Hierarchical clustering has been a popular method in various data analysis applications. It partitions a data set into a hierarchical collection of clusters, and can provide a global view of (cluster) structure behind data across different granularity levels. A hierarchical clustering (HC) of a data set can be naturally represented by a tree, called a HC-tree, where leaves correspond to input data and subtrees rooted at internal nodes correspond to clusters. Many hierarchical clustering algorithms used in practice are developed in a procedure manner. Dasgupta proposed to study the hierarchical clustering problem from an optimization point of view, and introduced an intuitive cost function for similarity-based hierarchical clustering with nice properties as well as natural approximation algorithms. We observe that while Dasgupta's cost function is effective at differentiating a good HC-tree from a bad one for a fixed graph, the value of this cost function does not reflect how well an input similarity graph is consistent to a hierarchical structure. In this paper, we present a new cost function, which is developed based on Dasgupta's cost function, to address this issue. The optimal tree under the new cost function remains the same as the one under Dasgupta's cost function. However, the value of our cost function is more meaningful. The new way of formulating the cost function also leads to a polynomial time algorithm to compute the optimal cluster tree when the input graph has a perfect HC-structure, or an approximation algorithm when the input graph 'almost' has a perfect HC-structure. Finally, we provide further understanding of the new cost function by studying its behavior for random graphs sampled from an edge probability matrix.
{"title":"An Improved Cost Function for Hierarchical Cluster Trees","authors":"Dingkang Wang, Yusu Wang","doi":"10.20382/jocg.v11i1a11","DOIUrl":"https://doi.org/10.20382/jocg.v11i1a11","url":null,"abstract":"Hierarchical clustering has been a popular method in various data analysis applications. It partitions a data set into a hierarchical collection of clusters, and can provide a global view of (cluster) structure behind data across different granularity levels. A hierarchical clustering (HC) of a data set can be naturally represented by a tree, called a HC-tree, where leaves correspond to input data and subtrees rooted at internal nodes correspond to clusters. Many hierarchical clustering algorithms used in practice are developed in a procedure manner. Dasgupta proposed to study the hierarchical clustering problem from an optimization point of view, and introduced an intuitive cost function for similarity-based hierarchical clustering with nice properties as well as natural approximation algorithms. \u0000We observe that while Dasgupta's cost function is effective at differentiating a good HC-tree from a bad one for a fixed graph, the value of this cost function does not reflect how well an input similarity graph is consistent to a hierarchical structure. In this paper, we present a new cost function, which is developed based on Dasgupta's cost function, to address this issue. The optimal tree under the new cost function remains the same as the one under Dasgupta's cost function. However, the value of our cost function is more meaningful. The new way of formulating the cost function also leads to a polynomial time algorithm to compute the optimal cluster tree when the input graph has a perfect HC-structure, or an approximation algorithm when the input graph 'almost' has a perfect HC-structure. Finally, we provide further understanding of the new cost function by studying its behavior for random graphs sampled from an edge probability matrix.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"40 1","pages":"283-331"},"PeriodicalIF":0.0,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89788009","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 : 2018-10-01DOI: 10.4230/LIPIcs.SoCG.2019.18
K. Bringmann, B. Chaudhury
In the classic polyline simplification problem we want to replace a given polygonal curve $P$, consisting of $n$ vertices, by a subsequence $P'$ of $k$ vertices from $P$ such that the polygonal curves $P$ and $P'$ are as close as possible. Closeness is usually measured using the Hausdorff or Fr'echet distance. These distance measures can be applied "globally", i.e., to the whole curves $P$ and $P'$, or "locally", i.e., to each simplified subcurve and the line segment that it was replaced with separately (and then taking the maximum). This gives rise to four problem variants: Global-Hausdorff (known to be NP-hard), Local-Hausdorff (in time $O(n^3)$), Global-Fr'echet (in time $O(k n^5)$), and Local-Fr'echet (in time $O(n^3)$). Our contribution is as follows. - Cubic time for all variants: For Global-Fr'echet we design an algorithm running in time $O(n^3)$. This shows that all three problems (Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet) can be solved in cubic time. All these algorithms work over a general metric space such as $(mathbb{R}^d,L_p)$, but the hidden constant depends on $p$ and (linearly) on $d$. - Cubic conditional lower bound: We provide evidence that in high dimensions cubic time is essentially optimal for all three problems (Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet). Specifically, improving the cubic time to $O(n^{3-epsilon} textrm{poly}(d))$ for polyline simplification over $(mathbb{R}^d,L_p)$ for $p = 1$ would violate plausible conjectures. We obtain similar results for all $p in [1,infty), p ne 2$. In total, in high dimensions and over general $L_p$-norms we resolve the complexity of polyline simplification with respect to Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet, by providing new algorithms and conditional lower bounds.
