1,182 research outputs found
Two-Level Rectilinear Steiner Trees
Given a set of terminals in the plane and a partition of into
subsets , a two-level rectilinear Steiner tree consists of a
rectilinear Steiner tree connecting the terminals in each set
() and a top-level tree connecting the trees . 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 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 and
().
Then, we apply any approximation algorithm for minimum rectilinear Steiner
trees in the plane to compute each and independently.
This gives us a -factor approximation with a running time of
suitable for fast practical computations. The
approximation factor reduces to by applying Arora's approximation scheme
in the plane
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
Optimal competitiveness for the Rectilinear Steiner Arborescence problem
We present optimal online algorithms for two related known problems involving
Steiner Arborescence, improving both the lower and the upper bounds. One of
them is the well studied continuous problem of the {\em Rectilinear Steiner
Arborescence} (). We improve the lower bound and the upper bound on the
competitive ratio for from and to
, where is the number of Steiner
points. This separates the competitive ratios of and the Symetric-,
two problems for which the bounds of Berman and Coulston is STOC 1997 were
identical. The second problem is one of the Multimedia Content Distribution
problems presented by Papadimitriou et al. in several papers and Charikar et
al. SODA 1998. It can be viewed as the discrete counterparts (or a network
counterpart) of . For this second problem we present tight bounds also in
terms of the network size, in addition to presenting tight bounds in terms of
the number of Steiner points (the latter are similar to those we derived for
)
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
Optimal competitiveness for Symmetric Rectilinear Steiner Arborescence and related problems
We present optimal competitive algorithms for two interrelated known problems
involving Steiner Arborescence. One is the continuous problem of the Symmetric
Rectilinear Steiner Arborescence (SRSA), studied by Berman and Coulston.
A very related, but discrete problem (studied separately in the past) is the
online Multimedia Content Delivery (MCD) problem on line networks, presented
originally by Papadimitriu, Ramanathan, and Rangan. An efficient content
delivery was modeled as a low cost Steiner arborescence in a grid of
network*time they defined. We study here the version studied by Charikar,
Halperin, and Motwani (who used the same problem definitions, but removed some
constraints on the inputs).
The bounds on the competitive ratios introduced separately in the above
papers are similar for the two problems: O(log N) for the continuous problem
and O(log n) for the network problem, where N was the number of terminals to
serve, and n was the size of the network. The lower bounds were Omega(sqrt{log
N}) and Omega(sqrt{log n}) correspondingly. Berman and Coulston conjectured
that both the upper bound and the lower bound could be improved.
We disprove this conjecture and close these quadratic gaps for both problems.
We first present an O(sqrt{log n}) deterministic competitive algorithm for MCD
on the line, matching the lower bound. We then translate this algorithm to
become a competitive optimal algorithm O(sqrt{log N}) for SRSA. Finally, we
translate the latter back to solve MCD problem, this time competitive optimally
even in the case that the number of requests is small (that is, O(min{sqrt{log
n},sqrt{log N}})). We also present a Omega(sqrt[3]{log n}) lower bound on the
competitiveness of any randomized algorithm. Some of the techniques may be
useful in other contexts
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
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Fully dynamic maintenance of euclidean minimum spanning trees
We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^5/6 log1^2/2 n) per update operation. No nontrivial dynamic geometric minimum spanning tree algorithm was previously known. We reduce the problem to maintaining bichromatic closest pairs, which we also solve in the same time bounds. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path
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