Square packing with $O(x^{0.6})$ wasted area
We show a new construction for square packing, and prove that it is more efficient than previous results.
cs.CG↗
arXiv subjects
Hong Duc Bui
Publications and source records attributed to Hong Duc Bui.
Improved All-Pairs Approximate Shortest Paths in Congested Clique
In this paper, we present a new randomized $O(1)$-approximation algorithm for the All-Pairs Shortest Paths (APSP) problem in weighted undirected graphs that runs in just $O(\log \log \log n)$ rounds in the Congested-Clique model. Before our work, the fastest algorithms achieving an $O(1)$-approximation for APSP in weighted undirected graphs required $\operatorname{poly}(\log n)$ rounds, as shown by Censor-Hillel, Dory, Korhonen, and Leitersdorf (PODC 2019 & Distributed Computing 2021). In the unweighted undirected setting, Dory and Parter (PODC 2020 & Journal of the ACM 2022) obtained $O(1)$-approximation in $\operatorname{poly}(\log \log n)$ rounds. By terminating our algorithm early, for any given parameter $t \geq 1$, we obtain an $O(t)$-round algorithm that guarantees an $O\left(\log^{1/2^t} n\right)$ approximation in weighted undirected graphs. This tradeoff between round complexity and approximation factor offers flexibility, allowing the algorithm to adapt to different requirements. In particular, for any constant $\varepsilon > 0$, an $O\left(\log^\varepsilon n\right)$-approximation can be obtained in $O(1)$ rounds. Previously, $O(1)$-round algorithms were only known for $O(\log n)$-approximation, as shown by Chechik and Zhang (PODC 2022). A key ingredient in our algorithm is a lemma that, under certain conditions, allows us to improve an $a$-approximation for APSP to an $O(\sqrt{a})$-approximation in $O(1)$ rounds. To prove this lemma, we develop several new techniques, including an $O(1)$-round algorithm for computing the $k$-nearest nodes, as well as new types of hopsets and skeleton graphs based on the notion of $k$-nearest nodes.
cs.DS↗
On existence of a compatible triangulation with the double circle order type
We show that the "double circle" order type and some of its generalizations have a compatible triangulation with any other order types with the same number of points and number of edges on convex hull, thus proving another special case of the conjecture in Aichholzer (2003).
math.CO↗
Straight-line Orthogonal Drawing of Complete Ternary Tree Requires $O(n^{1.032})$ Area
We resolve a conjecture posed by Covella, Frati and Patrignani by proving the straight-line orthogonal drawing of the complete ternary tree with $n$ nodes satisfying the subtree separation property with smallest area has area $Ω(n^{1.031})$. We also improve the upper bound of this area to $O(n^{1.032})$.
cs.CG↗
Square Packing with Asymptotically Smallest Waste Only Needs Good Squares
We consider the problem of packing a large square with nonoverlapping unit squares. Let $W(x)$ be the minimum wasted area when a large square of side length $x$ is packed with unit squares. In Roth and Vaughan's paper that proves the lower bound $W(x) \notin o(x^{1/2})$, a good square is defined to be a square with inclination at most $10^{-10}$ with respect to the large square. In this article, we prove that in calculating the asymptotic growth of the wasted space, it suffices to only consider packings with only good squares. This allows the lower bound proof in Roth and Vaughan's paper to be simplified by not having to handle bad squares.
cs.CG↗
Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials
We describe a framework that unifies the two types of polynomials introduced respectively by Bacher and Mouton and by Rutschmann and Wettstein to analyze the number of triangulations of point sets. Using this insight, we generalize the triangulation polynomials of chains to a wider class of near-edges, enabling efficient computation of the number of triangulations of certain families of point sets. We use the framework to try to improve the result in Rutschmann and Wettstein without success, suggesting that their result is close to optimal.
cs.CG↗