Comparing distributions and shapes using the kernel distance
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric computations on indecisive points
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Range counting coresets for uncertain data
Proceedings of the twenty-ninth annual symposium on Computational geometry
ACM Transactions on Database Systems (TODS) - Invited papers issue
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Consider a point set ${\mathcal{D}}$ with a measure function$\mu : {\mathcal{D}} \to \mathcal{R}$. Let ${\mathcal{A}}$ be theset of subsets of $\mathcal{D}$ induced by containment in a shapefrom some geometric family (e.g. axis-aligned rectangles, halfplanes, balls, k-oriented polygons). We say a range space$(\mathcal{D}, \mathcal{A})$ has anε-approximation P if$$\max_{R \in \mathcal{A}} \left| \frac{\mu(R \cap P)}{ \mu(P)}- \frac{\mu(R \cap \mathcal{D})}{ \mu(\mathcal{D})} \right| \leq\varepsilon.$$We describe algorithms for deterministically constructingdiscrete µ-approximations for continuous point setssuch as distributions or terrains. Furthermore, for certainfamilies of subsets $\mathcal{A}$, such as those described byaxis-aligned rectangles, we reduce the size of theε-approximations by almost a square root from$O(\frac{1}{\varepsilon^2} \log \frac{1}{\varepsilon})$ to . Thisis often the first step in transforming a continuous problem into adiscrete one for which combinatorial techniques can be applied. Wedescribe applications of this result in geospatial analysis,biosurveillance, and sensor networks.