STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Navigating nets: simple algorithms for proximity search
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
A computational framework for incremental motion
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Adaptively sampled particle fluids
ACM SIGGRAPH 2007 papers
An Optimal Dynamic Spanner for Doubling Metric Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Multi-dimensional online tracking
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Deformable spanners and applications
Computational Geometry: Theory and Applications
Kinetic convex hulls and delaunay triangulations in the black-box model
Proceedings of the twenty-seventh annual symposium on Computational geometry
Tracking moving objects with few handovers
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Competitive query strategies for minimising the ply of the potential locations of moving points
Proceedings of the twenty-ninth annual symposium on Computational geometry
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The problem of maintaining geometric structures for points in motion has been well studied over the years. Much theoretical work to date has been based on the assumption that point motion is continuous and predictable, but in practice, motion is typically presented incrementally in discrete time steps and may not be predictable. We consider the problem of maintaining a data structure for a set of points undergoing such incremental motion. We present a simple online model in which two agents cooperate to maintain the structure. One defines the data structure and provides a collection of certificates, which guarantee the structure's correctness. The other checks that the motion over time satisfies these certificates and notifies the first agent of any violations.We present efficient online algorithms for maintaining both nets and net trees for a point set undergoing incremental motion in a space of constant dimension. We analyze our algorithms' efficiencies by bounding their competitive ratios relative to an optimal algorithm. We prove a constant factor competitive ratio for maintaining a slack form of nets, and our competitive ratio for net trees is proportional to the square of the tree's height.