A linear-time algorithm for the center problem in weighted cycle graphs
We study the problem of computing the center of cycle graphs whose vertices are weighted. The distance from a vertex to a point of the graph is defined as the weight of the vertex times the length of the shortest path between the vertex and the point. The weighted center of the graph is a point of t...
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| Vydáno v: | Information processing letters Ročník 186; s. 106495 |
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Elsevier B.V
01.08.2024
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | We study the problem of computing the center of cycle graphs whose vertices are weighted. The distance from a vertex to a point of the graph is defined as the weight of the vertex times the length of the shortest path between the vertex and the point. The weighted center of the graph is a point of the graph such that the maximum distance of the vertices of the graph to the point is minimum among all points of the graph. We present an O(n)-time algorithm for the discrete and continuous weighted center problem on cycle graphs with n vertices. Our algorithm improves upon the best known algorithm that takes O(nlogn) time. Moreover, it is optimal for the weighted center problem of cycle graphs.
•We give an O(n)-time algorithm for the discrete and continuous weighted center problem on cycle graphs with n vertices. Our algorithm improves upon the best known algorithm by Rayco et al. that takes O(nlogn) time.•For a cycle graph with n vertices embedded in a unit circle S, we define the dominating interval of a vertex v with respect to a subset W of vertices as the set of points in S whose farthest vertex is v among the vertices in W.•Then we show that the weighted center can be computed using the active vertices in V and their dominating intervals.•The main algorithmic contribution lies in computing the sequence of active vertices in V in O(n) time. Compared to the previous algorithm, our algorithm is simple, straightforward to implement, and robust. |
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| AbstractList | We study the problem of computing the center of cycle graphs whose vertices are weighted. The distance from a vertex to a point of the graph is defined as the weight of the vertex times the length of the shortest path between the vertex and the point. The weighted center of the graph is a point of the graph such that the maximum distance of the vertices of the graph to the point is minimum among all points of the graph. We present an O(n)-time algorithm for the discrete and continuous weighted center problem on cycle graphs with n vertices. Our algorithm improves upon the best known algorithm that takes O(nlogn) time. Moreover, it is optimal for the weighted center problem of cycle graphs.
•We give an O(n)-time algorithm for the discrete and continuous weighted center problem on cycle graphs with n vertices. Our algorithm improves upon the best known algorithm by Rayco et al. that takes O(nlogn) time.•For a cycle graph with n vertices embedded in a unit circle S, we define the dominating interval of a vertex v with respect to a subset W of vertices as the set of points in S whose farthest vertex is v among the vertices in W.•Then we show that the weighted center can be computed using the active vertices in V and their dominating intervals.•The main algorithmic contribution lies in computing the sequence of active vertices in V in O(n) time. Compared to the previous algorithm, our algorithm is simple, straightforward to implement, and robust. |
| ArticleNumber | 106495 |
| Author | Ahn, Hee-Kap Eom, Taekang |
| Author_xml | – sequence: 1 givenname: Taekang surname: Eom fullname: Eom, Taekang email: tkeom0114@postech.ac.kr organization: Department of Computer Science and Engineering, Pohang University of Science and Technology, Cheongam-Ro 77, Pohang, 37673, Republic of Korea – sequence: 2 givenname: Hee-Kap surname: Ahn fullname: Ahn, Hee-Kap email: heekap@postech.ac.kr organization: Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, Cheongam-Ro 77, Pohang, 37673, Republic of Korea |
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| Cites_doi | 10.1016/S0305-0548(99)00019-2 10.1016/S0020-0190(99)00111-8 10.1137/18M1196522 10.1006/jagm.1996.0054 10.1137/0137040 10.1016/j.tcs.2007.02.033 10.1137/0212052 |
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| References_xml | – start-page: 299 year: 1991 end-page: 319 ident: br0030 article-title: Parametric search and locating supply centers in trees publication-title: Workshop on Algorithms and Data Structures – volume: 12 start-page: 759 year: 1983 end-page: 776 ident: br0070 article-title: Linear-time algorithms for linear programming in publication-title: SIAM J. Comput. – volume: 378 start-page: 237 year: 2007 end-page: 252 ident: br0010 article-title: Efficient algorithms for center problems in cactus networks publication-title: Theor. Comput. Sci. – volume: 71 start-page: 205 year: 1999 end-page: 212 ident: br0060 article-title: A linear-time algorithm for solving the center problem on weighted cactus graphs publication-title: Inf. Process. Lett. – volume: 21 start-page: 453 year: 1996 end-page: 475 ident: br0040 article-title: Off-line maintenance of planar configurations publication-title: J. Algorithms – volume: 50 start-page: 602 year: 2021 end-page: 635 ident: br0090 article-title: An publication-title: SIAM J. Comput. – volume: 26 start-page: 1113 year: 1999 end-page: 1124 ident: br0080 article-title: A publication-title: Comput. Oper. Res. – year: 2019 ident: br0020 article-title: The weighted publication-title: 30th International Symposium on Algorithms and Computation (ISAAC 2019) – volume: 37 start-page: 513 year: 1979 end-page: 538 ident: br0050 article-title: An algorithmic approach to network location problems. I: the publication-title: SIAM J. Appl. Math. – volume: 26 start-page: 1113 issue: 10–11 year: 1999 ident: 10.1016/j.ipl.2024.106495_br0080 article-title: A p-center grid-positioning aggregation procedure publication-title: Comput. Oper. Res. doi: 10.1016/S0305-0548(99)00019-2 – start-page: 299 year: 1991 ident: 10.1016/j.ipl.2024.106495_br0030 article-title: Parametric search and locating supply centers in trees – volume: 71 start-page: 205 issue: 5–6 year: 1999 ident: 10.1016/j.ipl.2024.106495_br0060 article-title: A linear-time algorithm for solving the center problem on weighted cactus graphs publication-title: Inf. Process. Lett. doi: 10.1016/S0020-0190(99)00111-8 – volume: 50 start-page: 602 issue: 2 year: 2021 ident: 10.1016/j.ipl.2024.106495_br0090 article-title: An O(nlogn)-time algorithm for the k-center problem in trees publication-title: SIAM J. Comput. doi: 10.1137/18M1196522 – year: 2019 ident: 10.1016/j.ipl.2024.106495_br0020 article-title: The weighted k-center problem in trees for fixed k – volume: 21 start-page: 453 issue: 3 year: 1996 ident: 10.1016/j.ipl.2024.106495_br0040 article-title: Off-line maintenance of planar configurations publication-title: J. Algorithms doi: 10.1006/jagm.1996.0054 – volume: 37 start-page: 513 issue: 3 year: 1979 ident: 10.1016/j.ipl.2024.106495_br0050 article-title: An algorithmic approach to network location problems. I: the p-centers publication-title: SIAM J. Appl. Math. doi: 10.1137/0137040 – volume: 378 start-page: 237 issue: 3 year: 2007 ident: 10.1016/j.ipl.2024.106495_br0010 article-title: Efficient algorithms for center problems in cactus networks publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2007.02.033 – volume: 12 start-page: 759 issue: 4 year: 1983 ident: 10.1016/j.ipl.2024.106495_br0070 article-title: Linear-time algorithms for linear programming in R3 and related problems publication-title: SIAM J. Comput. doi: 10.1137/0212052 |
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| Title | A linear-time algorithm for the center problem in weighted cycle graphs |
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