Towards a dynamic data structure for efficient bounded line range search

We present a data structure for efficient axis-aligned orthogonal range search on a set of n lines in a bounded plane. The algorithm requires O(log n+ k) time in the worst case to find all lines intersecting an axis aligned query rectangle R, where k is the number of lines in range. O(n + λ) space is required for the data structure used by the algorithm, where λ is the number of intersection points among the lines. Insertion of a new rightmost line ` or deletion of a leftmost line ` requires O(n) time in the worst case. For a sparse arrangement of lines (i.e., for λ = O(n)), insertion of a rightmost line ` or deletion of a leftmost line ` requires O( √ n) time, and O(log n + μ) expected time for μ the number of intersection points between ` and existing lines.