Estimating properties of discrete distributions is a fundamental problem in statistical learning.

We study the problem of generalized uniformity testing of a discrete probability distribution: Given samples from a probability distribution p over an unknown size discrete domain Ω, we want to distinguish, with probability at least 2/ 3, between the case that p is uniform on some subset of Ω versus null -far, in total variation distance, from any such uniform distribution. We establish tight bounds on the sample complexity of generalized uniformity testing. In more detail, we present a computationally efficient tester whose sample complexity is optimal, within constant factors, and a matching worst-case information-theoretic lower bound.

This paper considers the problem of searching for a point of interest (POI) moving along an urban road network with an uncrewed aerial vehicle (UAV). The UAV is modeled as a variable-speed Dubins vehicle with a line-of-sight sensor in an urban environment that may occlude the sensor's view of the POI. A search strategy is proposed that exploits a probabilistic visibility volume (VV) to plan its future motion with iterative deepening $A^\ast$. The probabilistic VV is a time-varying three-dimensional representation of the sensing constraints for a particular distribution of the POI's state. To find the path most likely to view the POI, the planner uses a heuristic to optimistically estimate the probability of viewing the POI over a time horizon. The probabilistic VV is max-pooled to create a variable-timestep planner that reduces the search space and balances long-term and short-term planning. The proposed path planning method is compared to prior work with a Monte-Carlo simulation and is shown to outperform the baseline methods in cluttered environments when the UAV's sensor has a higher false alarm probability.

Traditional vehicle routing systems efficiently optimize singular metrics like time or distance, and when considering multiple metrics, they need more processes to optimize . However, they lack the capability to interpret and integrate the complex, semantic, and dynamic contexts of human drivers, such as multi-step tasks, situational constraints, or urgent needs. This paper introduces and evaluates PAVe (Personalized Agentic Vehicular Routing), a hybrid agentic assistant designed to augment classical pathfinding algorithms with contextual reasoning. Our approach employs a Large Language Model (LLM) agent that operates on a candidate set of routes generated by a multi-objective (time, CO2) Dijkstra algorithm. The agent evaluates these options against user-provided tasks, preferences, and avoidance rules by leveraging a pre-processed geospatial cache of urban Points of Interest (POIs). In a benchmark of realistic urban scenarios, PAVe successfully used complex user intent into appropriate route modifications, achieving over 88% accuracy in its initial route selections with a local model. We conclude that combining classical routing algorithms with an LLM-based semantic reasoning layer is a robust and effective approach for creating personalized, adaptive, and scalable solutions for urban mobility optimization.

In the uniformity testing task, an algorithm is provided with samples from an unknown probability distribution over a (known) finite domain, and must decide whether it is the uniform distribution, or, alternatively, if its total variation distance from uniform exceeds some input distance parameter. This question has received a significant amount of interest and its complexity is, by now, fully settled. Yet, we argue that it fails to capture many scenarios of interest, and that its very definition as a gap problem in terms of a prespecified distance may lead to suboptimal performance. To address these shortcomings, we introduce the problem of uniformity tracking, whereby an algorithm is required to detect deviations from uniformity (however they may manifest themselves) using as few samples as possible, and be competitive against an optimal algorithm knowing the distribution profile in hindsight. Our main contribution is a $\operatorname{polylog}(\operatorname{opt})$-competitive uniformity tracking algorithm. We obtain this result by leveraging new structural results on Poisson mixtures, which we believe to be of independent interest.