TY - GEN

T1 - Dynamic connectivity

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

AU - Chan, Timothy M.

AU - Pǎtra̧cu, Mihai

AU - Roditty, Liam

PY - 2008

Y1 - 2008

N2 - Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems: Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by on vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in Õ(m2/3) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC'02], which required fast matrix multiplication and had an update time of O(m0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set ofn geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, Õ(n2/3), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries. In particular, we obtain the first sublinear update time for arbitrary 2D line segments: O*(n 9/10); for d-dimensional simplices: O* (n 1-1/d(2d+1); and for d-dimensional balls: O*(n 1-1/(d-1)(2d+3)).

AB - Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems: Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by on vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in Õ(m2/3) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC'02], which required fast matrix multiplication and had an update time of O(m0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set ofn geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, Õ(n2/3), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries. In particular, we obtain the first sublinear update time for arbitrary 2D line segments: O*(n 9/10); for d-dimensional simplices: O* (n 1-1/d(2d+1); and for d-dimensional balls: O*(n 1-1/(d-1)(2d+3)).

UR - http://www.scopus.com/inward/record.url?scp=57949083674&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.29

DO - 10.1109/FOCS.2008.29

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AN - SCOPUS:57949083674

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 95

EP - 104

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -