Dynamic Convex Hulls for Simple Paths

TL;DR

The paper tackles planar dynamic convex hull with restricted updates (deque-based) by developing worst-case optimal data structures for monotone-path and simple-path variants. It introduces a two-stack deque framework and incremental rebuilding to achieve constant-time updates and O(log n) hull-queries, with hull reporting at O(h+log n); one-sided monotone variations further tighten reporting to O(h) and queries to O(log h). For the simple-path case, a four-part path decomposition and advanced tangent-management enable similar update/query bounds and O(h+log n) reporting, supported by two key Guibas–Hershberger–Snoeyink lemmas. The authors prove matching lower bounds via reductions from set-inclusion, establishing the optimality of their results and extending prior amortized bounds to worst-case performance. Overall, the work advances dynamic convex hulls under restricted updates, offering practical, provable guarantees for real-time geometric maintenance with broad potential applications in computational geometry systems.

Abstract

We consider the planar dynamic convex hull problem. In the literature, solutions exist supporting the insertion and deletion of points in poly-logarithmic time and various queries on the convex hull of the current set of points in logarithmic time. If arbitrary insertion and deletion of points are allowed, constant time updates and fast queries are known to be impossible. This paper considers two restricted cases where worst-case constant time updates and logarithmic time queries are possible. We assume all updates are performed on a deque (double-ended queue) of points. The first case considers the monotonic path case, where all points are sorted in a given direction, say horizontally left-to-right, and only the leftmost and rightmost points can be inserted and deleted. The second case assumes that the points in the deque constitute a simple path. Note that the monotone case is a special case of the simple path case. For both cases, we present solutions supporting deque insertions and deletions in worst-case constant time and standard queries on the convex hull of the points in $O(\log n)$ time, where $n$ is the number of points in the current point set. The convex hull of the current point set can be reported in $O(h+\log n)$ time, where $h$ is the number of edges of the convex hull. For the 1-sided monotone path case, where updates are only allowed on one side, the reporting time can be reduced to $O(h)$, and queries on the convex hull are supported in $O(\log h)$ time. All our time bounds are worst case. In addition, we prove lower bounds that match these time bounds, and thus our results are optimal. For a quick comparison, the previous best update bounds for the simple path problem were amortized $O(\log n)$ time by Friedman, Hershberger, and Snoeyink [SoCG 1989].

Figures (5)

• Figure 1: The convex hull (dashed) of a simple path $p_1,\ldots,p_n$ (solid). Three types of convex hull queries are shown (dotted): the tangent points $t_1$ and $t_2$ with a query point $q$ outside the convex hull; the extreme point $p^\rho$ in direction $\rho$; and the two convex hull edges $e_1$ and $e_2$ intersecting a line $\ell$.
• Figure 2: Illustrating a deque tree, comprising two stack trees separated by the vertical line $\ell$.
• Figure 4: A schematic view of the deque path convex hull data structure $PH(\pi)$.
• Figure 5: Illustrating the walking procedure for Case 2. The dashed arcs illustrate $A_1$ and $A_2$ on $\partial \mathcal{H}(\pi_m)$ and $\partial\mathcal{H}(\pi_f)$, respectively
• Figure 6: Illustrating the convex hull $\mathcal{H}(\pi)$ in Case 6.

Theorems & Definitions (12)

• Lemma 1
• Lemma 3
• Lemma 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 8
• Lemma 9
• Lemma 10
• Lemma 11