Word Chain Generators for Prefix Normal Words

In 2011, Fici and Lipt\'ak introduced prefix normal words. A binary word is prefix normal if it has no factor (substring) that contains more occurrences of the letter 1 than the prefix of the same length. Among the open problems regarding this topic are the enumeration of prefix normal words and efficient testing methods. We show a range of characteristics of prefix normal words. These include properties of factors that are responsible for a word not being prefix normal. With word chains and generators, we introduce new ways of relating words of the same length to each other.

Fast and Safe Scheduling of Robots

In this paper, we present an experimental analysis of a fast heuristic algorithm that was designed to generate a fast, collision-free schedule for a set of robots on a path graph. The experiments confirm the algorithm's effectiveness in producing collision-free schedules as well as achieving the optimal solution when all tasks assigned to the robots are of equal duration. Additionally, we provide an integer linear programming formulation that guarantees an optimal solution for this scheduling problem on any input graph, at the expense of significantly greater computational resources. We prove the correctness of our integer linear program. By comparing the solutions of these two algorithms, including the time required by the schedule itself, and the run time of each algorithm, we show that the heuristic algorithm is optimal or near optimal in nearly all cases, with a far faster run time than the integer linear program.