A Synthesizing Optimal Switching Lattices

We study the combinatorial problem of switching lattice synthesis. The idea of using regular arrays of switches, each cell connected to its neighbours, is old and has many forms. For example, in the “rectangular logic arrays” of Akers [1972], each switch is a relatively complex 3-input 2-output unit implementing either the median-of-3 function or the Boolean if-then-else function, and switches are arranged so that a current can flow top-down and left-to-right only (for appropriate orientation of the array). The potentials of nanoscale technologies [Cui and Lieber 2001; Chen et al. 2003; Eshaghian-Wilner et al. 2006] have created interest in types of “crossbar” devices that can operate based on, say, molecular electronics. In this context a “switch” is ideally a very simple device, such as a four-terminal switch [Altun and Riedel 2012]. The four terminals of the switch link to the four neighbours of a lattice cell, so that these are either all connected (when the switch is on), or disconnected (when the switch is off). The direction of flow in the resulting lattice structure has fewer constraints, compared to the lattices considered by Akers, allowing for paths that meander through the lattice. In more detail, an n-by-m switching lattice is a wire mesh that results from laying n parallel wires perpendicularly on top of m parallel wires, with a four-terminal switch inserted wherever two wires cross. (We picture the n wires laid down horizontally, so that we have n rows and m columns of switches.) The switching lattice can be used to