6 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
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
SCIP-Jackâa solver for STP and variants with parallelization extensions
This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this recordâŻThe Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. While often
a strong relationship between different Steiner tree problem variants can be
observed, solution approaches employed so far have been prevalently problemspecific. In contrast, this paper introduces a general-purpose solver that can
be used to solve both the classical Steiner tree problem and many of its variants without modification. This versatility is achieved by transforming various
problem variants into a general form and solving them by using a state-ofthe-art MIP-framework. The result is a high-performance solver that can be
employed in massively parallel environments and is capable of solving previously unsolved instances.German Federal Ministry of Education and Researc
SCIP-Jack - A solver for STP and variants with parallelization extensions
The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. While often a strong relationship between different Steiner tree problem variants can be observed, solution approaches employed so far have been prevalently problem-specific. In contrast, this paper introduces a general-purpose solver that can be used to solve both the classical Steiner tree problem and many of its variants without modification. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances
Algorithm engineering in geometric network planning and data mining
The geometric nature of computational problems provides a rich source of solution strategies as well as complicating obstacles. This thesis considers three problems in the context of geometric network planning, data mining and spherical geometry.
Geometric Network Planning:
In the d-dimensional Generalized Minimum Manhattan Network problem (d-GMMN) one is interested in finding a minimum cost rectilinear network N connecting a given set of n pairs of points in â^d such that each pair is connected in N via a shortest Manhattan path. The decision version of this optimization problem is known to be NP-hard. The best known upper bound is an O(log^{d+1} n) approximation for d>2 and an O(log n) approximation for 2-GMMN.
In this work we provide some more insight in, whether the problem admits constant factor approximations in polynomial time. We develop two new algorithms, a `scale-diversity aware' algorithm with an O(D) approximation guarantee for 2-GMMN. Here D is a measure for the different `scales' that appear in the input, D â O(log n) but potentially much smaller, depending on the problem instance. The other algorithm is based on a primal-dual scheme solving a more general, combinatorial problem - which we call Path Cover. On 2-GMMN it performs well in practice with good a posteriori, instance-based approximation guarantees. Furthermore, it can be extended to deal with obstacle avoiding requirements. We show that the Path Cover problem is at least as hard to approximate as the Hitting Set problem. Moreover, we show that solutions of the primal-dual algorithm are 4Ï^2 approximations, where Ï â€ n denotes the maximum overlap of a problem instance. This implies that a potential proof of O(1)-inapproximability for 2-GMMN requires gadgets of many different scales and non-constant overlap in the construction.
Geometric Map Matching for Heterogeneous Data:
For a given sequence of location measurements, the goal of the geometric map matching is to compute a sequence of movements along edges of a spatially embedded graph which provides a `good explanation' for the measurements.
The problem gets challenging as real world data, like traces or graphs from the OpenStreetMap project, does not exhibit homogeneous data quality. Graph details and errors vary in areas and each trace has changing noise and precision. Hence, formalizing what a `good explanation' is becomes quite difficult.
We propose a novel map matching approach, which locally adapts to the data quality by constructing what we call dominance decompositions. While our approach is computationally more expensive than previous approaches, our experiments show that it allows for high quality map matching, even in presence of highly variable data quality without parameter tuning.
Rational Points on the Unit Spheres:
Each non-zero point in â^d identifies a closest point x on the unit sphere S^{d-1}. We are interested in computing an Δ-approximation y â â^d for x, that is exactly on S^{d-1} and has low bit-size. We revise lower bounds on rational approximations and provide explicit spherical instances.
We prove that floating-point numbers can only provide trivial solutions to the sphere equation in â^2 and â^3. However, we show how to construct a rational point with denominators of at most 10(d-1)/Δ^2 for any given Δ â (0, 1/8], improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation.
Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets, geo-referenced by latitude and longitude values.Die geometrische Gestalt von Berechnungsproblemen liefert vielfÀltige Lösungsstrategieen aber auch Hindernisse. Diese Arbeit betrachtet drei Probleme im Gebiet der geometrischen Netzwerk Planung, des geometrischen Data Minings und der sphÀrischen Geometrie.
