On the Stretch Factor of Polygonal Chains

Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain. We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space

On the Stretch Factor of Polygonal Chains

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain. The stretch factor of $P$ is the ratio between the total length of $P$ and the distance of its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For a parameter $c \geq 1$, we call $P$ a $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 \leq i<j<k \leq n$. The stretch factor is a global property: it measures how close $P$ is to a straight line, and it involves all the vertices of $P$; being a $c$-chain, on the other hand, is a fingerprint-property: it only depends on subsets of $O(1)$ vertices of the chain. We investigate how the $c$-chain property influences the stretch factor in the plane: (i) we show that for every $\varepsilon > 0$, there is a noncrossing $c$-chain that has stretch factor $\Omega(n^{1/2-\varepsilon})$, for sufficiently large constant $c=c(\varepsilon)$; (ii) on the other hand, the stretch factor of a $c$-chain $P$ is $O\left(n^{1/2}\right)$, for every constant $c\geq 1$, regardless of whether $P$ is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain $P$ in $\mathbb{R}^2$ with $n$ vertices, the minimum $c\geq 1$ for which $P$ is a $c$-chain in $O\left(n^{2.5}\ {\rm polylog}\ n\right)$ expected time and $O(n\log n)$ space.Comment: 16 pages, 11 figure

09451 Abstracts Collection -- Geometric Networks, Metric Space Embeddings and Spatial Data Mining

From November 1 to 6, 2009, the Dagstuhl Seminar 09451 ``Geometric Networks, Metric Space Embeddings and Spatial Data Mining\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Shortest Paths and Steiner Trees in VLSI Routing

Routing is one of the major steps in very-large-scale integration (VLSI) design. Its task is to find disjoint wire connections between sets of points on a chip, subject to numerous constraints. This problem is solved in a two-stage approach, which consists of so-called global and detailed routing steps. For each set of metal components to be connected, global routing reduces the search space by computing corridors in which detailed routing sequentially determines the desired connections as shortest paths. In this thesis, we present new theoretical results on Steiner trees and shortest paths, the two main mathematical concepts in routing. In the practical part, we give computational results of BonnRoute, a VLSI routing tool developed at the Research Institute for Discrete Mathematics at the University of Bonn. Interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the length of paths or direct delay measures should be taken into account when constructing rectilinear Steiner trees. We consider the problem of finding a rectilinear Steiner minimum tree (RSMT) that --- as a secondary objective --- minimizes a signal delay related objective. Given a source we derive some structural properties of RSMTs for which the weighted sum of path lengths from the source to the other terminals is minimized. Also, we present an exact algorithm for constructing RSMTs with weighted sum of path lengths as secondary objective, and a heuristic for various secondary objectives. Computational results for industrial designs are presented. We further consider the problem of finding a shortest rectilinear Steiner tree in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if it intersects an obstacle, then no connected component of the induced subtree must be longer than a given fixed length. This kind of length restriction is motivated by its application in VLSI routing where a large Steiner tree requires the insertion of repeaters which must not be placed on top of obstacles. We show that there are optimal length-restricted Steiner trees with a special structure. In particular, we prove that a certain graph (called augmented Hanan grid) always contains an optimal solution. Based on this structural result, we give an approximation scheme for the special case that all obstacles are of rectangular shape or are represented by at most a constant number of edges. Turning to the shortest paths problem, we present a new generic framework for Dijkstra's algorithm for finding shortest paths in digraphs with non-negative integral edge lengths. Instead of labeling individual vertices, we label subgraphs which partition the given graph. Much better running times can be achieved if the number of involved subgraphs is small compared to the order of the original graph and the shortest path problems restricted to these subgraphs is computationally easy. As an application we consider the VLSI routing problem, where we need to find millions of shortest paths in partial grid graphs with billions of vertices. Here, the algorithm can be applied twice, once in a coarse abstraction (where the labeled subgraphs are rectangles), and once in a detailed model (where the labeled subgraphs are intervals). Using the result of the first algorithm to speed up the second one via goal-oriented techniques leads to considerably reduced running time. We illustrate this with the routing program BonnRoute on leading-edge industrial chips. Finally, we present computational results of BonnRoute obtained on real-world VLSI chips. BonnRoute fulfills all requirements of modern VLSI routing and has been used by IBM and its customers over many years to produce more than one thousand different chips. To demonstrate the strength of BonnRoute as a state-of-the-art industrial routing tool, we show that it performs excellently on all traditional quality measures such as wire length and number of vias, but also on further criteria of equal importance in the every-day work of the designer

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

On-line algorithms for robot navigation and server problems

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 83-88).by Jon Michael Kleinberg.M.S

