Lower Bounds for Semialgebraic Range Searching and Stabbing Problems

In the semialgebraic range searching problem, we are given a set of n points in \({\mathbb {R}}^d \) and want to preprocess such that for any query belonging family constant complexity sets (Tarski cells), all intersecting can be reported or counted efficiently. When ranges composed simplices, problem is well-understood: it solved using S ( ) space with Q time \(S(n)Q(n)^d = \tilde{O}(n^d) where \(\tilde{O}(\cdot) notation hides polylogarithmic factors this trade-off tight (up o (1) factors). particular, there exist “low space” structures use O 1 − 1/ d [8, 25] “fast query” (log [9]. However, general ranges, only solutions known, but best [7] match same curve as simplex queries, \(\tilde{O}(n^{1-1/d}) time. It has been conjectured could done case open stayed unresolved. Here, disprove conjecture. We give first nontrivial lower bounds other related problems. More precisely, show data structure reporting between two concentric circles, call 2D annulus reporting, must \(S(n)={\overset{{\scriptscriptstyle o}}{\Omega }}(n^3/Q(n)^5) \({\overset{{\scriptscriptstyle }}(\cdot) factors, meaning, log , }}(n^3) used. addition, study subset input polynomial slab defined by \(\lbrace (x,y)\in {\mathbb {R}}^2:P(x)\le y\le P(x)+w\rbrace \(P(x)=\sum _{i=0}^\Delta a_i x^i univariate degree Δ \(a_0, \cdots, a_\Delta, w at time, reporting. For this, bound }}(n^{\Delta +1}/Q(n)^{(\Delta +3)\Delta /2}) which implies +1}) space. also consider dual stabbing problems searching, present them. linear space, solves Ω 2/3 Note almost matches upper obtained lifting annuli 3D. Like Again, our size case.