Abstract
Fan proposed a set of hyperuniversal switch blocks (HUSBs), called Fan’s (4, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${w}$ </tex-math></inline-formula> )-HUSBs in this paper. He also proved that his design with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${w}$ </tex-math></inline-formula> less than 6 was optimum. That is, they used the least switches to achieve hyperuniversal routability. However, we find a counter example to show that there is an unnecessary switch in Fan’s (4, 4)-HUSB. Namely, we propose a new (4, 4)-HUSB, water-molecule-shaped switch block (WMSB), proving (4, 4)-WMSB with 24 switches is optimal. Furthermore, we also propose optimal (4, 7)-WMSB, which is a (4, 7)-HUSB with 42 switches.
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