Geometric Dilation and Halving Distance

Let us consider the network of streets of a city represented by a geometric graph G in the plane. The vertices of G represent the crossroads and the edges represent the streets. The latter do not have to be straight line segments, they may be curved. If one wants to drive from a place p to some other place q, normally the length of the shortest path along streets, d_G(p,q), is bigger than the airline distance (Euclidean distance) |pq|. The (relative) DETOUR is defined as delta_G(p,q) := d_G(p,q)/|pq|. The supremum of all these ratios is called the GEOMETRIC DILATION of G. It measures the quality of the network. A small dilation value guarantees that there is no bigger detour between any two points. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the DILATION of S and denote it by delta(S). The main results of this thesis are - a general upper bound to the dilation of any finite point set S, delta(S) - a lower bound for a specific set P, delta(P)>(1+10^(-11))pi/2, which approximately equals 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the HALVING PAIRS, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the HALVING DISTANCE. A transformation technique based on halving pairs, the HALVING PAIR TRANSFORMATION, and the curve formed by the midpoints of the halving pairs, the MIDPOINT CURVE, help us to derive lower bounds to dilation. For constructing graphs of small dilation, we use ZINDLER CURVES. These are closed curves of constant halving distance. To give a structured overview, the mathematical apparatus for deriving the main results of this thesis includes - upper bound: * the construction of certain Zindler curves to generate a periodic graph of small dilation * an embedding argument based on a number theoretical result by Dirichlet - lower bound: * the formulation and analysis of the halving pair transformation * a stability result for the dilation of closed curves based on this transformation and the midpoint curve * the application of a disk-packing result In addition, this thesis contains - a detailed analysis of the dilation of closed curves - a collection of inequalities, which relate halving distance to other important quantities from convex geometry, and their proofs; including four new inequalities - the rediscovery of Zindler curves and a compact presentation of their properties - a proof of the applied disk packing result.Geometrische Dilation und Halbierungsabstand Man kann das von den Straßen einer Stadt gebildete Netzwerk durch einen geometrischen Graphen in der Ebene darstellen. Die Knoten dieses Graphen repräsentieren die Kreuzungen und die Kanten sind die Straßen. Letztere müssen nicht geradlinig sein, sondern können beliebig gekrümmt sein. Wenn man nun von einem Ort p zu einem anderen Ort q fahren möchte, dann ist normalerweise die Länge des kürzesten Pfades über Straßen, d_G(p,q), länger als der Luftlinienabstand (euklidischer Abstand) |pq|. Der (relative) UMWEG (DETOUR) ist definiert als delta_G(p,q) := d_G(p,q)/|pq|. Das Supremum all dieser Brüche wird GEOMETRISCHE DILATION (GEOMETRIC DILATION) von G genannt. Es ist ein Maß für die Qualität des Straßennetzes. Ein kleiner Dilationswert garantiert, dass es keinen größeren Umweg zwischen beliebigen zwei Punkten gibt. Für eine gegebene endliche Punktmenge S würden wir nun gerne bestimmen, was der kleinste Dilationswert ist, den wir mit einem Graphen erreichen können, der die gegebenen Punkte auf seinen Kanten enthält. Dieses Infimum nennen wir die DILATION von S und schreiben kurz delta(S). Die Haupt-Ergebnisse dieser Arbeit sind - eine allgemeine obere Schranke für die Dilation jeder beliebigen endlichen Punktmenge S: delta(S) - eine untere Schranke für eine bestimmte Menge P: delta(P)>(1+10^(-11))pi/2, was ungefähr der Zahl 1.571 entspricht Um diese Ergebnisse zu erreichen, betrachten wir zunächst geschlossene Kurven. Ihre Dilation hängt von sogenannten HALBIERUNGSPAAREN (HALVING PAIRS) ab. Das sind Punktpaare, die die geschlossene Kurve in zwei Teile gleicher Länge teilen. Besonders der Abstand der beiden Punkte ist von Bedeutung, der HALBIERUNGSABSTAND (HALVING DISTANCE). Eine auf den Halbierungspaaren aufbauende Transformation, die HALBIERUNGSPAARTRANSFORMATION (HALVING PAIR TRANSFORMATION), und die von den Mittelpunkten der Halbierungspaare gebildete Kurve, die MITTELPUNKTKURVE (MIDPOINT CURVE), helfen uns untere Dilationsschranken herzuleiten. Zur Konstruktion von Graphen mit kleiner Dilation benutzen wir ZINDLERKURVEN (ZINDLER CURVES). Dies sind geschlossene Kurven mit konstantem Halbierungspaarabstand. Die mathematischen Hilfsmittel, mit deren Hilfe wir schließlich die Hauptresultate beweisen, sind unter anderem - obere Schranke: * die Konstruktion von bestimmten Zindlerkurven, mit denen periodische Graphen kleiner Dilation gebildet werden können * ein Einbettungsargument, das einen zahlentheoretischen Satz von Dirichlet benutzt - untere Schranke: * die Definition und Analyse der Halbierungspaartransformation * ein Stabilitätsresultat für die Dilation geschlossener Kurven, das auf dieser Transformation und der Mittelpunktkurve basiert * die Anwendung eines Kreispackungssatzes Zusätzlich enthält diese Dissertation - eine detaillierte Analyse der Dilation geschlossener Kurven - eine Sammlung von Ungleichungen, die den Halbierungsabstand zu anderen wichtigen Größen der Konvexgeometrie in Beziehung setzen, und ihre Beweise; inklusive vier neuer Ungleichungen - die Wiederentdeckung von Zindlerkurven und eine kompakte Darstellung ihrer Eigenschaften - einen Beweis des angewendeten Kreispackungssatzes