Pub Date : 2016-12-07DOI: 10.4230/LIPIcs.ISAAC.2017.9
S. Bae, M. D. Berg, O. Cheong, Joachim Gudmundsson, C. Levcopoulos
Abstract Let C be the unit circle in R 2 . We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place k ⩾ 1 shortcuts on C such that the diameter of the resulting graph is minimized. We analyze for each k with 1 ⩽ k ⩽ 7 what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is 2 + Θ ( 1 / k 2 3 ) for any k.
{"title":"Shortcuts for the Circle","authors":"S. Bae, M. D. Berg, O. Cheong, Joachim Gudmundsson, C. Levcopoulos","doi":"10.4230/LIPIcs.ISAAC.2017.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2017.9","url":null,"abstract":"Abstract Let C be the unit circle in R 2 . We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place k ⩾ 1 shortcuts on C such that the diameter of the resulting graph is minimized. We analyze for each k with 1 ⩽ k ⩽ 7 what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is 2 + Θ ( 1 / k 2 3 ) for any k.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"14 1","pages":"37-54"},"PeriodicalIF":0.0,"publicationDate":"2016-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83355128","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 : 2016-08-31DOI: 10.1007/978-3-319-50106-2_20
Alessio Arleo, Carla Binucci, E. D. Giacomo, W. Evans, L. Grilli, G. Liotta, H. Meijer, Fabrizio Montecchiani, S. Whitesides, S. Wismath
{"title":"Visibility Representations of Boxes in 2.5 Dimensions","authors":"Alessio Arleo, Carla Binucci, E. D. Giacomo, W. Evans, L. Grilli, G. Liotta, H. Meijer, Fabrizio Montecchiani, S. Whitesides, S. Wismath","doi":"10.1007/978-3-319-50106-2_20","DOIUrl":"https://doi.org/10.1007/978-3-319-50106-2_20","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"18 1","pages":"19-33"},"PeriodicalIF":0.0,"publicationDate":"2016-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88307954","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 : 2016-07-25DOI: 10.1007/978-3-319-62127-2_1
Ankush Acharyya, S. Nandy, S. Pandit, Sasanka Roy
{"title":"Covering Segments with Unit Squares","authors":"Ankush Acharyya, S. Nandy, S. Pandit, Sasanka Roy","doi":"10.1007/978-3-319-62127-2_1","DOIUrl":"https://doi.org/10.1007/978-3-319-62127-2_1","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"27 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2016-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73596920","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 : 2016-05-01DOI: 10.1016/j.comgeo.2016.03.002
S. Dobrev, M. Hesari, Fraser MacQuarie, Ján Manuch, Oscar Morales-Ponce, L. Narayanan, J. Opatrny, L. Stacho
{"title":"Connectivity with directional antennas in the symmetric communication model","authors":"S. Dobrev, M. Hesari, Fraser MacQuarie, Ján Manuch, Oscar Morales-Ponce, L. Narayanan, J. Opatrny, L. Stacho","doi":"10.1016/j.comgeo.2016.03.002","DOIUrl":"https://doi.org/10.1016/j.comgeo.2016.03.002","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"1 1","pages":"1-25"},"PeriodicalIF":0.0,"publicationDate":"2016-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85281530","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 : 2016-04-11DOI: 10.1007/978-3-662-49529-2_48
Eunjin Oh, S. Bae, Hee-Kap Ahn
{"title":"Computing a Geodesic Two-Center of Points in a Simple Polygon","authors":"Eunjin Oh, S. Bae, Hee-Kap Ahn","doi":"10.1007/978-3-662-49529-2_48","DOIUrl":"https://doi.org/10.1007/978-3-662-49529-2_48","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"44 1","pages":"45-59"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87531405","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 : 2016-04-11DOI: 10.1007/978-3-662-49529-2_9
S. Bae, C. Shin, A. Vigneron
{"title":"Tight Bounds for Beacon-Based Coverage in Simple Rectilinear Polygons","authors":"S. Bae, C. Shin, A. Vigneron","doi":"10.1007/978-3-662-49529-2_9","DOIUrl":"https://doi.org/10.1007/978-3-662-49529-2_9","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"2015 1","pages":"40-52"},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87796955","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 : 2015-11-17DOI: 10.1016/j.comgeo.2017.10.004
P. Györgyi, Bálint Hujter, Sándor Kisfaludi-Bak
{"title":"On the number of touching pairs in a set of planar curves","authors":"P. Györgyi, Bálint Hujter, Sándor Kisfaludi-Bak","doi":"10.1016/j.comgeo.2017.10.004","DOIUrl":"https://doi.org/10.1016/j.comgeo.2017.10.004","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"28 1","pages":"29-37"},"PeriodicalIF":0.0,"publicationDate":"2015-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73232798","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 : 2015-09-14DOI: 10.1007/978-3-319-48532-4_16
E. Demaine, Matias Korman, Jason S. Ku, Joseph S. B. Mitchell, Y. Otachi, André van Renssen, Marcel Roeloffzen, Ryuhei Uehara, Yushi Uno
{"title":"Symmetric Assembly Puzzles are Hard, Beyond a Few Pieces","authors":"E. Demaine, Matias Korman, Jason S. Ku, Joseph S. B. Mitchell, Y. Otachi, André van Renssen, Marcel Roeloffzen, Ryuhei Uehara, Yushi Uno","doi":"10.1007/978-3-319-48532-4_16","DOIUrl":"https://doi.org/10.1007/978-3-319-48532-4_16","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"7 1","pages":"101648"},"PeriodicalIF":0.0,"publicationDate":"2015-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73941892","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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