Before revision:  The first is a data structure for answering point-location queries among a family of semi-algebraic sets in \mathbb{R}^d in O(\log n) time ; the second is data structure for answering range search queries with semi-algebraic ranges in  O(\log n) time ;
After revision:  The first is a data structure for answering point-location queries among a family of semi-algebraic sets in \mathbb{R}^d in O(\log n) time , with storage complexity and expected preprocessing time of O(n^{d+\epsilon data structure for answering range search queries with semi-algebraic ranges in O(\log n) time ;