The Bichromatic Rectangle Problem in High Dimensions

Given a set of blue points and a set of red points in ddimensional space, we show how to find an axis-aligned hyperrectangle that contains no red points and as many blue points as possible. Our algorithm enumerates the set of relevant hyperrectangles (inclusion maximal axisaligned hyperrectangles that do not contain a red point) and counts the number of blue points in each one. The runtime of our algorithm depends on the total number of relevant hyperrectangles. We prove asymptotically tight bounds on this quantity in the worst case. The techniques developed directly apply to the maximum empty rectangle problem in high dimensions.

• Type: Conference
• Year: 2009
• Where: CCCG
• Authors: Jonathan Backer, J. Mark Keil
• Keywords: Blue Points, CCCG 2007, CCCG 2009, Red Point, Relevant Hyperrectangles