Pub Date : 2009-11-25DOI: 10.1007/978-3-642-10631-6_106
S. Bae, Y. Okamoto
{"title":"Querying Two Boundary Points for Shortest Paths in a Polygonal Domain","authors":"S. Bae, Y. Okamoto","doi":"10.1007/978-3-642-10631-6_106","DOIUrl":"https://doi.org/10.1007/978-3-642-10631-6_106","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"23 1","pages":"284-293"},"PeriodicalIF":0.0,"publicationDate":"2009-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87307309","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 present cache-oblivious solutions to two important variants of range searching: range reporting and approximate range counting. The main contribution of our paper is a general approach for constructing cache-oblivious data structures that provide relative (1+ε)-approximations for a general class of range counting queries. This class includes three-sided range counting, 3-d dominance counting, and 3-d halfspace range counting. Our technique allows us to obtain data structures that use linear space and answer queries in the optimal query bound of O(logB(N/K)) block transfers in the worst case, where K is the number of points in the query range. Using the same technique, we also obtain the first approximate 3-d halfspace range counting and 3-d dominance counting data structures with a worst-case query time of O(log(N/K)) in internal memory. An easy but important consequence of our main result is the existence of O(N log N)-space cache-oblivious data structures with an optimal query bound of O(logBN + K/B) block transfers for the reporting versions of the above problems. Using standard reductions, these data structures allow us to obtain the first cache-oblivious data structures that use near-linear space and achieve the optimal query bound for circular range reporting and K-nearest neighbour searching in the plane, as well as for orthogonal range reporting in three dimensions.
{"title":"A general approach for cache-oblivious range reporting and approximate range counting","authors":"P. Afshani, Chris H. Hamilton, N. Zeh","doi":"10.1145/1542362.1542413","DOIUrl":"https://doi.org/10.1145/1542362.1542413","url":null,"abstract":"We present cache-oblivious solutions to two important variants of range searching: range reporting and approximate range counting. The main contribution of our paper is a general approach for constructing cache-oblivious data structures that provide relative (1+ε)-approximations for a general class of range counting queries. This class includes three-sided range counting, 3-d dominance counting, and 3-d halfspace range counting. Our technique allows us to obtain data structures that use linear space and answer queries in the optimal query bound of O(logB(N/K)) block transfers in the worst case, where K is the number of points in the query range. Using the same technique, we also obtain the first approximate 3-d halfspace range counting and 3-d dominance counting data structures with a worst-case query time of O(log(N/K)) in internal memory.\u0000 An easy but important consequence of our main result is the existence of O(N log N)-space cache-oblivious data structures with an optimal query bound of O(logBN + K/B) block transfers for the reporting versions of the above problems. Using standard reductions, these data structures allow us to obtain the first cache-oblivious data structures that use near-linear space and achieve the optimal query bound for circular range reporting and K-nearest neighbour searching in the plane, as well as for orthogonal range reporting in three dimensions.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"303 1","pages":"700-712"},"PeriodicalIF":0.0,"publicationDate":"2009-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76860838","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}
G. Borradaile, James R. Lee, Anastasios Sidiropoulos
It was shown in [Indyk-Sidiropoulos 07] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2O(g). By removing all $g$ handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [Erickson-HarPeled 04] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [Lee-Sidiropoulos 08].
{"title":"Randomly removing g handles at once","authors":"G. Borradaile, James R. Lee, Anastasios Sidiropoulos","doi":"10.1145/1542362.1542425","DOIUrl":"https://doi.org/10.1145/1542362.1542425","url":null,"abstract":"It was shown in [Indyk-Sidiropoulos 07] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2O(g). By removing all $g$ handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [Erickson-HarPeled 04] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [Lee-Sidiropoulos 08].","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"15 1","pages":"655-662"},"PeriodicalIF":0.0,"publicationDate":"2009-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76693254","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 topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint. These conjectures are shown to be true apart from log* n and log2 n factors, respectively. We also settle the conjectures for some special cases.
{"title":"On grids in topological graphs","authors":"Eyal Ackerman, J. Fox, J. Pach, Andrew Suk","doi":"10.1145/1542362.1542430","DOIUrl":"https://doi.org/10.1145/1542362.1542430","url":null,"abstract":"A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges.\u0000 We conjecture that this statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint.\u0000 These conjectures are shown to be true apart from log* n and log2 n factors, respectively. We also settle the conjectures for some special cases.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"2015 1","pages":"710-723"},"PeriodicalIF":0.0,"publicationDate":"2009-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86847996","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 : 2009-06-01DOI: 10.1016/j.comgeo.2011.05.002
Shawn Martin, J. Watson
{"title":"Non-manifold surface reconstruction from high-dimensional point cloud data","authors":"Shawn Martin, J. Watson","doi":"10.1016/j.comgeo.2011.05.002","DOIUrl":"https://doi.org/10.1016/j.comgeo.2011.05.002","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"31 1","pages":"427-441"},"PeriodicalIF":0.0,"publicationDate":"2009-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86523595","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 : 2009-05-01DOI: 10.1016/j.comgeo.2008.09.005
Rodrigo I. Silveira, M. V. Kreveld
{"title":"Towards a Definition of Higher Order Constrained Delaunay Triangulations","authors":"Rodrigo I. Silveira, M. V. Kreveld","doi":"10.1016/j.comgeo.2008.09.005","DOIUrl":"https://doi.org/10.1016/j.comgeo.2008.09.005","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"334 1","pages":"322-337"},"PeriodicalIF":0.0,"publicationDate":"2009-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77098774","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 : 2009-05-01DOI: 10.1016/j.comgeo.2008.06.003
J. Egeblad, Benny K. Nielsen, M. Brazil
{"title":"Translational packing of arbitrary polytopes","authors":"J. Egeblad, Benny K. Nielsen, M. Brazil","doi":"10.1016/j.comgeo.2008.06.003","DOIUrl":"https://doi.org/10.1016/j.comgeo.2008.06.003","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"35 1","pages":"269-288"},"PeriodicalIF":0.0,"publicationDate":"2009-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81383924","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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