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

Let P be a set of m points in ℝ^2, let Σ be a set of n semi-algebraic sets of constant complexity in ℝ^2, 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 time O^*( m^2s/5s-4n^5s-6/5s-4 + m^2/3n^2/3 + 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 on-line manner. The latter takes O^*(m^s/2s-1n^2s-2/2s-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 ΣΦ P = { (σ,p) ∈Σ× P |Φ(σ,p)=1}. Our algorithm actually computes a partition ℬ_Φ of ΣΦ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) O^*( m^2s/5s-4n^5s-6/5s-4 + m^2/3n^2/3 + 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, ∑_σ∈Σ: p ∈ση(σ), for every point p∈ P, can be computed from ℬ_Φ in a straightforward manner. A recent work of Chan et al. solves the online version of this dual point enclosure problem within the same performance bound as our off-line solution. We also mention a few other applications of computing ℬ_Φ.