在经典的折线简化问题中，我们想要用$P$的$k$顶点的子序列$P'$来替换由$n$顶点组成的给定多边形曲线$P$，从而使多边形曲线$P$和$P'$尽可能接近。亲密度通常用Hausdorff或fracimchet距离来衡量。这些距离度量可以应用于“全局”，即整个曲线$P$和$P'$，或“局部”，即每个简化的子曲线和它被单独替换的线段(然后取最大值)。这就产生了四种问题变体:Global-Hausdorff(已知为NP-hard)、Local-Hausdorff(时间上$O(n^3)$)、global - fr(时间上$O(k n^5)$)和local - fr(时间上$O(n^3)$)。我们的贡献如下。-所有变体的立方时间:对于global - fr我们设计了一个算法运行在时间$O(n^3)$。这表明所有三个问题(Local-Hausdorff, local - frachimet和global - frachimet)都可以在三次时间内解决。所有这些算法都在一般的度量空间(如$(mathbb{R}^d,L_p)$)上工作，但是隐藏常数依赖于$p$和(线性地)依赖于$d$。-三次条件下界:我们提供了证据，在高维三次时间本质上是最优的所有三个问题(Local-Hausdorff, local - fr和global - fr)。具体来说，将折线简化的三次时间提高到$O(n^{3-epsilon} textrm{poly}(d))$而不是$p = 1$的$(mathbb{R}^d,L_p)$将违反貌似合理的猜想。我们对所有$p in [1,infty), p ne 2$都得到了类似的结果。总的来说，在高维和一般$L_p$ -范数上，我们通过提供新的算法和条件下界，解决了关于Local-Hausdorff, local - fr和global - fr的折线简化的复杂性。
{"title":"Polyline Simplification has Cubic Complexity","authors":"K. Bringmann, B. Chaudhury","doi":"10.4230/LIPIcs.SoCG.2019.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2019.18","url":null,"abstract":"In the classic polyline simplification problem we want to replace a given polygonal curve $P$, consisting of $n$ vertices, by a subsequence $P'$ of $k$ vertices from $P$ such that the polygonal curves $P$ and $P'$ are as close as possible. Closeness is usually measured using the Hausdorff or Fr'echet distance. These distance measures can be applied \"globally\", i.e., to the whole curves $P$ and $P'$, or \"locally\", i.e., to each simplified subcurve and the line segment that it was replaced with separately (and then taking the maximum). This gives rise to four problem variants: Global-Hausdorff (known to be NP-hard), Local-Hausdorff (in time $O(n^3)$), Global-Fr'echet (in time $O(k n^5)$), and Local-Fr'echet (in time $O(n^3)$). \u0000Our contribution is as follows. \u0000- Cubic time for all variants: For Global-Fr'echet we design an algorithm running in time $O(n^3)$. This shows that all three problems (Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet) can be solved in cubic time. All these algorithms work over a general metric space such as $(mathbb{R}^d,L_p)$, but the hidden constant depends on $p$ and (linearly) on $d$. \u0000- Cubic conditional lower bound: We provide evidence that in high dimensions cubic time is essentially optimal for all three problems (Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet). Specifically, improving the cubic time to $O(n^{3-epsilon} textrm{poly}(d))$ for polyline simplification over $(mathbb{R}^d,L_p)$ for $p = 1$ would violate plausible conjectures. We obtain similar results for all $p in [1,infty), p ne 2$. \u0000In total, in high dimensions and over general $L_p$-norms we resolve the complexity of polyline simplification with respect to Local-Hausdorff, Local-Fr'echet, and Global-Fr'echet, by providing new algorithms and conditional lower bounds.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"62 1","pages":"94-130"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87700816","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}
S. Schleimer, A. D. Mesmay, J. Purcell, E. Sedgwick
We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mari~no.