Geometrische Netzwerk Planung:
Im d-dimensionalen Generalized Minimum Manhattan Network Problem (d-GMMN) möchte man ein gĂŒnstigstes geradliniges Netzwerk finden, welches jedes der gegebenen n Punktepaare aus â^d mit einem kĂŒrzesten Manhattan Pfad verbindet. Es ist bekannt, dass die Entscheidungsvariante dieses Optimierungsproblems NP-hart ist. Die beste bekannte obere Schranke ist eine O(log^{d+1} n) Approximation fĂŒr d>2 und eine O(log n) Approximation fĂŒr 2-GMMN.
Durch diese Arbeit geben wir etwas mehr Einblick, ob das Problem eine Approximation mit konstantem Faktor in polynomieller Zeit zulĂ€sst. Wir entwickeln zwei neue Algorithmen. Ersterer nutzt die `SkalendiversitĂ€t' und hat eine O(D) ApproximationsgĂŒte fĂŒr 2-GMMN. Hierbei ist D ein MaĂ fĂŒr die in Eingaben auftretende `Skalen'. D â O(log n), aber potentiell deutlichen kleiner fĂŒr manche Problem Instanzen. Der andere Algorithmus basiert auf einem Primal-Dual Schema zur Lösung eines allgemeineren, kombinatorischen Problems, welches wir Path Cover nennen. Die praktisch erzielten a posteriori ApproximationsgĂŒten auf Instanzen von 2-GMMN verhalten sich gut. Dieser Algorithmus kann fĂŒr Netzwerk Planungsprobleme mit Hindernis-Anforderungen angepasst werden. Wir zeigen, dass das Path Cover Problem mindestens so schwierig zu approximieren ist wie das Hitting Set Problem. DarĂŒber hinaus zeigen wir, dass Lösungen des Primal-Dual Algorithmus 4Ï^2 Approximationen sind, wobei Ï â€ n die maximale Ăberlappung einer Probleminstanz bezeichnet. Daher mĂŒssen potentielle Beweise, die konstante Approximationen fĂŒr 2-GMMN ausschlieĂen möchten, Instanzen mit vielen unterschiedlichen Skalen und nicht konstanter Ăberlappung konstruieren.
Geometrisches Map Matching fĂŒr heterogene Daten:
FĂŒr eine gegebene Sequenz von Positionsmessungen ist das Ziel des geometrischen Map Matchings eine Sequenz von Bewegungen entlang Kanten eines rĂ€umlich eingebetteten Graphen zu finden, welche eine `gute ErklĂ€rung' fĂŒr die Messungen ist.
Das Problem wird anspruchsvoll da reale Messungen, wie beispielsweise Traces oder Graphen des OpenStreetMap Projekts, keine homogene DatenqualitÀt aufweisen. Graphdetails und -fehler variieren in Gebieten und jeder Trace hat wechselndes Rauschen und Messgenauigkeiten. Zu formalisieren, was eine `gute ErklÀrung' ist, wird dadurch schwer.
Wir stellen einen neuen Map Matching Ansatz vor, welcher sich lokal der DatenqualitĂ€t anpasst indem er sogenannte Dominance Decompositions berechnet. Obwohl unser Ansatz teurer im Rechenaufwand ist, zeigen unsere Experimente, dass qualitativ hochwertige Map Matching Ergebnisse auf hoch variabler DatenqualitĂ€t erzielbar sind ohne vorher Parameter kalibrieren zu mĂŒssen.
Rationale Punkte auf EinheitssphÀren:
Jeder, von Null verschiedene, Punkt in â^d identifiziert einen nĂ€chsten Punkt x auf der EinheitssphĂ€re S^{d-1}. Wir suchen eine Δ-Approximation y â â^d fĂŒr x zu berechnen, welche exakt auf S^{d-1} ist und niedrige Bit-GröĂe hat. Wir wiederholen untere Schranken an rationale Approximationen und liefern explizite, sphĂ€rische Instanzen.
Wir beweisen, dass Floating-Point Zahlen nur triviale Lösungen zur SphĂ€ren-Gleichung in â^2 und â^3 liefern können. Jedoch zeigen wir die Konstruktion eines rationalen Punktes mit Nennern die maximal 10(d-1)/Δ^2 sind fĂŒr gegebene Δ â (0, 1/8], was ein bekanntes Resultat verbessert. DarĂŒber hinaus profitiert die Methode von Algorithmen fĂŒr simultane Diophantische Approximationen.
Unsere quell-offene Implementierung und die Experimente demonstrieren die PraktikabilitĂ€t unseres Ansatzes fĂŒr sehr groĂe, durch geometrische LĂ€ngen- und Breitengrade referenzierte, DatensĂ€tze