Two-Level Rectilinear Steiner Trees

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

Two-Level Rectilinear Steiner Trees

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 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

Paradigmenwechsel der Planung und Steuerung von Wertschöpfungsnetzen

In komplexer werdenden Wertschöpfungsnetzen müssen Entscheidungen zukünftig schneller und fundierter getroffen werden. Im Zuge dessen ist es unabdingbar neue Methoden zur Entscheidungsunterstützung zu entwickeln. Vor dem Hintergrund des Paradigmenwechsels hin zur Industrie 4.0 wird die Planung und Steuerung entsprechender Netze gravierend von Ansätzen der Modellierung und Simulation profitieren