Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
Maintenance of a minimum spanning forest in a dynamic plane graph
Journal of Algorithms
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
Sparsification—a technique for speeding up dynamic graph algorithms
Journal of the ACM (JACM)
Journal of Computer and System Sciences
Nearest common ancestors: a survey and a new distributed algorithm
Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures
Integer Sorting in 0(n sqrt (log log n)) Expected Time and Linear Space
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Deterministic sorting in O(nlog logn) time and linear space
Journal of Algorithms
Time-space trade-offs for predecessor search
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Dynamic ordered sets with exponential search trees
Journal of the ACM (JACM)
Dynamic Subgraph Connectivity with Geometric Applications
SIAM Journal on Computing
Planning for Fast Connectivity Updates
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Oracles for Distances Avoiding a Failed Node or Link
SIAM Journal on Computing
Dual-failure distance and connectivity oracles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Dynamic Connectivity for Axis-Parallel Rectangles
Algorithmica
A nearly optimal oracle for avoiding failed vertices and edges
Proceedings of the forty-first annual ACM symposium on Theory of computing
Connectivity oracles for failure prone graphs
Proceedings of the forty-second ACM symposium on Theory of computing
New data structures for subgraph connectivity
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Dynamic Connectivity: Connecting to Networks and Geometry
SIAM Journal on Computing
Connectivity oracles for planar graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Connectivity oracles for planar graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
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We consider dynamic subgraph connectivity problems for planar undirected graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either "off" (failed) or "on" (recovered). We wish to answer connectivity queries with respect to the "on" subgraph. The model has two natural variants, one in which there are d edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a d-failure connectivity oracle for planar graphs that processes any d edge/vertex failures in sort(d,n) time so that connectivity queries can be answered in pred(d,n) time. (Here sort and pred are the time for integer sorting and integer predecessor search over a subset of [n] of size d.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette's theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Pǎtraşcu and Thorup's lower bound on predecessor search is that no d-failure connectivity oracle (even on trees) can beat pred(d,n) query time. We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in O(logn) time (amortized), whereas in general planar graphs both bounds become O(log2n).