Average lengths of wire routing under M-architecture and X-architecture

The X-architecture is a new integrated-circuit wiring technique in the physical design. Compared with the currently used M-architecture, which uses either horizontal or vertical routing, it is based on the pervasive use of diagonal wires. The experimental studies show that the X-architecture demonstrates a wire length reduction of more than 10-20% and better performance of timing. In this paper, we make a theoretical study on the wire lengths under these two architectures and obtain their expected values for the cases of two and three terminals, respectively. Our theoretical study confirms the wire length reduction as previous experimental studies claimed, but the reduction for three terminals is not as significant as for two terminals. Our analysis shows that the wire length reduction tends to become smaller as the number of terminals turns larger. We also estimate the lower and upper bounds on the expected wire lengths of M-architecture and X-architecture for arbitrary number of terminals