1,017 research outputs found
Routing for analog chip designs at NXP Semiconductors
During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP.
This resulted in an heuristic approach, which we presented at the end of the
week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach
Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound
Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a
1.39-approximation for the Steiner tree problem, using a hypergraph-based
linear programming relaxation. They also upper-bounded its integrality gap by
1.55. We describe a shorter proof of the same integrality gap bound, by
applying some of their techniques to a randomized loss-contracting algorithm
Rectilinear Steiner Trees in Narrow Strips
A rectilinear Steiner tree for a set of points in is a
tree that connects the points in using horizontal and vertical line
segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear
Steiner tree with minimal total length. We investigate how the complexity of
Minimal Rectilinear Steiner Tree for point sets inside the strip
depends on the strip width . We
obtain two main results. 1) We present an algorithm with running time
for sparse point sets, that is, point sets where each
rectangle inside the strip contains points. 2) For
random point sets, where the points are chosen randomly inside a rectangle of
height and expected width , we present an algorithm that is
fixed-parameter tractable with respect to and linear in . It has an
expected running time of .Comment: 21 pages, 13 figure
Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions
Dynamic programming on tree decompositions is a frequently used approach to
solve otherwise intractable problems on instances of small treewidth. In recent
work by Bodlaender et al., it was shown that for many connectivity problems,
there exist algorithms that use time, linear in the number of vertices, and
single exponential in the width of the tree decomposition that is used. The
central idea is that it suffices to compute representative sets, and these can
be computed efficiently with help of Gaussian elimination.
In this paper, we give an experimental evaluation of this technique for the
Steiner Tree problem. A comparison of the classic dynamic programming algorithm
and the improved dynamic programming algorithm that employs the table reduction
shows that the new approach gives significant improvements on the running time
of the algorithm and the size of the tables computed by the dynamic programming
algorithm, and thus that the rank based approach from Bodlaender et al. does
not only give significant theoretical improvements but also is a viable
approach in a practical setting, and showcases the potential of exploiting the
idea of representative sets for speeding up dynamic programming algorithms
Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems
A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time
A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}
We revisit the classic task of finding the shortest tour of points in -dimensional Euclidean space, for any fixed constant . We determine the optimal dependence on in the running time of an algorithm that computes a -approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in time. This improves the previously smallest dependence on in the running time of the algorithm by Rao and Smith (STOC 1998). We also show that a algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree
Further improvements of Steiner tree approximations
The Steiner tree problem requires to find a shortest tree connecting a given set of terminal points in a metric space. We suggest a better and fast heuristic for the Steiner problem in graphs and in rectilinear plane. This heuristic finds a Steiner tree at most 1.757 and 1.267 times longer than the optimal solution in graphs and rectilinear plane, respectively
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