The Cartesian Product Problem and Implementing Production Systems on Reconfigurable Meshes

Let A and B be two groups of up to n elements distributed on the rst row of an n n reconng-urable mesh, and CA;B a subset of the cartesian product A B satisfying some unknown condition C. Only one broadcasting step is needed in order to compute CA;B's elements. However, the problem of moving CA;B's elements to the rst row in optimal time (so that they can be further processed) is not trivial. The conditional cartesian product (CCP) problem is to move CA;B's elements to the rst row in O(jC A;B j n) steps. This requires optimizing the cartesian product operation such that CA;B's elements will be optimally scattered in the mesh, so that O(n) elements can be retrieved in a single step (as opposed to !(p jCA;B j) elements needed if the cartesian product is not optimized). We give a determin-istic algorithm that for any A; B; C solves this problem in O(log log n jC A;B j n) steps, and an \adaptive" randomized algorithm whose optimality is veriied by experimental results. Note that the CCP is a case where we overcome the inherent limitation of the reconngurable mesh, namely, the inability to perform fast routing of packets located in a small area. We also present the model of production systems, in which computation is realized by executing cartesian products of subsets in a common element space. Production systems are useful for database applications, expert systems, and can even be used as a general parallel programming language. Solving the CCP problem allows us to devise an eecient implementation for production systems on the reconng-urable mesh. In this way the reconngurable mesh is shown to be an attractive architecture for database machines and for parallel programming as well.