The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph. In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist $n$-vertex planar graphs whose planar edge-length ratio is in $Omega(n)$; this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs.
{"title":"On the planar edge-length ratio of planar graphs","authors":"Manuel Borrazzo, Fabrizio Frati","doi":"10.20382/jocg.v11i1a6","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a6","url":null,"abstract":"The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph. \u0000In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist $n$-vertex planar graphs whose planar edge-length ratio is in $Omega(n)$; this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"81 1","pages":"137-155"},"PeriodicalIF":0.0,"publicationDate":"2019-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91292835","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 : 2019-07-02DOI: 10.1007/978-3-030-80879-2_6
Stav Ashur, O. Filtser, M. J. Katz
{"title":"A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon","authors":"Stav Ashur, O. Filtser, M. J. Katz","doi":"10.1007/978-3-030-80879-2_6","DOIUrl":"https://doi.org/10.1007/978-3-030-80879-2_6","url":null,"abstract":"","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"71 1","pages":"128-144"},"PeriodicalIF":0.0,"publicationDate":"2019-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86355523","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 : 2019-07-01DOI: 10.4230/LIPIcs.SoCG.2020.45
K. Fox, Jiashuai Lu
The geometric transportation problem takes as input a set of points $P$ in $d$-dimensional Euclidean space and a supply function $mu : P to mathbb{R}$. The goal is to find a transportation map, a non-negative assignment $tau : P times P to mathbb{R}_{geq 0}$ to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., $sum_{r in P} tau(q, r) - sum_{p in P} tau(p, q) = mu(q)$ for all points $q in P$. The goal is to minimize the weighted sum of Euclidean distances for the pairs, $sum_{(p, q) in P times P} tau(p, q) cdot ||q - p||_2$. We describe the first algorithm for this problem that returns, with high probability, a $(1 + epsilon)$-approximation to the optimal transportation map in $O(n:text{poly}(1 / epsilon):text{polylog}{n})$ time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of $P$ and the magnitude of its real-valued supplies.
几何运输问题以$d$维欧氏空间中的一组点$P$和一个供给函数$mu : P to mathbb{R}$作为输入。目标是找到一个交通地图，对点对的非负分配$tau : P times P to mathbb{R}_{geq 0}$，所以每个点的总分配等于它的供给，即对所有点$q in P$的总分配为$sum_{r in P} tau(q, r) - sum_{p in P} tau(p, q) = mu(q)$。目标是最小化对欧几里得距离的加权和，$sum_{(p, q) in P times P} tau(p, q) cdot ||q - p||_2$。我们描述了该问题的第一种算法，它以高概率返回$O(n:text{poly}(1 / epsilon):text{polylog}{n})$时间内最优交通地图的$(1 + epsilon)$ -近似值。与此问题的先前最佳算法相比，我们的近线性运行时间界限独立于$P$的传播及其实值供应的大小。
{"title":"A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread","authors":"K. Fox, Jiashuai Lu","doi":"10.4230/LIPIcs.SoCG.2020.45","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2020.45","url":null,"abstract":"The geometric transportation problem takes as input a set of points $P$ in $d$-dimensional Euclidean space and a supply function $mu : P to mathbb{R}$. The goal is to find a transportation map, a non-negative assignment $tau : P times P to mathbb{R}_{geq 0}$ to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., $sum_{r in P} tau(q, r) - sum_{p in P} tau(p, q) = mu(q)$ for all points $q in P$. The goal is to minimize the weighted sum of Euclidean distances for the pairs, $sum_{(p, q) in P times P} tau(p, q) cdot ||q - p||_2$. \u0000We describe the first algorithm for this problem that returns, with high probability, a $(1 + epsilon)$-approximation to the optimal transportation map in $O(n:text{poly}(1 / epsilon):text{polylog}{n})$ time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of $P$ and the magnitude of its real-valued supplies.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"228 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75771520","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 : 2019-03-07DOI: 10.4230/LIPIcs.SoCG.2019.40
X. Goaoc, Andreas F. Holmsen, C. Nicaud
We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in $mathbb{R}^3$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be "lifted" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.
