Construction of non-rectangular floor plans for properly triangulated planar graphs

Abstract

A majority of previous research on the problem of floor planning has been limited to constructing floor plans with rectangular exterior boundaries. It includes rectangular floor plans (RFPs) (Kozminski and Kinnen in IEEE Trans-Actions Circuits Syst 35:1401–1416, 1988, [1]) for properly triangulated plane graphs (PTPGs) having four or fewer corner implying paths (CIPs), where both the modules and exterior boundary are considered rectangular, and orthogonal floor plans (OFPs) (Liao et al in J Algorithms 2:441–451, 2003, [2]) for the remaining graphs (which do not possess RFPs), where the exterior boundary is rectangular but modules are taken of rectilinear shapes (L-shaped, T-shaped, Z-shaped, etc.). As an alternative to OFPs, sometimes floor plans containing rectilinear external boundaries and rectangular modules can also be obtained, known as non-rectangular floor plans (Raveena Shekhawat in Theor Comput Sci 942:57–92, 2023, [3]). This work aims to investigate non-rectangular floor plans (NRFPs) providing the best alternative solution in terms of the least number of concave corners (comparing with the number of bends in OFPs) for the PTPGs having more than four CIPs. We present a linear time algorithm to construct NRFPs with the least possible number of concave corners at the exterior boundary corresponding to the PTPGs with more than four CIPs. Further, we claim that the obtained NRFPs are non-trivial. An NRFP is considered non-trivial if the count of concave corners at its exterior boundary cannot be lowered without disrupting the horizontal and vertical adjacencies of the modules. In addition, we demonstrate that it is always feasible to produce an NRFP with precisely concave corners for any PTPG with k; CIPs.