{"title":"On the tree-width of knot diagrams","authors":"S. Schleimer, A. D. Mesmay, J. Purcell, E. Sedgwick","doi":"10.20382/jocg.v10i1a6","DOIUrl":"https://doi.org/10.20382/jocg.v10i1a6","url":null,"abstract":"We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mari~no.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"56 1","pages":"164-180"},"PeriodicalIF":0.0,"publicationDate":"2018-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79974383","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}
It is shown that if a $d$-dimensional cube is decomposed into $n$ cubes, the side lengths of which belong to the interval $(1 − frac{n}{1/d 1 +1} , 1]$, then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.
{"title":"A stability theorem on cube tessellations","authors":"J. Pach, P. Frankl","doi":"10.20382/jocg.v9i1a13","DOIUrl":"https://doi.org/10.20382/jocg.v9i1a13","url":null,"abstract":"It is shown that if a $d$-dimensional cube is decomposed into $n$ cubes, the side lengths of which belong to the interval $(1 − frac{n}{1/d 1 +1} , 1]$, then $n$ is a perfect $d$-th power and all cubes are of the same size. This result is essentially tight.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"69 1","pages":"387-390"},"PeriodicalIF":0.0,"publicationDate":"2018-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82555341","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 : 2018-05-19DOI: 10.4230/LIPIcs.ICALP.2018.31
Timothy M. Chan, Yakov Nekrich, S. Rahul, Konstantinos Tsakalidis
In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. -We give the first linear-space data structure that supports 3-d point location queries on $n$ disjoint axis-aligned boxes with optimal $Oleft( log nright)$ query time in the (arithmetic) pointer machine model. This improves the previous $Oleft( log^{3/2} n right)$ bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. -We give the first linear-space data structure that supports 3-d $4$-sided and $5$-sided rectangle stabbing queries in optimal $O(log_wn+k)$ time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-$k$ rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.
{"title":"Orthogonal Point Location and Rectangle Stabbing Queries in 3-d","authors":"Timothy M. Chan, Yakov Nekrich, S. Rahul, Konstantinos Tsakalidis","doi":"10.4230/LIPIcs.ICALP.2018.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.ICALP.2018.31","url":null,"abstract":"In this work, we present a collection of new results on two fundamental problems in geometric data structures: orthogonal point location and rectangle stabbing. \u0000-We give the first linear-space data structure that supports 3-d point location queries on $n$ disjoint axis-aligned boxes with optimal $Oleft( log nright)$ query time in the (arithmetic) pointer machine model. This improves the previous $Oleft( log^{3/2} n right)$ bound of Rahul [SODA 2015]. We similarly obtain the first linear-space data structure in the I/O model with optimal query cost, and also the first linear-space data structure in the word RAM model with sub-logarithmic query time. \u0000-We give the first linear-space data structure that supports 3-d $4$-sided and $5$-sided rectangle stabbing queries in optimal $O(log_wn+k)$ time in the word RAM model. We similarly obtain the first optimal data structure for the closely related problem of 2-d top-$k$ rectangle stabbing in the word RAM model, and also improved results for 3-d 6-sided rectangle stabbing. \u0000For point location, our solution is simpler than previous methods, and is based on an interesting variant of the van Emde Boas recursion, applied in a round-robin fashion over the dimensions, combined with bit-packing techniques. For rectangle stabbing, our solution is a variant of Alstrup, Brodal, and Rauhe's grid-based recursive technique (FOCS 2000), combined with a number of new ideas.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79509961","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}
{"title":"An Efficient Line Clipping Algorithm for Circular Windows Using Vector Calculus and Parallelization","authors":"P. Kumar, Fenil Patel, R. Kanna","doi":"10.5121/IJCGA.2018.8201","DOIUrl":"https://doi.org/10.5121/IJCGA.2018.8201","url":null,"abstract":"","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"8 1","pages":"01-08"},"PeriodicalIF":0.0,"publicationDate":"2018-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.5121/IJCGA.2018.8201","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47161650","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 : 2018-04-25DOI: 10.4230/LIPIcs.ICALP.2018.89
G. Barequet, D. Eppstein, M. Goodrich, Nil Mamano
We study algorithms and combinatorial complexity bounds for emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or "appetite" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work on the stable-matching Voronoi diagram provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. In this paper, we show that a stable-matching Voronoi diagram of $n$ point sites has $O(n^{2+varepsilon})$ faces and edges, for any $varepsilon>0$, and show that this bound is almost tight by giving a family of diagrams with $Theta(n^2)$ faces and edges. We also provide a discrete algorithm for constructing it in $O(n^3log n+n^2f(n))$ time in the real-RAM model of computation, where $f(n)$ is the runtime of a geometric primitive (which we define) that can be approximated numerically, but cannot, in general, be performed exactly in an algebraic model of computation. We show, however, how to compute the geometric primitive exactly for polygonal convex distance functions.