{"title":"An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting","authors":"X. Goaoc, Andreas F. Holmsen, C. Nicaud","doi":"10.4230/LIPIcs.SoCG.2019.40","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2019.40","url":null,"abstract":"We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in $mathbb{R}^3$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be \"lifted\" to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"41 1","pages":"131-161"},"PeriodicalIF":0.0,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81001845","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}
Let $P$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $P$ under the $L_1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $n_s$ "gateways" for $s$ and a set of $n_t$ "gateways" for $t$ such that there exist a shortest $s-t$ path containing a gateway of $s$ and a gateway of $t$, and then compute a shortest $s-t$ path using these gateways. Previous algorithms all take quadratic $O(n_s n_t)$ time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in $O(n_s + n_tlog n_s)$ time. As a consequence, we construct a data structure of $O(n+(h^2 log^3 h / loglog h))$ size in $O(n+(h^2 log^4 h / loglog h))$ time such that each query can be answered in $O(log n)$ time.
设P是一个有h个洞和n个顶点的多边形定义域。对于任意两个查询点$s$和$t$，我们研究了在$L_1$度量下，在$P$中可以计算$s$和$t$之间的最短路径的数据结构问题。为此，标准方法是首先为$s$找到一组$n_s$“网关”，为$t$找到一组$n_t$“网关”，使得存在包含$s$网关和$t$网关的最短$s-t$路径，然后使用这些网关计算最短$s-t$路径。以前的算法都需要二次$O(n_s n_t)$的时间来解决这个问题。在本文中，我们提出了一种分而治之的技术，在$O(n_s + n_tlog n_s)$时间内解决了这个问题。因此，我们在$O(n+(h^2 log^3 h / loglog h))$ time内构造了$O(n+(h^2 log^4 h / loglog h))$ size的数据结构，这样每个查询都可以在$O(log n)$ time内得到回答。
{"title":"A Divide-and-Conquer Algorithm for Two-Point L1 Shortest Path Queries in Polygonal Domains","authors":"Haitao Wang","doi":"10.20382/JOCG.V11I1A10","DOIUrl":"https://doi.org/10.20382/JOCG.V11I1A10","url":null,"abstract":"Let $P$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $P$ under the $L_1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $n_s$ \"gateways\" for $s$ and a set of $n_t$ \"gateways\" for $t$ such that there exist a shortest $s-t$ path containing a gateway of $s$ and a gateway of $t$, and then compute a shortest $s-t$ path using these gateways. Previous algorithms all take quadratic $O(n_s n_t)$ time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in $O(n_s + n_tlog n_s)$ time. As a consequence, we construct a data structure of $O(n+(h^2 log^3 h / loglog h))$ size in $O(n+(h^2 log^4 h / loglog h))$ time such that each query can be answered in $O(log n)$ time.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"26 1","pages":"235-282"},"PeriodicalIF":0.0,"publicationDate":"2019-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85087072","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 : 2019-01-06DOI: 10.4230/LIPICS.SOCG.2019.17
K. Bringmann, Marvin Künnemann, A. Nusser
The Fr'echet distance provides a natural and intuitive measure for the popular task of computing the similarity of two (polygonal) curves. While a simple algorithm computes it in near-quadratic time, a strongly subquadratic algorithm cannot exist unless the Strong Exponential Time Hypothesis fails. Still, fast practical implementations of the Fr'echet distance, in particular for realistic input curves, are highly desirable. This has even lead to a designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge was to implement a near-neighbor data structure under the Fr'echet distance. The bottleneck of the top three implementations turned out to be precisely the decision procedure for the Fr'echet distance. In this work, we present a fast, certifying implementation for deciding the Fr'echet distance, in order to (1) complement its pessimistic worst-case hardness by an empirical analysis on realistic input data and to (2) improve the state of the art for the GIS Cup challenge. We experimentally evaluate our implementation on a large benchmark consisting of several data sets (including handwritten characters and GPS trajectories). Compared to the winning implementation of the GIS Cup, we obtain running time improvements of up to more than two orders of magnitude for the decision procedure and of up to a factor of 30 for queries to the near-neighbor data structure.
