Các phân đoạn giao với đường được lưu trữ trong cả hai. Quá trình này tiếp tục cho đến khi chỉ có một vài phân đoạn của một màu, và các thuật toán sức mạnh vũ phu có thể được chạy cho họ. R-tree cho phép Ο ((n + k) * log (n + k)) chạy thời gian và Ο (n + k) không gian trong trường hợp xấu nhất (van Oosterom, 1994). Nó dựa trên cấu trúc dữ liệu không gian R-Tree (Guttman, 1984). | Map Overlay Problem lie strictly either on the left or right side of the line are stored in the corresponding subtree. The segments that intersect the line are stored in both. The process continues until there are only a few segments of one color and the brute force algorithm can be run for them. R-tree allows o n k log n k running time and o n k space in the worst case van Oosterom 1994 . It is based on the spatial data structure R-Tree Guttman 1984 . It could also be used to deal with distributed data in a non-uniform way. The insertion algorithm creates more nodes in the areas with a larger number of line segments. Thus it will automatically adapt to the distribution of the data. The R-Tree has been designed to work efficiently with large amounts of data that do not fit in primary memory. Therefore it can handle the expensive operations of swapping memory pages between primary and secondary memory. Spatial Ordering Algorithms Because algorithms based on plane partitions divide the space without considering the topology of the input they run in quadratic time. However we can consider spatial ordering algorithms that use an aboveness ordering to reduce the number of comparisons between segments. A segment A is above B if some vertical line intersects A at a greater y-coordinate than B. Some examples are Andrews et al. 1994 Bentley-Ottman sweep has a running time in the worst and average case of o n log n k log n and a o n size space Bentley Ottman 1979 . This algorithm is based on the plane sweep technique. A vertical line is moved across the plane stopping at each node event of the subdivisions. It uses two data structures a priority queue for storing the nodes of both subdivisions ordered by the x-coordinate in aboveness order and a binary search tree for the segments that intersect the line at the events if corresponding. The initial subdivisions are stored in a Doubly-Connected Edge List DCEL frequently used for representing planar subdivisions of the space .
