Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane

Let P be a set of m points in R², let Σ be a set of n semi-algebraic sets of constant complexity in R², let (S, +) be a semigroup, and let w : P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P ∩ σ) for every σ ∈ Σ in overall expected 2s time O^{∗(m5s−4 n5 5 s s − − 6 4} + m²/³n²/³ + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^∗(·) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an 2s−2 s on-line manner. The latter takes O^∗(m²s⁻¹ n²s⁻¹ + m + n) time. Let Φ: Σ × P → {0, 1} be the Boolean predicate (of constant complexity) such that Φ(σ, p) = 1 if p ∈ σ and 0 otherwise, and let Σ Φ P = {(σ, p) ∈ Σ × P | Φ(σ, p) = 1}. Our algorithm actually computes a partition B_Φ of Σ Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes 2s of the vertex sets of its bicliques) O^{∗(m5s−4 n5 5 s s − − 4 6} + m²/³n²/³ + m + n). It is straightforward to compute w(P ∩ σ) for all σ ∈ Σ from B_Φ. Similarly, if η : Σ → S is a weight function on the regions of Σ, ^P_σ∈_Σ:p∈σ η(σ), for every point p ∈ P, can be computed from B_Φ in a straightforward manner.