514 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
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
Optimal Flood Control
A mathematical model for optimal control of the water levels in a chain of
reservoirs is studied. Some remarks regarding sensitivity with respect to the time horizon, terminal cost and forecast of inflow are made
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
Rectilinear Steiner Tree Construction
The Minimum Rectilinear Steiner Tree (MRST) problem is to find the minimal spanning tree of a set of points (also called terminals) in the plane that interconnects all the terminals and some extra points (called Steiner points) introduced by intermediate junctions, and in which edge lengths are measured in the L1 (Manhattan) metric. This is one of the oldest optimization problems in mathematics that has been extensively studied and has been proven to be NP-complete, thus efficient approximation heuristics are more applicable than exact algorithms.
In this thesis, we present a new heuristic to construct rectilinear Steiner trees (RSTs) with a close approximation of minimum length in Î(n log n) time. To this end, we recursively divide a plane into a set of sub-planes of which optimal rectilinear Steiner trees (optRSTs) can be generated by a proposed exact algorithm called Const_optRST. By connecting all the optRSTs of the sub-planes, a sub-optimal MRST is eventually constructed.
We show experimentally that for topologies with up to 100 terminals, the heuristic is 1.06 to 3.45 times faster than RMST, which is an efficient algorithm based on Primâs method, with accuracy improvements varying from 1.31 % to 10.21 %
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