Two-Level Rectilinear Steiner Trees

TL;DR

This paper introduces algorithms for constructing two-level rectilinear Steiner trees that optimize total length, including a PTAS for bounded cases and a practical 2.37-approximation for general cases, relevant in chip design.

Contribution

It presents a polynomial time approximation scheme for bounded cases and a fast 2.37-approximation algorithm for the general two-level rectilinear Steiner tree problem.

Findings

PTAS based on Arora's method for bounded k

A 2.37-approximation algorithm for general cases

Reduction of approximation factor to 1.63 with Arora's scheme

Abstract

Given a set $P$ of terminals in the plane and a partition of $P$ into $k$ subsets $P_1, ..., P_k$, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree $T_i$ connecting the terminals in each set $P_i$ ($i=1,...,k$) and a top-level tree $T_{top}$ connecting the trees $T_1, ..., T_k$. The goal is to minimize the total length of all trees. This problem arises naturally in the design of low-power physical implementations of parity functions on a computer chip. For bounded $k$ we present a polynomial time approximation scheme (PTAS) that is based on Arora's PTAS for rectilinear Steiner trees after lifting each partition into an extra dimension. For the general case we propose an algorithm that predetermines a connection point for each $T_i$ and $T_{top}$ ($i=1,...,k$). Then, we apply any approximation algorithm for minimum rectilinear Steiner trees in the plane to compute each $T_i$ and $T_{top}$ independently. This gives us a $2.37$-factor approximation with a running time of $\mathcal{O}(|P|\log|P|)$ suitable for fast practical computations. The approximation factor reduces to $1.63$ by applying Arora's approximation scheme in the plane.