We investigate the following problem in the visibility-based discrete-time model of pursuit evasion in the plane: how many pursuers are needed to capture an evader in a polygonal environment with obstacles under the minimalist assumption that pursuers and the evader have the same maximum speed? When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ (√n) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω (n2/3) and an upper bound of O(n5/6) for the number of pursuers that are needed in the worst-case, where n is the total number of vertices including the hole boundaries. More precisely, if the polygon contains h holes, our upper bound is O(n1/2 h1/4), for h ≤ n2/3, and O(n1/3 h1/2) otherwise. These bounds show that capture with minimal assumptions requires significantly more pursuers than what is possible either for visibility detection where pursuers win if one of them can see the evader [Guibas et al. 1999], or for capture when players' movement speed is small compared to "features" of the environment [Klein and Suri, 2012].
本文研究了平面上基于可见性的离散时间追逐逃避模型中的以下问题:在具有障碍物的多边形环境中，在追逐者和逃避者具有相同最大速度的极简假设下，需要多少追逐者才能捕获一个逃避者?当环境是一个n个顶点的单连通(无孔)多边形时，我们证明了Θ(√n)跟踪器在最坏情况下既是必要的也是充分的。当环境是一个有洞的多边形时，我们证明了最坏情况下需要的追踪者数量的下界为Ω (n2/3)，上界为O(n2/ 6)，其中n为包含洞边界的顶点总数。更准确地说，如果多边形包含h个孔，则当h≤n2/3时，上界为O(n1/2 h /4)，否则为O(n1/3 h /2)。这些界限表明，与能见度检测相比，最小假设下的捕获需要更多的追捕者(如果其中一人可以看到逃避者，追捕者就会获胜)[guet al. 1999]，或者当玩家的移动速度与环境的“特征”相比较小时捕获[Klein and Suri, 2012]。
{"title":"Capture bounds for visibility-based pursuit evasion","authors":"Kyle Klein, S. Suri","doi":"10.1145/2462356.2462364","DOIUrl":"https://doi.org/10.1145/2462356.2462364","url":null,"abstract":"We investigate the following problem in the visibility-based discrete-time model of pursuit evasion in the plane: how many pursuers are needed to capture an evader in a polygonal environment with obstacles under the minimalist assumption that pursuers and the evader have the same maximum speed? When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ (√n) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω (n2/3) and an upper bound of O(n5/6) for the number of pursuers that are needed in the worst-case, where n is the total number of vertices including the hole boundaries. More precisely, if the polygon contains h holes, our upper bound is O(n1/2 h1/4), for h ≤ n2/3, and O(n1/3 h1/2) otherwise. These bounds show that capture with minimal assumptions requires significantly more pursuers than what is possible either for visibility detection where pursuers win if one of them can see the evader [Guibas et al. 1999], or for capture when players' movement speed is small compared to \"features\" of the environment [Klein and Suri, 2012].","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"1 1","pages":"205-220"},"PeriodicalIF":0.0,"publicationDate":"2013-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88467246","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 most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has a Θ(kn2) worst-case running time, where n is the number of input vertices and k is the number of input segments. We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses, and demonstrate with a performance comparison that it is faster than gift-wrapping for segments that cross many edges. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(n log n + n log2 k) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O(n log n) time, incremental CDT construction is easier to program and competitive in practice. Moreover, the ability to incrementally update a CDT by inserting a segment is useful in itself.
在平面上构造约束Delaunay三角剖分(CDT)最常用的方法是先构造Delaunay三角剖分，然后逐个增量地插入输入段。对于段插入的典型实现，该方法的最坏情况运行时间为Θ(kn2)，其中n是输入顶点的数量，k是输入段的数量。我们给出了一种随机算法，用于在预期的时间线性内将一个片段插入到CDT中，并通过性能比较证明，对于跨越许多边的片段，它比礼物包装更快。Agarwal, Arge和Yi的结果表明，通过我们的片段插入算法随机增量构建cdt需要预期的O(n log n + n log2k)时间。我们通过推导一个匹配的下界来证明这个下界是紧的。虽然有保证在O(n log n)时间内运行的CDT构造算法，但增量CDT构造更容易编程，并且在实践中具有竞争力。此外，通过插入段来增量地更新CDT的能力本身是有用的。
{"title":"Fast segment insertion and incremental construction of constrained delaunay triangulations","authors":"J. Shewchuk, Brielin C. Brown","doi":"10.1145/2462356.2462389","DOIUrl":"https://doi.org/10.1145/2462356.2462389","url":null,"abstract":"The most commonly implemented method of constructing a constrained Delaunay triangulation (CDT) in the plane is to first construct a Delaunay triangulation, then incrementally insert the input segments one by one. For typical implementations of segment insertion, this method has a Θ(kn2) worst-case running time, where n is the number of input vertices and k is the number of input segments. We give a randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses, and demonstrate with a performance comparison that it is faster than gift-wrapping for segments that cross many edges. A result of Agarwal, Arge, and Yi implies that randomized incremental construction of CDTs by our segment insertion algorithm takes expected O(n log n + n log2 k) time. We show that this bound is tight by deriving a matching lower bound. Although there are CDT construction algorithms guaranteed to run in O(n log n) time, incremental CDT construction is easier to program and competitive in practice. Moreover, the ability to incrementally update a CDT by inserting a segment is useful in itself.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"43 1","pages":"554-574"},"PeriodicalIF":0.0,"publicationDate":"2013-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77797178","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}
D. Attali, Ulrich Bauer, O. Devillers, M. Glisse, A. Lieutier
We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology of some complex H*(X) with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.
