arXiv subjects

John Stuart

Publications and source records attributed to John Stuart.

4 recordsLinked to original sources

arXiv2026

The Spanning Ratio of the Directed $\Theta_6$-Graph is 5

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vec{\Theta}_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vec{\Theta}_6(P)$-graph is defined as follows: the plane around each point $u\in P$ is partitioned into $6$ equiangular cones with apex $u$, and in each cone, $u$ is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the $\vec{\Theta}_6(P)$-graph contains an edge from $u$ to $v$ exactly when the interior of $\nabla_u^v$ is disjoint from $P$, where $\nabla_u^v$ is the unique equilateral triangle containing $u$ on a corner, $v$ on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the $\vec{\Theta}_6(P)$-graph is between $4$ and $7$ in the worst case (Akitaya, Biniaz, and Bose \emph{Comput. Geom.}, 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any $\vec{\Theta}_k(P)$-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.

cs.CG↗

arXiv2025

Tight Routing and Spanning Ratios of Arbitrary Triangle Delaunay Graphs

A Delaunay graph built on a planar point set has an edge between two vertices when there exists a disk with the two vertices on its boundary and no vertices in its interior. When the disk is replaced with an equilateral triangle, the resulting graph is known as a Triangle-Distance Delaunay Graph or TD-Delaunay for short. A generalized $\text{TD}_{\theta_1,\theta_2}$-Delaunay graph is a TD-Delaunay graph whose empty region is a scaled translate of a triangle with angles of $\theta_1,\theta_2,\theta_3:=\pi-\theta_1-\theta_2$ with $\theta_1\leq\theta_2\leq\theta_3$. We prove that $\frac{1}{\sin(\theta_1/2)}$ is a lower bound on the spanning ratio of these graphs which matches the best known upper bound (Lubiw & Mondal, J. Graph Algorithms Appl., 23(2):345-369). Then we provide an online local routing algorithm for $\text{TD}_{\theta_1,\theta_2}$-Delaunay graphs with a routing ratio that is optimal in the worst case. When $\theta_1=\theta_2=\frac{\pi}{3}$, our expressions for the spanning ratio and routing ratio evaluate to $2$ and $\frac{\sqrt{5}}{3}$, matching the known tight bounds for TD-Delaunay graphs.

cs.CG↗

arXiv2023

Zero-Knowledge MIPs using Homomorphic Commitment Schemes

A Zero-Knowledge Protocol (ZKP) allows one party to convince another party of a fact without disclosing any extra knowledge except the validity of the fact. For example, it could be used to allow a customer to prove their identity to a potentially malicious bank machine without giving away private information such as a personal identification number. This way, any knowledge gained by a malicious bank machine during an interaction cannot be used later to compromise the client's banking account. An important tool in many ZKPs is bit commitment, which is essentially a digital way for a sender to put a message in a lock-box, lock it, and send it to the receiver. Later, the key is sent for the receiver to open the lock box and read the message. This way, the message is hidden from the receiver until they receive the key, and the sender is unable to change their mind after sending the lock box. In this paper, the homomorphic properties of a particular multi-party commitment scheme are exploited to allow the receiver to perform operations on commitments, resulting in polynomial time ZKPs for two NP-Complete problems: the Subset Sum Problem and 3SAT. These ZKPs are secure with no computational restrictions on the provers, even with shared quantum entanglement. In terms of efficiency, the Subset Sum ZKP is competitive with other practical quantum-secure ZKPs in the literature, with less rounds required, and fewer computations.

quant-ph↗

arXiv2020

Frobenius nilHecke algebras

To any Frobenius superalgebra $A$ we associate towers of Frobenius nilCoxeter algebras and Frobenius nilHecke algebras. These act naturally, via Frobenius divided difference operators, on Frobenius polynomial algebras. When $A$ is the ground ring, our algebras recover the classical nilCoxeter and nilHecke algebras. When $A$ is the two-dimensional Clifford algebra, they are Morita equivalent to the odd nilCoxeter and odd nilHecke algebras.

math.RT↗