No. 318 on SWAT 88: 1st Scandinavian workshop on algorithm theory
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Rectangular matrix multiplication revisited
Journal of Complexity
Fast rectangular matrix multiplication and applications
Journal of Complexity
Kinetic connectivity of rectangles
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Randomized fully dynamic graph algorithms with polylogarithmic time per operation
Journal of the ACM (JACM)
Near-optimal fully-dynamic graph connectivity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Simplified kinetic connectivity for rectangles and hypercubes
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Dynamic subgraph connectivity with geometric applications
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On Computing Connected Components of Line Segments
IEEE Transactions on Computers
A fully dynamic algorithm for maintaining the transitive closure
Journal of Computer and System Sciences - STOC 1999
Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Detecting short directed cycles using rectangular matrix multiplication and dynamic programming
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Testing bipartiteness of geometric intersection graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
A fully dynamic reachability algorithm for directed graphs with an almost linear update time
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A new approach to dynamic all pairs shortest paths
Journal of the ACM (JACM)
Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O(n2 barrier
Journal of the ACM (JACM)
Worst-case update times for fully-dynamic all-pairs shortest paths
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Fast sparse matrix multiplication
ACM Transactions on Algorithms (TALG)
A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Logarithmic Lower Bounds in the Cell-Probe Model
SIAM Journal on Computing
Combinatorica
Dynamic connectivity for axis-parallel rectangles
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Planning for Fast Connectivity Updates
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the twenty-sixth annual symposium on Computational geometry
New data structures for subgraph connectivity
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Analyzing graph structure via linear measurements
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Connectivity oracles for planar graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Hi-index | 0.01 |
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 us 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 $\widetilde{O}(m^{2/3})$ amortized time, where $m$ denotes the number of edges in the graph. This greatly improves upon the previous result [T. M. Chan, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 7-13], which required fast matrix multiplication and had an update time of $O(m^{0.94})$. The new data structure is also simpler. Geometric connectivity asks us to maintain a dynamic set of $n$ 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, $\widetilde{O}(n^{2/3})$, 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 two-dimensional line segments: $O^*(n^{9/10})$; for $d$-dimensional simplices: $O^*(n^{1-\frac{1}{d(2d+1)}})$; and for $d$-dimensional balls: $O^*(n^{1-\frac{1}{(d+1)(2d+3)}})$.