{"title":"Walking the Dog Fast in Practice: Algorithm Engineering of the Fréchet Distance","authors":"K. Bringmann, Marvin Künnemann, A. Nusser","doi":"10.4230/LIPICS.SOCG.2019.17","DOIUrl":"https://doi.org/10.4230/LIPICS.SOCG.2019.17","url":null,"abstract":"The Fr'echet distance provides a natural and intuitive measure for the popular task of computing the similarity of two (polygonal) curves. While a simple algorithm computes it in near-quadratic time, a strongly subquadratic algorithm cannot exist unless the Strong Exponential Time Hypothesis fails. Still, fast practical implementations of the Fr'echet distance, in particular for realistic input curves, are highly desirable. This has even lead to a designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge was to implement a near-neighbor data structure under the Fr'echet distance. The bottleneck of the top three implementations turned out to be precisely the decision procedure for the Fr'echet distance. \u0000In this work, we present a fast, certifying implementation for deciding the Fr'echet distance, in order to (1) complement its pessimistic worst-case hardness by an empirical analysis on realistic input data and to (2) improve the state of the art for the GIS Cup challenge. We experimentally evaluate our implementation on a large benchmark consisting of several data sets (including handwritten characters and GPS trajectories). Compared to the winning implementation of the GIS Cup, we obtain running time improvements of up to more than two orders of magnitude for the decision procedure and of up to a factor of 30 for queries to the near-neighbor data structure.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"31 1","pages":"70-108"},"PeriodicalIF":0.0,"publicationDate":"2019-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81772206","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. Kalyanaraman, M. Kamruzzaman, Bala Krishnamoorthy
Given a high dimensional point cloud of data with functions defined on the points, the mapper algorithm produces a compact summary in the form of a simplicial complex connecting the points. We study the problem of quantifying the interestingness of subpopulations in a given mapper complex. First, we create a weighted directed graph G = (V,E) using the 1-skeleton of the mapper complex. We use the average values at the vertices of a target function (dependent variable) to direct the edges from low to high values, and assign the difference (high−low) as the weight of the edge. Covariation of the remaining h functions (independent variables) is captured by a h-bit binary signature assigned to the edge. An interesting path in G is a directed path whose edges all have the same signature. The interestingness score of such a path as a sum of its edge weights multiplied by a nonlinear function of their corresponding ranks, i.e., the depths of the edges along the path. Such a nonlinear function could model application use-cases where the growth in the dependent variable values is expected to be concentrated in specific intervals of a path. Second, we study three optimization problems on this graph G to quantify interesting subpopulations. In the problem Max-IP, the goal is to find the most interesting path in G, i.e., an interesting path with the maximum interestingness score. For the case where G is a directed acyclic graph (DAG), we show that Max-IP can be solved in polynomial time. In the more general problem IP, the goal is to find a collection of interesting paths that are edge-disjoint, and the sum of interestingness scores of all paths is maximized. We also study a variant of IP termed k-IP, where the goal is to identify a collection of edgedisjoint interesting paths each with k edges, and the total interestingness score of all paths is maximized. While k-IP can be solved in polynomial time for k ≤ 2, we show k-IP is NP-complete for k ≥ 3 even when G is a DAG. We develop heuristics for IP and k-IP on DAGs, which use the algorithm for Max-IP on DAGs as a subroutine. We have released open source implementations of our algorithms to find interesting paths. We also present a detailed experimental evaluation of this software framework on a real-world maize plant phenomics data set. We use interesting paths identified on several mapper graphs to explain how the genotype and environmental factors influence the growth rate, both in isolation as well as in combinations. ∗School of Electrical Engineering and Computer Science, Washington State University, Pullman, USA †Department of Mathematics and Statistics, Washington State University, Vancouver, USA {ananth,md.kamruzzaman,kbala}@wsu.edu
{"title":"Interesting paths in the mapper complex","authors":"A. Kalyanaraman, M. Kamruzzaman, Bala Krishnamoorthy","doi":"10.20382/jocg.v10i1a17","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a17","url":null,"abstract":"Given a high dimensional point cloud of data with functions defined on the points, the mapper algorithm produces a compact summary in the form of a simplicial complex connecting the points. We study the problem of quantifying the interestingness of subpopulations in a given mapper complex. First, we create a weighted directed graph G = (V,E) using the 1-skeleton of the mapper complex. We use the average values at the vertices of a target function (dependent variable) to direct the edges from low to high values, and assign the difference (high−low) as the weight of the edge. Covariation of the remaining h functions (independent variables) is captured by a h-bit binary signature assigned to the edge. An interesting path in G is a directed path whose edges all have the same signature. The interestingness score of such a path as a sum of its edge weights multiplied by a nonlinear function of their corresponding ranks, i.e., the depths of the edges along the path. Such a nonlinear function could model application use-cases where the growth in the dependent variable values is expected to be concentrated in specific intervals of a path. Second, we study three optimization problems on this graph G to quantify interesting subpopulations. In the problem Max-IP, the goal is to find the most interesting path in G, i.e., an interesting path with the maximum interestingness score. For the case where G is a directed acyclic graph (DAG), we show that Max-IP can be solved in polynomial time. In the more general problem IP, the goal is to find a collection of interesting paths that are edge-disjoint, and the sum of interestingness scores of all paths is maximized. We also study a variant of IP termed k-IP, where the goal is to identify a collection of edgedisjoint interesting paths each with k edges, and the total interestingness score of all paths is maximized. While k-IP can be solved in polynomial time for k ≤ 2, we show k-IP is NP-complete for k ≥ 3 even when G is a DAG. We develop heuristics for IP and k-IP on DAGs, which use the algorithm for Max-IP on DAGs as a subroutine. We have released open source implementations of our algorithms to find interesting paths. We also present a detailed experimental evaluation of this software framework on a real-world maize plant phenomics data set. We use interesting paths identified on several mapper graphs to explain how the genotype and environmental factors influence the growth rate, both in isolation as well as in combinations. ∗School of Electrical Engineering and Computer Science, Washington State University, Pullman, USA †Department of Mathematics and Statistics, Washington State University, Vancouver, USA {ananth,md.kamruzzaman,kbala}@wsu.edu","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"24 1","pages":"500-531"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76095962","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 give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already nondegenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.
{"title":"Simplices modelled on spaces of constant curvature","authors":"R. Dyer, G. Vegter, M. Wintraecken","doi":"10.20382/jocg.v10i1a9","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a9","url":null,"abstract":"We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already nondegenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"2 1","pages":"223-256"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90080445","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}
Annette Ebbers-Baumann, R. Klein, Christian Knauer, G. Rote
Given three points in the plane, we construct the plane geometric network of smallest geometric dilation that connects them. The geometric dilation of a plane network is defined as the maximum dilation (distance along the network divided by Euclidean distance) between any two points on its edges. We show that the optimum network is either a line segment, a Steiner tree, or a curve consisting of two straight edges and a segment of a logarithmic spiral.
{"title":"The geometric dilation of three points","authors":"Annette Ebbers-Baumann, R. Klein, Christian Knauer, G. Rote","doi":"10.20382/jocg.v10i1a18","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a18","url":null,"abstract":"Given three points in the plane, we construct the plane geometric network of smallest geometric dilation that connects them. The geometric dilation of a plane network is defined as the maximum dilation (distance along the network divided by Euclidean distance) between any two points on its edges. We show that the optimum network is either a line segment, a Steiner tree, or a curve consisting of two straight edges and a segment of a logarithmic spiral.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"2 1","pages":"532-549"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74493958","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}
Ellen Gasparovic, Maria Gommel, Emilie Purvine, R. Sazdanovic, Bei Wang, Yusu Wang, Lori Ziegelmeier
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.
{"title":"The Relationship Between the Intrinsic Cech and Persistence Distortion Distances for Metric Graphs","authors":"Ellen Gasparovic, Maria Gommel, Emilie Purvine, R. Sazdanovic, Bei Wang, Yusu Wang, Lori Ziegelmeier","doi":"10.20382/jocg.v10i1a16","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a16","url":null,"abstract":"Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"22 1","pages":"477-499"},"PeriodicalIF":0.0,"publicationDate":"2018-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78053260","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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