1,304 research outputs found
VLSI Routing for Advanced Technology
Routing is a major step in VLSI design, the design process of complex integrated circuits (commonly known as chips). The basic task in routing is to connect predetermined locations on a chip (pins) with wires which serve as electrical connections. One main challenge in routing for advanced chip technology is the increasing complexity of design rules which reflect manufacturing requirements. In this thesis we investigate various aspects of this challenge. First, we consider polygon decomposition problems in the context of VLSI design rules. We introduce different width notions for polygons which are important for width-dependent design rules in VLSI routing, and we present efficient algorithms for computing width-preserving decompositions of rectilinear polygons into rectangles. Such decompositions are used in routing to allow for fast design rule checking. A main contribution of this thesis is an O(n) time algorithm for computing a decomposition of a simple rectilinear polygon with n vertices into O(n) rectangles, preseverving two-dimensional width. Here the two-dimensional width at a point of the polygon is defined as the edge length of a largest square that contains the point and is contained in the polygon. In order to obtain these results we establish a connection between such decompositions and Voronoi diagrams. Furthermore, we consider implications of multiple patterning and other advanced design rules for VLSI routing. The main contribution in this context is the detailed description of a routing approach which is able to manage such advanced design rules. As a main algorithmic concept we use multi-label shortest paths where certain path properties (which model design rules) can be enforced by defining labels assigned to path vertices and allowing only certain label transitions. The described approach has been implemented in BonnRoute, a VLSI routing tool developed at the Research Institute for Discrete Mathematics, University of Bonn, in cooperation with IBM. We present experimental results confirming that a flow combining BonnRoute and an external cleanup step produces far superior results compared to an industry standard router. In particular, our proposed flow runs more than twice as fast, reduces the via count by more than 20%, the wiring length by more than 10%, and the number of remaining design rule errors by more than 60%. These results obtained by applying our multiple patterning approach to real-world chip instances provided by IBM are another main contribution of this thesis. We note that IBM uses our proposed combined BonnRoute flow as the default tool for signal routing
Algorithms for distance problems in planar complexes of global nonpositive curvature
CAT(0) metric spaces and hyperbolic spaces play an important role in
combinatorial and geometric group theory. In this paper, we present efficient
algorithms for distance problems in CAT(0) planar complexes. First of all, we
present an algorithm for answering single-point distance queries in a CAT(0)
planar complex. Namely, we show that for a CAT(0) planar complex K with n
vertices, one can construct in O(n^2 log n) time a data structure D of size
O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x
and the query point y can be computed in linear time. Our second algorithm
computes the convex hull of a finite set of points in a CAT(0) planar complex.
This algorithm is based on Toussaint's algorithm for computing the convex hull
of a finite set of points in a simple polygon and it constructs the convex hull
of a set of k points in O(n^2 log n + nk log k) time, using a data structure of
size O(n^2 + k)
An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures
We study algorithmic aspects of bending wires and sheet metal into a
specified structure. Problems of this type are closely related to the question
of deciding whether a simple non-self-intersecting wire structure (a
carpenter's ruler) can be straightened, a problem that was open for several
years and has only recently been solved in the affirmative.
If we impose some of the constraints that are imposed by the manufacturing
process, we obtain quite different results. In particular, we study the variant
of the carpenter's ruler problem in which there is a restriction that only one
joint can be modified at a time. For a linkage that does not self-intersect or
self-touch, the recent results of Connelly et al. and Streinu imply that it can
always be straightened, modifying one joint at a time. However, we show that
for a linkage with even a single vertex degeneracy, it becomes NP-hard to
decide if it can be straightened while altering only one joint at a time. If we
add the restriction that each joint can be altered at most once, we show that
the problem is NP-complete even without vertex degeneracies.
In the special case, arising in wire forming manufacturing, that each joint
can be altered at most once, and must be done sequentially from one or both
ends of the linkage, we give an efficient algorithm to determine if a linkage
can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry -
Theory and Application
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
A multifacility location problem on median spaces
AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem
Engineering Algorithms for Route Planning in Multimodal Transportation Networks
Practical algorithms for route planning in transportation networks are a showpiece of successful Algorithm Engineering. This has produced many speedup techniques, varying in preprocessing time, space, query performance, simplicity, and ease of implementation. This thesis explores solutions to more realistic scenarios, taking into account, e.g., traffic, user preferences, public transit schedules, and the options offered by the many modalities of modern transportation networks
Massively Parallel Computation and Sublinear-Time Algorithms for Embedded Planar Graphs
While algorithms for planar graphs have received a lot of attention, few
papers have focused on the additional power that one gets from assuming an
embedding of the graph is available. While in the classic sequential setting,
this assumption gives no additional power (as a planar graph can be embedded in
linear time), we show that this is far from being the case in other settings.
We assume that the embedding is straight-line, but our methods also generalize
to non-straight-line embeddings. Specifically, we focus on sublinear-time
computation and massively parallel computation (MPC).
Our main technical contribution is a sublinear-time algorithm for computing a
relaxed version of an -division. We then show how this can be used to
estimate Lipschitz additive graph parameters. This includes, for example, the
maximum matching, maximum independent set, or the minimum dominating set. We
also show how this can be used to solve some property testing problems with
respect to the vertex edit distance.
In the second part of our paper, we show an MPC algorithm that computes an
-division of the input graph. We show how this can be used to solve various
classical graph problems with space per machine of for
some , and while performing rounds. This includes for
example approximate shortest paths or the minimum spanning tree. Our results
also imply an improved MPC algorithm for Euclidean minimum spanning tree
- …