The Turnpike problem aims to reconstruct a set of one-dimensional points from their unordered pairwise distances. Turnpike arises in biological applications such as molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes. Under noisy observation of the distances, the Turnpike problem is NP-hard and can take exponential time and space to solve when using traditional algorithms. To address this, we reframe the noisy Turnpike problem through the lens of optimization, seeking to simultaneously find the unknown point set and a permutation that maximizes similarity to the input distances. Our core contribution is a suite of algorithms that robustly solve this new objective. This includes a bilevel optimization framework that can efficiently solve Turnpike instances with up to 100,000 points. We show that this framework can be extended to scenarios with domain-specific constraints that include duplicated, missing, and partially labeled distances. Using these, we also extend our algorithms to work for points distributed on a circle (the Beltway problem). For small-scale applications that require global optimality, we formulate an integer linear program (ILP) that (i) accepts an objective from a generic family of convex functions and (ii) uses an extended formulation to reduce the number of binary variables. On synthetic and real partial digest data, our bilevel algorithms achieved state-of-the-art scalability across challenging scenarios with performance that matches or exceeds competing baselines. On small-scale instances, our ILP efficiently recovered ground-truth assignments and produced reconstructions that match or exceed our alternating algorithms. Our implementations are available at https://github.com/Kingsford-Group/turnpikesolvermm.

The Beltway and Turnpike problems entail the reconstruction of circular and linear one-dimensional point sets from unordered pairwise distances. These problems arise in computational biology when the measurements provide distances but do not associate those distances with the entities that gave rise to them. Such applications include molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes (since sequencing and mass spec technologies can give lengths or weights, usually without connecting them to endpoints). Practical algorithms for Turnpike are known when the distance measurements are accurate, but both problems become strongly NP-hard under any level of measurement uncertainty. This is problematic since all known applications experience some degree of uncertainty from uncontrollable factors. Traditional algorithms cope with this complexity by exploring a much larger solution space, leading to exponential blowup in terms of both time and space. To alleviate both issues, we propose a novel alternating optimization algorithm that can scale to large, uncertain distance sets with as many as 100,000 points. This algorithm is space and time-efficient, with each step running in 𝒪(m log (m)) time and requiring only 𝒪(√(m)) working space for a distance set of size m. Evaluations of this approach on synthetic and partial digest data showcase improved accuracy and scalability in the presence of uncertain, duplicated, and missing distances. Our implementation of the algorithm is available at https://github.com/Kingsford-Group/turnpikesolvermm .

Robotic liquid handlers play a crucial role in automating laboratory tasks such as sample preparation, high-throughput screening, and assay development. Manually designing protocols takes significant effort, and can result in inefficient protocols and involve human error. We investigates the application of reinforcement learning to automate the protocol design process resulting in reduced human labor and further automation in liquid handling. We develop a reinforcement learning agent that can automatically output the step-by-step protocol based on the initial state of the deck, reagent types and volumes, and the desired state of the reagents after the protocol is finished. We show that finding the optimal protocol for solving a liquid handler instance is NP-complete, and we present a reinforcement learning algorithm that can solve the planning problem practically for cases with a deck of up to 20 x 20 wells and four different types of reagents. We design and implement an actor-critic approach, and we train our agent using the Impala algorithm. Our findings demonstrate that reinforcement learning can be used to automatically program liquid handler robotic arms, enabling more precise and efficient planning for the liquid handler and laboratory automation.

Finding associations between pairs of variables in large datasets is crucial for various disciplines. The brute force method for solving this problem requires computing the mutual information between ((2)(N)) pairs. In this paper, we consider the problem of finding pairs of variables with high mutual information in sub-quadratic complexity. This problem is analogous to the nearest neighbor search, where the goal is to find pairs among N variables that are similar to each other. To solve this problem, we develop a new algorithm for finding associations based on constructing a decision tree that assigns a hash to each variable, in a way that for pairs with higher mutual information, the chance of having the same hash is higher. For any 1 <= lambda <= 2, we prove that in the case of binary data, we can reduce the number of necessary mutual information computations for finding all pairs satisfying I(X, Y) > 2 - lambda from O(N-2) to O(N-lambda), where I(X, Y) is the empirical mutual information between variables X and Y. Finally, we confirmed our theory by experiments on simulated and real data.