I/O-Efficient Rectangular Segment Search

Abstract-We consider the I/O-efficient rectangular segment search problem in 2D. The problem involves storing a given set S of N line segments in a data structure such that an axis aligned rectangular range query R can be performed efficiently; i.e., report all line segments in S which intersect R. We give a data structure requiring space O(N(N/B)²) disk blocks that can answer a range query R using O(Iog_BN + K/B) I/Os, where B is the number of line segments transferred in one I/O, and K is the number of line segments intersecting R. Search complexity of O(log_B(N/B) + K/B) I/Os can be achieved with reduced storage if the set S contains only non-intersecting line segments, or if set S contains only horizontal and vertical line segments. In the former case the space complexity is O((N/B)²) disk blocks and in the latter case the space complexity is O(N log N/log log_B N). We also consider the problem of finding all the line segments which are entirely within the rectangle R if the set S contains only vertical and horizontal line segments. For this problem, an optimal data structure is presented with size O(N log N/log log_B N) disk blocks that requires O(log_B(N/B) + K/B) I/Os to answer the query.