Traversing the machining graph of a pocket

A simple and linear-time algorithm is presented for solving the problem of traversing a machining graph with minimum retractions encountered in zigzag pocket machining and other applications. This algorithm finds a traversal of the machining graph of a general pocket P with Nh holes, such that the number of retractions in the traversal is no greater than OPT þ Nh þ Nr, where OPT is the (unknown) minimum number of retractions required by any algorithm and Nr is the number of reducible blocks in P (to be defined in the paper). When the step-over distance is small enough relative to the size of P, Nr becomes zero, and our result deviates from OPT by at most the number of holes in P,a significant improvement over the upper bound 5OPT þ 6Nh achieved [Proceedings of the Seventh ACM-SIAM Symposium on Discrete Algorithms, 1996; Algorithmica 2000 (26) 19]. In particular, if Nh is zero as well, i.e. when P has no holes, the proposed algorithm outputs an optimal solution. A novel computational modeling tool called block transition graph is introduced to formulate the traversal problem in a compact and concise form. Efficient algorithms are then presented for traversing this graph, which in turn gives rise to the major result