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• W48645953 abstract "In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an n-vertex graph G = (V,E,w) with nonnegative edge weights w : E → R. Our goal is to maintain a tree T which is a good approximation of the minimum Steiner tree in G when the terminal set S ⊆ V changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). The Steiner tree problem is one of the core problems in combinatorial optimization. It has been studied in many different settings, starting from classical approximation algorithms, through online and stochastic models, ending with game theoretic approaches. However, almost nothing is known about the Steiner tree problem in a dynamic setting. The reason for this seems to be the lack of strong enough tools in both online approximation algorithms and dynamic algorithms. In this paper we develop appropriate methods that contribute to these both areas. The first ingredient is not restricted to planar graphs and contributes to the area of online algorithms. We prove that the maintained tree T after each update does not change too much and that it can be recomputed using a vertex-to-label distance oracle. A vertex-to-label oracle allows one to label the vertices and ask for the distance between a given vertex and the set of vertices marked with a particular label. A dynamic vertex-to-label oracle additionally supports merge and split operations on the label sets. We show that in incremental scenario O(log n) calls to the oracle per update operation suffice, in decremental O(1) calls are sufficient, whereas in fully dynamic scenario O(logD) calls are enough. Here, D denotes the stretch of the metric induced by G. The second ingredient, which contributes to the area of dynamic algorithms, is a construction of a fast (1+ )-approximate vertex-to-label distance oracles for planar graphs. In the case when labels’ sets can only grow (incremental scenario) we give an O( −1 log n logD) time data-structure. When the full set of operations is supported (i.e., merge and split on labels’ sets) we give O( −1 √ n log n logD) time data-structure. As an interesting side result, we present an exact incremental variant of the oracle working in O( √ n log n) amortized time and a fully-dynamic variant working in O(n log n) worst case time. The above oracles imply the first known sublinear time algorithm for dynamic Steiner tree problem in planar graphs. In particular we develop a polylogarithmic time incremental algorithm, an O( √ n log n logD) time decremental algorithm and an O( √ n log n logD) time fully-dynamic algorithm. ∗Institute of Informatics, University of Warsaw, Poland, j.lacki@mimuw.edu.pl. Jakub Łącki is a recipient of the Google Europe Fellowship in Graph Algorithms, and this research is supported in part by this Google Fellowship. †Institute of Informatics, University of Warsaw, Poland, netkuba@gmail.com. Partially supported by ERC grant PAAl no. 259515. ‡Institute of Informatics, University of Warsaw, Poland, malcin@mimuw.edu.pl. Partially supported by ERC grant PAAl no. 259515 and the Foundation for Polish Science. §Institute of Informatics, University of Warsaw, Poland, sank@mimuw.edu.pl. Partially supported by ERC grant PAAl no. 259515 and the Foundation for Polish Science. ¶Institute of Informatics, University of Warsaw, Poland, anka@mimuw.edu.pl. Partially supported by ERC grant PAAl no. 259515. ar X iv :1 30 8. 33 36 v2 [ cs .D S] 1 6 A ug 2 01 3" @default.
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• W48645953 date "2013-08-15" @default.
• W48645953 modified "2023-09-27" @default.
• W48645953 title "Dynamic Steiner Tree in Planar Graphs." @default.
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