Instance-Optimal Imprecise Convex Hull

TL;DR

This work tackles the problem of reconstructing the convex hull order for imprecise point sets where each point lies in a known region. It introduces an instance-optimal reconstruction strategy that simultaneously minimizes retrievals and achieves polylogarithmic time per retrieval for families of simple polygons, extending to simple k-gons and disks with radii constrained by k. Central to the approach are the Witness-based reconstruction framework, the augmentation of the Partial Hull Tree (PHT), and the Median Cut Decomposition (MCD) data structure, which together enable efficient handling of overlapping regions in 2D and enable near-optimal preprocessing and query performance. The results significantly generalize prior instance-optimal work by supporting overlapping regions in 2D and provide practical data-structures to realize polylog per-retrieval costs, with linear dependence on k for disk families.

Abstract

Imprecise measurements of a point set P = (p1, ..., pn) can be modelled by a family of regions F = (R1, ..., Rn), where each imprecise region Ri contains a unique point pi. A retrieval models an accurate measurement by replacing an imprecise region Ri with its corresponding point pi. We construct the convex hull of an imprecise point set in the plane, where regions in F may be retrieved at unit cost. The goal is to determine the cyclic ordering of the convex hull vertices of P as efficiently as possible. Here, efficiency is interpreted in two ways: (i) minimising the number of retrievals, and (ii) computing each retrieval location quickly. Prior works focused on only one of these two aspects: either minimising retrievals or optimising algorithmic runtime. Our contribution is the first to simultaneously achieve both. Let r(F, P) denote the minimal number of retrievals required by any algorithm to determine the convex hull of P for a given instance (F, P). For a family F of n constant-complexity polygons, our main result is a reconstruction algorithm that performs O(r(F, P)) retrievals in O(r(F, P) log^3 n) time. Compared to previous approaches that achieve optimal retrieval counts, we improve the runtime per retrieval by a exponential factor, from polynomial to polylogarithmic. Compared to near-linear time algorithms, we significantly reduce the number of retrievals used, and broaden the input families to include overlapping regions. We further extend our results to simple k-gons and to pairwise disjoint disks with radii in [1,k], where our runtime scales linearly with k.

Figures (7)

• Figure 1: (a) A family of regions $F = R_1,\ldots,R_5$ and a sequence $P = (p_1,\ldots,p_5)$ with $P \sim F$. (b) If we retrieve $R_3$ and $R_4$ to obtain $F'$ then for all $P' \sim F'$ the convex hull equals $(p_1, p_3, p_2, p_4)$. Note that if $p_3$ would lie in $R_2 \cap R_3$ instead, then retrieving only $R_3$ and $R_4$ does not suffice.
• Figure 2: (a) Regions may have bends and sharp corners, and, be points. (b) Regions may not be unbounded, non-simple, connected by a line, or have infinitely many sharp corners. (c) Vertices between regions may coincide, and point regions ($R_3$) may coincide with vertices of other regions.
• Figure 3: Three examples of a band (blue) between regions (green, brown).
• Figure 4: We illustrate all (sub) cases of vertex pairs corresponding to Definition \ref{['def:edge_label']}: All pairs that include $t$ are non-canonical, as $t$ is a vertex of more than one non-point region.All pairs in $\{r,s,u,v, w\}^2$ are all canonical, as $v$ is exclusively a vertex of two point regions.The pair $(u,v)$ is non-dividing, as $R_3$ and $R_4$ are not vertically separated.The pairs $(s, u)$, $(r,u)$, $(r,v)$, $(s,v)$, and any pair in $\{ r, s, u, v \} \times \{ w \}$ are dividing because their regions are vertically separated. $(r,s)$ is dividing because they are vertices of the same region.The pair $(r,s)$ is not occupied since $\mathop{\mathrm{band}}\nolimits(R_1,R_1)$ only contains vertices of $R_1$. The pair $(v,w)$ is not occupied since $\mathop{\mathrm{band}}\nolimits(R_4,R_6)$ only contains vertices $v$ and $w$.The pair $(u, w)$ is occupied as $V(F - R_3 - R_6)$ contains the vertex $t \in R_2$. The pairs $(s, u)$, $(r, u)$, $(r, v)$, $(s, v)$, $(r, w)$, and $(s, w)$ are also occupied.
• Figure 5: The convex chain from $q$ to $t$ is spanning. The chain from $r$ to $v$ is not spanning.

Theorems & Definitions (59)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 6
• Definition 7: PHT from DBLP:conf/stoc/OvermarsV80
• Definition 8: Band. Fig. \ref{['fig:definition_band']}
• Definition 9: Fig. \ref{['fig:canonical-dividing-occupied']}
• Definition 10: Spanning chain. Fig. \ref{['fig:spanning-chain']}
• Lemma 10