Timothy M. Chan

Timothy M. Chan, Isaac M. Hair

Given two convex polygons $P$ and $Q$ with $n$ and $m$ edges, the maximum overlap problem is to find a translation of $P$ that maximizes the area of its intersection with $Q$. We give the first randomized algorithm for this problem with linear running time. Our result improves the previous two-and-a-half-decades-old algorithm by de Berg, Cheong, Devillers, van Kreveld, and Teillaud (1998), which ran in $O((n+m)\log(n+m))$ time, as well as multiple recent algorithms given for special cases of the problem.

Timothy M. Chan, Hsien-Chih Chang, Jie Gao et al.

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [SoCG '22] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter $Δ$ is at least $Ω(\log n)$, hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: - A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. - A truly subquadratic time lower bound for \Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be truly subquadratic hard when the diameter is $Ω(\log n)$. - A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D. - A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

Timothy M. Chan

We consider the problem of triangulating a polygon with $n$ vertices and $h$ holes, or relatedly the problem of computing the trapezoidal decomposition of a collection of $h$ disjoint simple polygonal chains with $n$ vertices total. Clarkson, Cole, and Tarjan (1992) and Seidel (1991) gave randomized algorithms running in $O(n\log^*n + h\log h)$ time, while Bar-Yehuda and Chazelle (1994) described deterministic algorithms running in $O(n+h\log^{1+\varepsilon}h)$ or $O((n+h\log h)\log\log h)$ time, for an arbitrarily small positive constant $\varepsilon$. No improvements have been reported since. We describe a new $O(n + h\log h)$-time algorithm, which is optimal and deterministic. More generally, when the given polygonal chains are not necessarily simple and may intersect each other, we show how to compute their trapezoidal decomposition (and in particular, compute all intersections) in optimal $O(n + h\log h)$ deterministic time when the number of intersections is at most $n^{1-\varepsilon}$. To obtain these results, Chazelle's linear-time algorithm for triangulating a simple polygon is used as a black box.

Timothy M. Chan, Hsien-Chih Chang, Jie Gao et al.

Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time, while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis. We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include: 1. The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant. 2. An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms rely on succinct representation assuming bounded VC-dimension; for such a strategy $Ω(n^{7/4})$ time is an inherent barrier. 3. An $\tilde{O}(n^{4/3})$-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time $\tilde{O}(n^{2-1/9})$. 4. Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds only hold when deciding if diameter is at most 3.

Timothy M. Chan, Yuancheng Yu

We present an algorithm that computes the girth of the intersection graph of $n$ given line segments in the plane in $O(n^{1.483})$ expected time. This is the first such algorithm with $O(n^{3/2-\varepsilon})$ running time for a positive constant $\varepsilon$, and makes progress towards an open question posed by Chan (SODA 2023). The main techniques include (i)~the usage of recent subcubic algorithms for bounded-difference min-plus matrix multiplication, and (ii)~an interesting variant of the planar graph separator theorem. The result extends to intersection graphs of connected algebraic curves or semialgebraic sets of constant description complexity.

Sepideh Aghamolaei, Kevin Buchin, Timothy M. Chan et al.

Computing the convex hull of a planar $n$-point set $P$ is one of the most fundamental problems in computational geometry. It has an $Ω(n \log n)$ lower bound in the algebraic computation tree model, and many convex hull algorithms match this bound. Classical results show that, under special input assumptions, sub-$O(n \log n)$ algorithms are possible. For instance, when the points are given in lexicographic or angular order, the convex hull can be computed in linear time. Even under the weaker assumption that the sequence of points corresponds to the ordered vertices of a simple polygonal chain, linear-time algorithms exist. This naturally raises the question: can the convex hull of a point set be computed in sub-$O(n \log n)$ time under weaker input assumptions? We answer this positively. Under the promise that the input sequence contains the convex hull as a subsequence, we give a deterministic $O(n \sqrt{\log n})$-time algorithm to compute the convex hull of $P$. With randomisation, we achieve expected running time $O(n \log^{\varepsilon} n)$ for any constant $\varepsilon > 0$. We find this surprising, as points not on the convex hull may behave adversarially toward our convex hull construction algorithm. Yet the promise that \emph{only} the hull points are sorted suffices for $o(n \log n)$-time algorithms. Finally, we show that this promise is tight: if it is even slightly broken, i.e., allowing just one hull point to appear out of order, we prove an adversarial $Ω(n \log n)$-time lower bound. Consequently, the promise cannot be verified with fewer than $Ω(n \log n)$ comparisons. This also negatively resolves an open problem of Löffler and Raichel, who conjectured sub-$O(n \log n)$-time algorithms for computing the convex hull of a supersequence containing the hull as a subsequence.