Let N be a set of points in the plane. Informally, the Steiner tree problem asks us to construct a network, known as a Steiner minimum network or Steiner minimum tree, interconnecting the points in N, such that the network has the shortest length possible with respect to a given distance function. Unlike minimum spanning trees, a Steiner minimum tree may contain vertices of degree 3 or more not in N. These points are usually referred to as Steiner points, and the inclusion of these points makes the Steiner problem an NP-hard problem.

Steiner Minimum Trees in Uniform Orientation Metrics

Hexagonal three-layer channel routing

Interconnection problems have natural applications in the design of integrated circuits (or chips). A modern chip consists of billions of trans istors that are connected by metal wires on the surface of the chip. These metal w ires are routed on a (fairly small) number of layers in such a way that electri cally independent nets do not intersect each other. Traditional manufacturin g technology limits the orientations of the wires to be either horizontal or vertica l — and is known as Manhattan architecture. Over the last decade there has been a growing interest in gene ral architectures, where more than two perpendicular orientations can be used f or routing. This development has made fixed orientation interconnection pro blems (where an arbitrary set of fixed orientations can be used) interesting fr om a research point of view. In particular, the problem of computing minimum lengt h networks with fixed orientations — the so-called fixed orientation Steiner tr e problem — has received significant attention. This doctoral dissertation is a collection of twelve resear ch papers and a survey on the fixed orientation Steiner tree problem and some of its gen eralizations. One of the main contributions is a linear time algorithm for comput ing a Steiner minimum tree for a given full topology. Also, a linear programming fo rmulation is presented for the problem. For the general problem an exact algorithm t at computes optimal solutions to problem instances with thousands of poin ts is described and implemented. A novel paradigm for routing a chip using a gene ral architecture is implemented and tested on a set of benchmark instances; the a pproach documents the advantages of using more than two fixed orientations in ch ip design. The last part of the dissertation is concerned with generali zations that are motivated by chip design. Firstly, a catalog of problems that can be solved on the so-called Hanan grid is presented. Next, generalizations r elated to signal delay and group interconnections are studied, and finally, proper ties of the rotational Steiner tree problem are given. The results of the dissertation represent a significant step forward, both concerning theory and algorithms, for the fixed orientation Steiner tre e problem. In addition, the work maintains a close link to applications and generali zations motivated by chip design.

http://findresearcher.sdu.dk/portal/files/113838506/dissertation_final_all.pdf

Fixed Orientation Interconnection Problems: Theory, Algorithms and Applications

Routing multiterminal nets on a hexagonal grid

Interconnect planning is an important issue in VLSI designs. Routing congestion is an important problem during placement. This paper presents a novel approach to estimating the routing congestion on the hexagonal model with bounded-length detours. It is the first probabilistic estimation work on hexagonal topologies. A combinatorial estimation algorithm is presented. The experimental results demonstrate the effectiveness of the method.

Congestion estimation for hexagonal routing

Channel routing is an important, time-consuming, and difficult problem in VLSI layout design, In this paper, we consider the two-terminal channel routing problem in the new routing model, called times square model (TSM); see, Lodj, Info. Proc. Lett., vol. 35, p. 41, 1990.; where the grid is composed of horizontal tracks, right tracks (with slope +60/spl deg/) and left tracks (with slope -60/spl deg/). We show a new lower bound [2d/3]-1 to the width of channel and present an optimal algorithm for two-terminal channel routing problems, which obtains [2d/3]+2 as an upper bound to the channel width where d is the channel density. The algorithm not only utilizes the horizontal tracks, but also the zig-zag connections, thus achieving the routing optimality. >

An optimal channel-routing algorithm in the times square model

In this paper, we study the net adding problem arising in VLSI layout process. Given a layout H and a new net N, we attempt to add net N to layout H without changing H. We present an efficient method to find a solution or report there is no solution to the given problem. Our solution is optimal in minimizing the number of vias required in two-terminal cases, and nearly optimal in multi-terminal

Xuehou Tan

Papers

An optimal channel-routing algorithm in the times square model

The net adding problem