Approximate Voronoi Cell Computation on Geometric Data Streams

Several studies have exploited the properties of Voronoi diagrams to improve variations of the nearest neighbor search on stored datasets. However, the significance of Voronoi diagrams and their basic building blocks, Voronoi cells, has been neglected when the geometry data is incrementally becoming available as a data stream. In this paper, we study the problem of Voronoi cell computation for fixed 2-d site points when the locations of the neighboring sites arrive as geometric data streams. We show that the non-streaming solution to the problem does not meet the memory requirements of streaming applications over a sliding window. Hence, we propose AVC and AVC-SW, two approximate streaming algorithms that compute ε-approximations to the actual Voronoi cell in O(κ) using O(κ) space where κ is their sample size. With the sliding window model, we prove both theoretically and experimentally that AVC-SW significantly reduces the average memory requirements of the classic algorithm, specially when the window size w is large, which is the case in real-world scenarios.