In the previous lecture, we covered a series of online/offline edge-weighted Steiner tree/forest problems. This lecture extends the discussion to the node-weighted scope. In particular, we will study the nodeweighted Steiner tree/forest problem and introduce an offline O(logn)−approximation polynomial-time algorithm[KR95]. It is well known there is no polynormial-time algorithm that achieves o(...

The “chicken-egg” dilemma between VLSI interconnect timing optimization and delay calculation suggests an iterative approach. We separate interconnect timing transformation as Hanan grafting and non-Hanan sliding, and reveal generally negligible contribution of non-Hanan sliding. We propose a greedy iterative interconnect timing optimization algorithm called Q-Tree. Our experimental results sho...

We give a new lower bound on the length of the minimal Steiner tree with a given topology joining given terminals in Euclidean space, in terms of toroidal images. The lower bound is equal to the length when the topology is full. We use the lower bound to prove bounds on the “error” e in the length of an approximate Steiner tree, in terms of the maximum deviation d of an interior angle of the tr...

In the generalized Steiner tree problem, we find a minimumcost set of edges to connect a given set of source-sink pairs. In the online version of this problem, the source-sink pairs arrive over time. Agrawal, Klein, and Ravi [1] give a 2-approximation algorithm for the offline problem; Berman and Coulston [3] give an O(logn)-competitive algorithm for the online problem. Goemans and Williamson [...

Most heuristics for the Steiner tree problem in the Euclidean plane perform a series of iterative improvements using the minimum spanning tree as an initial solution. We may therefore characterize them as local search heuristics. In this paper, we rst give a survey of existing heuristic approaches from a local search perspective, by setting up solution spaces and neighbourhood structures. Secon...

This paper presents an algorithm for interconnect layout optimization with bu er insertion. Given a source and n sinks of a signal net, with given positions and a required arrival time associated with each sink, the algorithm nds a bu ered Steiner tree so that the required arrival time (or timing slack) at the source is maximized. In the algorithm, Steiner routing tree construction and bu er in...

The Steiner tree problem requires a shortest tree spanning a given vertex subset S within graph G = @‘, El. Two heuristics finding approximate Steiner trees no longer than 11/6 times of the optima1 length appeared recently. They run in time O(CY + KS2 + S4) and O(a + l@ + S3s), respectively, where LY is the time complexity of the all-pairs-shortest-paths problem. We reduce the runtime of the fi...

The history of the Euclidean Steiner tree problem, which is the problem of constructing a shortest possible network interconnecting a set of given points in the Euclidean plane, goes back to Gergonne in the early 19th century. We present a detailed account of the mathematical contributions of some of the earliest papers on the Euclidean Steiner tree problem. Furthermore, we link these initial c...

On Approximate Min-Max Theorems for Graph Connectivity Problems Lap Chi Lau Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2006 Given an undirected graph G and a subset of vertices S ⊆ V (G), we call the vertices in S the terminal vertices and the vertices in V (G) − S the Steiner vertices. In this thesis, we study two problems whose goals are to achieve high...

In reoptimization problems, one is given an optimal solution to a problem instance and a local modification of the instance. The goal is to obtain a solution for the modified instance. The additional information about the instance provided by the given solution plays a central role: we aim to use that information in order to obtain better solutions than we are able to compute from scratch. In t...