials. We also present a couple of applications of $k$-non-degenerate sets: (i) We consider an extension of the three-dimensional unit distances problem, in which we are given a set $D$ of $k$ distinct distances and ask for a three-dimensional set of $m$ points that maximizes the number of pairs of points that span a distance from $D$. By relying on $k$-non-degenerate sets of spheres, we prove an upper bound of $O(m^{236/149+varepsilon}k^{125/149})$ for the problem (which improves the trivial bound for large values of $k$). Â Â Â (ii) We consider the maximum number of incidences between a three-dimensional set of $n$ planes (without any restrictions) and a set of $m$ points, such that no $k$ points are collinear. Our bound for $k$-non-degenerate planes immediately implies a bound of $O(n^{4/5+varepsilon}m^{3/5}k^{2/5} + m + nk)$ for this problem, generalizing the previous bound $O(n^{4/5}m^{3/5} + nlog m)$ for the specific case where no three points are collinear (up to the $varepsilon$ in the exponent).
{"title":"Incidences with k-non-degenerate sets and their applications","authors":"A. Basit, Adam Sheffer","doi":"10.20382/jocg.v5i1a14","DOIUrl":"https://doi.org/10.20382/jocg.v5i1a14","url":null,"abstract":"We study point-sphere and point-plane incidences in the three-dimensional space. In particular, for $1 0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ spheres is [ O(m^{3/4+varepsilon}n^{3/4}k^{1/4}+n+mk).] Similarly, we prove that, for every $varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ planes is [ O(m^{4/5+varepsilon}n^{3/5}k^{2/5} + n + mk). ] These bounds are obtained by using the polynomial partitioning technique, recently introduced by Guth and Katz. More specifically, in our proofs we use a pair of constant-degree partitioning polynomials. We also present a couple of applications of $k$-non-degenerate sets: (i) We consider an extension of the three-dimensional unit distances problem, in which we are given a set $D$ of $k$ distinct distances and ask for a three-dimensional set of $m$ points that maximizes the number of pairs of points that span a distance from $D$. By relying on $k$-non-degenerate sets of spheres, we prove an upper bound of $O(m^{236/149+varepsilon}k^{125/149})$ for the problem (which improves the trivial bound for large values of $k$). Â Â Â (ii) We consider the maximum number of incidences between a three-dimensional set of $n$ planes (without any restrictions) and a set of $m$ points, such that no $k$ points are collinear. Our bound for $k$-non-degenerate planes immediately implies a bound of $O(n^{4/5+varepsilon}m^{3/5}k^{2/5} + m + nk)$ for this problem, generalizing the previous bound $O(n^{4/5}m^{3/5} + nlog m)$ for the specific case where no three points are collinear (up to the $varepsilon$ in the exponent).","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"42 1","pages":"284-302"},"PeriodicalIF":0.0,"publicationDate":"2014-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84700825","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}
Benjamin A. Burton, M. Elder, A. Kalka, Stephan Tillmann
We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.
证明了2流形的同胚问题可以在对数空间中确定。该证明依赖于Reingold对图中无向$s， $ t -连通性问题的对数空间解决方案。
{"title":"2-manifold Recognition Is in Logspace","authors":"Benjamin A. Burton, M. Elder, A. Kalka, Stephan Tillmann","doi":"10.20382/jocg.v7i1a4","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a4","url":null,"abstract":"We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"2 1","pages":"70-85"},"PeriodicalIF":0.0,"publicationDate":"2014-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75583302","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 : 2014-12-01DOI: 10.1007/978-3-662-48971-0_3
P. Bose, Rolf Fagerberg, André van Renssen, S. Verdonschot
{"title":"Competitive Local Routing with Constraints","authors":"P. Bose, Rolf Fagerberg, André van Renssen, S. Verdonschot","doi":"10.1007/978-3-662-48971-0_3","DOIUrl":"https://doi.org/10.1007/978-3-662-48971-0_3","url":null,"abstract":"","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"1 1","pages":"125-152"},"PeriodicalIF":0.0,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76170070","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}
Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $mathbb{R}^2$.
{"title":"Flat norm decomposition of integral currents","authors":"Sharif Ibrahim, B. Krishnamoorthy, K. Vixie","doi":"10.20382/jocg.v7i1a14","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a14","url":null,"abstract":"Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. \u0000Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. \u0000We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $mathbb{R}^2$.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"621 ","pages":"285-307"},"PeriodicalIF":0.0,"publicationDate":"2014-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72442781","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}
The Map Maker algorithm which converts survey data into geometric data with 2-dimensional Cartesian coordinates has been previously published. Analysis of the performance of this algorithm is continuing. The algorithm is suitable for generating 2D maps and it would be helpful to have this algorithm generalized to generate 3D and higher dimensional coordinates. The trigonometric approach of the Map Maker algorithm does not extend well into higher dimensions however this paper reports on an algebraic approach which solves the problem. A similar algorithm called the Coordinatizator algorithm has been published which converts survey data defining a higher dimensional space of measured sites into the lowest dimensionalcoordinatization accurately fitting the data. Therefore the Coordinatizator algorithm is not a projection transformation whereas the n-dimensional Map Maker algorithm is.
