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Step Chemical Reaction Networks (step CRNs) are an augmentation of the Chemical Reaction Network (CRN) model where additional species may be introduced to the system in a sequence of “steps.” We study step CRN systems using a weak subset of reaction rules, void rules, in which molecular species can only be deleted. We demonstrate that step CRNs with only void rules of size (2,0) can simulate threshold formulas (TFs) under linear resources. These limited systems can also simulate threshold circuits (TCs) by modifying the volume of the system to be exponential. We then prove a matching exponential lower bound on the required volume for simulating threshold circuits in a step CRN with (2,0)-size rules under a restricted gate-wise simulation, thus showing our construction is optimal for simulating circuits in this way. 

Route reconstruction is an important application for Geographic Information Systems (GIS) that rely heavily upon GPS data and other location data from IoT devices. Many of these techniques rely on geometric methods involving the \frechet\ distance to compare curve similarity. The goal of reconstruction, or map matching, is to find the most similar path within a given graph to a given input curve, which is often approximate location data. This process can be approximated by sampling the curves and using the \dfd. Due to power and coverage constraints, the GPS data itself may be sparse causing improper constraints along the edges during the reconstruction if only the continuous \frechet\ distance is used. Here, we look at two variations of discrete map matching: one constraining the walk length and the other limiting the number of vertices visited in the graph. %, and the constraint that the walk may not self-intersect. We give an efficient algorithm to solve the question based on walk length showing it is in \textbf{P}. We prove the other problem is \npc\ and the minimization variant is \apx\ while also giving a parameterized algorithm to solve the problem.