Fast Continuous Haar and Fourier Transforms of Rectilinear Polygons from VLSI Layouts

TL;DR

This paper introduces efficient algorithms for continuous Haar and Fourier transforms tailored for rectilinear polygons in VLSI layouts, significantly reducing computational complexity and accelerating design processes.

Contribution

The paper presents novel pruned continuous Haar and Fourier algorithms specifically optimized for rectilinear polygons in VLSI, with proven speed improvements over traditional discrete methods.

Findings

Pruned continuous Haar transform is 5 to 25 times faster.

Fast continuous Fourier series is 1.5 to 3 times faster.

Algorithms are effective for polygons with few vertices, common in VLSI layouts.

Abstract

We develop the pruned continuous Haar transform and the fast continuous Fourier series, two fast and efficient algorithms for rectilinear polygons. Rectilinear polygons are used in VLSI processes to describe design and mask layouts of integrated circuits. The Fourier representation is at the heart of many of these processes and the Haar transform is expected to play a major role in techniques envisioned to speed up VLSI design. To ensure correct printing of the constantly shrinking transistors and simultaneously handle their increasingly large number, ever more computationally intensive techniques are needed. Therefore, efficient algorithms for the Haar and Fourier transforms are vital. We derive the complexity of both algorithms and compare it to that of discrete transforms traditionally used in VLSI. We find a significant reduction in complexity when the number of vertices of the polygons is small, as is the case in VLSI layouts. This analysis is completed by an implementation and a benchmark of the continuous algorithms and their discrete counterpart. We show that on tested VLSI layouts the pruned continuous Haar transform is 5 to 25 times faster, while the fast continuous Fourier series is 1.5 to 3 times faster.