{"title":"THE N-DIMENSIONAL MAP MAKER ALGORITHM","authors":"J. Rankin","doi":"10.5121/IJCGA.2014.4402","DOIUrl":"https://doi.org/10.5121/IJCGA.2014.4402","url":null,"abstract":"The Map Maker algorithm which converts survey data into geometric data with 2-dimensional Cartesian coordinates has been previously published. Analysis of the performance of this algorithm is continuing. The algorithm is suitable for generating 2D maps and it would be helpful to have this algorithm generalized to generate 3D and higher dimensional coordinates. The trigonometric approach of the Map Maker algorithm does not extend well into higher dimensions however this paper reports on an algebraic approach which solves the problem. A similar algorithm called the Coordinatizator algorithm has been published which converts survey data defining a higher dimensional space of measured sites into the lowest dimensionalcoordinatization accurately fitting the data. Therefore the Coordinatizator algorithm is not a projection transformation whereas the n-dimensional Map Maker algorithm is.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"4 1","pages":"19-26"},"PeriodicalIF":0.0,"publicationDate":"2014-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.5121/IJCGA.2014.4402","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70617685","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}
For producing a single high dynamic range image (HDRI), multiple low dynamic range images (LDRIs) are captured with different exposures and combined. In high dynamic range (HDR) imaging, local motion of objects and noise in a set of LDRIs can influence a final HDRI: local motion of objects causes the ghost artifact and LDRIs, especially captured with under-exposure, make the final HDRI noisy. In this paper, we propose a ghost and noise removal method for HDRI using exposure fusion with subband architecture, in which Haar wavelet filter is used. The proposed method blends weight map of exposure fusion in the subband pyramid, where the weight map is produced for ghost artifact removal as well as exposure fusion. Then, the noise is removed using multi-resolution bilateral filtering. After removing the ghost artifact and noise in subband images, details of the images are enhanced using a gain control map. Experimental results with various sets of LDRIs show that the proposed method effectively removes the ghost artifact and noise, enhancing the contrast in a final HDRI.
{"title":"GHOST AND NOISE REMOVAL IN EXPOSURE FUSION FOR HIGH DYNAMIC RANGE IMAGING","authors":"Dong-Kyu Lee, Rae-Hong Park, Soonkeun Chang","doi":"10.5121/IJCGA.2014.4401","DOIUrl":"https://doi.org/10.5121/IJCGA.2014.4401","url":null,"abstract":"For producing a single high dynamic range image (HDRI), multiple low dynamic range images (LDRIs) are captured with different exposures and combined. In high dynamic range (HDR) imaging, local motion of objects and noise in a set of LDRIs can influence a final HDRI: local motion of objects causes the ghost artifact and LDRIs, especially captured with under-exposure, make the final HDRI noisy. In this paper, we propose a ghost and noise removal method for HDRI using exposure fusion with subband architecture, in which Haar wavelet filter is used. The proposed method blends weight map of exposure fusion in the subband pyramid, where the weight map is produced for ghost artifact removal as well as exposure fusion. Then, the noise is removed using multi-resolution bilateral filtering. After removing the ghost artifact and noise in subband images, details of the images are enhanced using a gain control map. Experimental results with various sets of LDRIs show that the proposed method effectively removes the ghost artifact and noise, enhancing the contrast in a final HDRI.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"4 1","pages":"1-18"},"PeriodicalIF":0.0,"publicationDate":"2014-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.5121/IJCGA.2014.4401","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70617299","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}
Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.
{"title":"Determining hyperbolicity of compact orientable 3-manifolds with torus boundary","authors":"R. Haraway","doi":"10.20382/jocg.v11i1a5","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a5","url":null,"abstract":"Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. \u0000A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"5 1","pages":"125-136"},"PeriodicalIF":0.0,"publicationDate":"2014-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84442275","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}
Por and Wood conjectured that for all $k,l ge 2$ there exists $n ge 2$ with the following property: whenever $n$ points, no $l + 1$ of which are collinear, are chosen in the plane and each of them is assigned one of $k$ colours, then there must be a line (that is, a maximal set of collinear points) all of whose points have the same colour. The conjecture is easily seen to be true for $l = 2$ (by the pigeonhole principle) and in the case $k = 2$ it is an immediate corollary of the Motzkin-Rabin theorem. In this note we show that the conjecture is false for $k, l ge 3$.
{"title":"A counterexample to a geometric Hales-Jewett type conjecture","authors":"V. Gruslys","doi":"10.20382/jocg.v5i1a11","DOIUrl":"https://doi.org/10.20382/jocg.v5i1a11","url":null,"abstract":"Por and Wood conjectured that for all $k,l ge 2$ there exists $n ge 2$ with the following property: whenever $n$ points, no $l + 1$ of which are collinear, are chosen in the plane and each of them is assigned one of $k$ colours, then there must be a line (that is, a maximal set of collinear points) all of whose points have the same colour. The conjecture is easily seen to be true for $l = 2$ (by the pigeonhole principle) and in the case $k = 2$ it is an immediate corollary of the Motzkin-Rabin theorem. In this note we show that the conjecture is false for $k, l ge 3$.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"38 1","pages":"245-249"},"PeriodicalIF":0.0,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78723593","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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