{"title":"Homological reconstruction and simplification in R3","authors":"D. Attali, Ulrich Bauer, O. Devillers, M. Glisse, A. Lieutier","doi":"10.1145/2462356.2462373","DOIUrl":"https://doi.org/10.1145/2462356.2462373","url":null,"abstract":"We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology of some complex H*(X) with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"13 1","pages":"606-621"},"PeriodicalIF":0.0,"publicationDate":"2013-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89321093","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 : 2013-02-19DOI: 10.1007/978-3-642-45346-5_15
Matias Korman, M. Löffler, Rodrigo I. Silveira, Darren Strash
{"title":"On the Complexity of Barrier Resilience for Fat Regions","authors":"Matias Korman, M. Löffler, Rodrigo I. Silveira, Darren Strash","doi":"10.1007/978-3-642-45346-5_15","DOIUrl":"https://doi.org/10.1007/978-3-642-45346-5_15","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"120 1","pages":"34-51"},"PeriodicalIF":0.0,"publicationDate":"2013-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87828630","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 : 2012-07-04DOI: 10.1007/978-3-642-31155-0_3
Takehiro Ito, Shin-Ichi Nakano, Y. Okamoto, Y. Otachi, Ryuhei Uehara, T. Uno, Yushi Uno
{"title":"A Polynomial-Time Approximation Scheme for the Geometric Unique Coverage Problem on Unit Squares","authors":"Takehiro Ito, Shin-Ichi Nakano, Y. Okamoto, Y. Otachi, Ryuhei Uehara, T. Uno, Yushi Uno","doi":"10.1007/978-3-642-31155-0_3","DOIUrl":"https://doi.org/10.1007/978-3-642-31155-0_3","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"1 1","pages":"25-39"},"PeriodicalIF":0.0,"publicationDate":"2012-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82381815","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 : 2012-04-01DOI: 10.1016/J.COMGEO.2011.09.002
R. Nishat, Debajyoti Mondal, M. S. Rahman
{"title":"Point-set embeddings of plane 3-trees","authors":"R. Nishat, Debajyoti Mondal, M. S. Rahman","doi":"10.1016/J.COMGEO.2011.09.002","DOIUrl":"https://doi.org/10.1016/J.COMGEO.2011.09.002","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"28 1","pages":"88-98"},"PeriodicalIF":0.0,"publicationDate":"2012-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89923687","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 : 2012-04-01DOI: 10.1016/j.comgeo.2011.09.003
Joondong Kim, Joseph S. B. Mitchell, V. Polishchuk, Shang Yang, Jingyu Zou
{"title":"Routing multi-class traffic flows in the plane","authors":"Joondong Kim, Joseph S. B. Mitchell, V. Polishchuk, Shang Yang, Jingyu Zou","doi":"10.1016/j.comgeo.2011.09.003","DOIUrl":"https://doi.org/10.1016/j.comgeo.2011.09.003","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"45 1","pages":"99-114"},"PeriodicalIF":0.0,"publicationDate":"2012-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88137941","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 : 2012-02-27DOI: 10.1016/j.comgeo.2017.10.006
L. Devroye, Pat Morin
{"title":"A Note on Interference in Random Networks","authors":"L. Devroye, Pat Morin","doi":"10.1016/j.comgeo.2017.10.006","DOIUrl":"https://doi.org/10.1016/j.comgeo.2017.10.006","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"480 1","pages":"2-10"},"PeriodicalIF":0.0,"publicationDate":"2012-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80144149","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 : 2011-10-01DOI: 10.1016/j.comgeo.2011.04.003
Y. Disser, Matús Mihalák, P. Widmayer
{"title":"A polygon is determined by its angles","authors":"Y. Disser, Matús Mihalák, P. Widmayer","doi":"10.1016/j.comgeo.2011.04.003","DOIUrl":"https://doi.org/10.1016/j.comgeo.2011.04.003","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"3 1","pages":"418-426"},"PeriodicalIF":0.0,"publicationDate":"2011-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87678462","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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