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

TL;DR

A randomized algorithm for computing w(P\cap\sigma) for every $\sigma\in\Sigma$ in overall expected time is described, which solves the online version of this dual point enclosure problem within the same performance bound as the off-line solution.

Abstract

Let $P$ be a set of $m$ points in ${\mathbb R}^2$, let $Σ$ be a set of $n$ semi-algebraic sets of constant complexity in ${\mathbb R}^2$, let $(S,+)$ be a semigroup, and let $w: P \rightarrow S$ be a weight function on the points of $P$. We describe a randomized algorithm for computing $w(P\capσ)$ for every $σ\inΣ$ in overall expected time $O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$, where $s>0$ is a constant that bounds the maximum complexity of the regions of $Σ$, and where the $O^*(\cdot)$ notation hides subpolynomial factors. For $s\ge 3$, surprisingly, this bound is smaller than the best-known bound for answering $m$ such queries in an on-line manner. The latter takes $O^*(m^{\frac{s}{2s-1}}n^{\frac{2s-2}{2s-1}}+m+n)$ time. Let $Φ: Σ\times P \rightarrow \{0,1\}$ be the Boolean predicate (of constant complexity) such that $Φ(σ,p) = 1$ if $p\inσ$ and $0$ otherwise, and let $Σ\mathopΦ P = \{ (σ,p) \in Σ\times P \mid Φ(σ,p)=1\}$. Our algorithm actually computes a partition ${\mathcal B}_Φ$ of $Σ\mathopΦ P$ into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) $O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$. It is straightforward to compute $w(P\capσ)$ for all $σ\in Σ$ from ${\mathcal B}_Φ$. Similarly, if $η: Σ\rightarrow S$ is a weight function on the regions of $Σ$, $\sum_{σ\in Σ: p \in σ} η(σ)$, for every point $p\in P$, can be computed from ${\mathcal B}_Φ$ 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 ${\mathcal B}_Φ$.

Theorems & Definitions (9)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Lemma 3.1
• Lemma 4.1: Matoušek and Patáková MP
• Lemma 4.2: Barone and Basu BB12