On Semialgebraic Range Reporting
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In the problem of semialgebraic range searching, we are to preprocess a set of points in ℝ^D such that the subset of points inside a semialgebraic region described by O(1) polynomial inequalities of degree Δ can be found efficiently. Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" structures [AMS13,MP15] with almost optimal query time of Q(n)=O(n^1-1/D+o(1)) were obtained. For "fast query" structures (i.e., when Q(n)=n^o(1)), it was conjectured that a structure with space S(n) = O(n^D+o(1)) is possible. The conjecture was refuted recently by Afshani and Cheng [AC21]. In the plane, they proved that S(n) = Ω(n^Δ+1 - o(1)/Q(n)^(Δ+3)Δ/2) which shows Ω(n^Δ+1-o(1)) space is needed for Q(n) = n^o(1). While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of n or Q(n) seem to be tight even for D=2, as the current upper bound is S(n) = O(n^m+o(1)/Q(n)^(m-1)D/(D-1)) where m=D+ΔD-1 = Ω(Δ^D) is the maximum number of parameters to define a monic degree-Δ D-variate polynomial, for any D,Δ=O(1). In this paper, we resolve two of the issues: we prove a lower bound in D-dimensions and show that when Q(n)=n^o(1)+O(k), S(n)=Ω(n^m-o(1)), which is almost tight as far as the exponent of n is considered in the pointer machine model. When considering the exponent of Q(n), we show that the analysis in [AC21] is tight for D=2, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or a new fundamentally different input set is needed to get a better lower bound.
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