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We present the first algorithm for finding holes in high dimensional data that runs in polynomial time with respect to the number of dimensions. Previous algorithms are exponential. Finding large empty rectangles or boxes in a set of points in 2D and 3D space has been well studied. Efficient algorithms exist to identify the empty regions in these low-dimensional spaces. Unfortunately such efficiency is lacking in higher dimensions where the problem has been shown to be NP-complete when the dimensions are included in the input. Applications for algorithms that find large empty spaces include big data analysis, recommender systems, automated knowledge discovery, and query optimization. Our Monte Carlo-based algorithm discovers interesting maximal empty hyper-rectangles in cases where dimensionality and input size would otherwise make analysis impractical. The run-time is polynomial in the size of the input and the number of dimensions. We apply the algorithm on a 39-dimensional data set for protein structures and discover interesting properties that we think could not be inferred otherwise.

Big Holes in Big Data: A Monte Carlo Algorithm for Detecting Large Hyper-Rectangles in High Dimensional Data