Optimal partition trees

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG’92, Matouˇsek gave a partition tree method for ddimensional simplex range searching achieving O(n) space and O(n1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n1+ε ) preprocessing time for any fixed ε > 0. An earlier method by Matouˇsek (SoCG’91) requires O(n log n) preprocessing time but O(n1−1/d logO(1) n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1−1/d ) query time with high probability. Our method has several advantages: • It is conceptually simpler than Matouˇsek’s SoCG’92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children’s cells at each node). • It leads to more efficient multileve...

• Type: Conference
• Year: 2010
• Where: COMPGEOM
• Authors: Timothy M. Chan
• Keywords: COMPGEOM 2010, Partition Tree Method, Partition Trees, Query Time