{"title":"Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms","authors":"G. Barequet, D. Eppstein, M. Goodrich, Nil Mamano","doi":"10.4230/LIPIcs.ICALP.2018.89","DOIUrl":"https://doi.org/10.4230/LIPIcs.ICALP.2018.89","url":null,"abstract":"We study algorithms and combinatorial complexity bounds for emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or \"appetite\" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work on the stable-matching Voronoi diagram provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. In this paper, we show that a stable-matching Voronoi diagram of $n$ point sites has $O(n^{2+varepsilon})$ faces and edges, for any $varepsilon>0$, and show that this bound is almost tight by giving a family of diagrams with $Theta(n^2)$ faces and edges. We also provide a discrete algorithm for constructing it in $O(n^3log n+n^2f(n))$ time in the real-RAM model of computation, where $f(n)$ is the runtime of a geometric primitive (which we define) that can be approximated numerically, but cannot, in general, be performed exactly in an algebraic model of computation. We show, however, how to compute the geometric primitive exactly for polygonal convex distance functions.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"33 1","pages":"26-59"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89825362","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 : 2018-04-25DOI: 10.4230/LIPICS.SOCG.2020.9
Alan Arroyo, Julien Bensmail, R. Richter
A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if the edges can be extended to an arrangement of pseudolines. In the recent study of crossing numbers, pseudolinear drawings have played an important role as they are a natural combinatorial extension of rectilinear drawings. A characterization of the pseudolinear drawings of $K_n$ was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.
{"title":"Extending Drawings of Graphs to Arrangements of Pseudolines","authors":"Alan Arroyo, Julien Bensmail, R. Richter","doi":"10.4230/LIPICS.SOCG.2020.9","DOIUrl":"https://doi.org/10.4230/LIPICS.SOCG.2020.9","url":null,"abstract":"A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if the edges can be extended to an arrangement of pseudolines. In the recent study of crossing numbers, pseudolinear drawings have played an important role as they are a natural combinatorial extension of rectilinear drawings. A characterization of the pseudolinear drawings of $K_n$ was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"65 1","pages":"3-24"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77785621","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 : 2018-03-15DOI: 10.4230/LIPIcs.SoCG.2018.15
M. Buchet, Emerson G. Escolar
While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.
{"title":"Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension","authors":"M. Buchet, Emerson G. Escolar","doi":"10.4230/LIPIcs.SoCG.2018.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.SoCG.2018.15","url":null,"abstract":"While persistent homology has taken strides towards becoming a wide-spread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87880252","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}
P. Bose, Matias Korman, André van Renssen, S. Verdonschot
We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.
{"title":"Routing on the Visibility Graph","authors":"P. Bose, Matias Korman, André van Renssen, S. Verdonschot","doi":"10.20382/jocg.v9i1a15","DOIUrl":"https://doi.org/10.20382/jocg.v9i1a15","url":null,"abstract":"We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"124 1","pages":"430-453"},"PeriodicalIF":0.0,"publicationDate":"2018-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87839343","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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