Convexity Helps Iterated Search in 3D

TL;DR

This work addresses the problem of iterated search in 3D where each graph vertex holds a hyperplane set and queries must determine, for a connected subgraph, whether a point lies below the associated lower envelopes. The authors develop LEIS3D, a linear-space data structure delivering query times near \\mathcal{O}(\\log n) plus a sublinear term in the size of the queried subgraph, by combining shallow cuttings with 3D convex-overlay point-location techniques. They extend the framework to on-demand and anchored variants and demonstrate broad applicability, notably advancing 3D halfspace max and related geometric querying tasks with improved time bounds. The results challenge planar lower-bound assumptions by exploiting convexity and 3D structure, offering practical tools for high-dimensional geometric data processing and related data-structure problems.

Abstract

Inspired by the classical fractional cascading technique, we introduce new techniques to speed up the following type of iterated search in 3D: The input is a graph $\mathbf{G}$ with bounded degree together with a set $H_v$ of 3D hyperplanes associated with every vertex of $v$ of $\mathbf{G}$. The goal is to store the input such that given a query point $q\in \mathbb{R}^3$ and a connected subgraph $\mathbf{H}\subset \mathbf{G}$, we can decide if $q$ is below or above the lower envelope of $H_v$ for every $v\in \mathbf{H}$. We show that using linear space, it is possible to answer queries in roughly $O(\log n + |\mathbf{H}|\sqrt{\log n})$ time which improves trivial bound of $O(|\mathbf{H}|\log n)$ obtained by using planar point location data structures. Our data structure can in fact answer more general queries (it combines with shallow cuttings) and it even works when $\mathbf{H}$ is given one vertex at a time. We show that this has a number of new applications and in particular, we give improved solutions to a set of natural data structure problems that up to our knowledge had not seen any improvements. We believe this is a very surprising result because obtaining similar results for the planar point location problem was known to be impossible.

Figures (2)

• Figure 1: A schematic drawing of the LEIS Problem (with hyperplanes in $\mathbb{R}^2$ rather than $\mathbb{R}^3$). Each vertex $v$ of $\mathbf{G}\xspace$ is associated with a set of hyperplanes $H_v$. For each vertex $v$ in the query subgraph $\mathbf{H}\xspace \subset \mathbf{G}\xspace$ we wish to find the triangle in the $k$-shallow cutting $\mathscr{C}_v$ above the query point $q$.
• Figure 2: The inner and outer simplifications are shown in 2D, for clarity. The central polytope is being simplified here. To the left, we have its inner simplification after deleting the vertices marked with red and to the right, we have the outer simplification after deleting the edges marked with blue.

Theorems & Definitions (7)

• Lemma 1
• Theorem 2
• Lemma 2
• Lemma 2: basic point location
• Lemma 2: advanced point location
• Lemma 3
• Lemma 3