Polylogarithmic Approximation for Generalized Minimum Manhattan Networks

Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axis-parallel segments whose total length equals the pair's Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d \ge 3$, the problem is APX-hard; it is known to admit, for any \eps > 0, an O(n^\eps)-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(\log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d \ge 2$ dimensions and an $O(\log n)$-approximation algorithm for 2D. We show that an existing $O(\log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.Comment: 14 pages, 5 figures; added appendix and figure

A Fast 2-Approximation of Minimum Manhattan Networks

Given a set P of n points in the plane, a Manhattan network of P is a network that contains a rectilinear shortest path between every pair of points of P. A minimum Manhattan network of P is a Manhattan network of minimum total length. It is unknown whether it is NP-hard to construct a minimum Manhattan network. The best approximations published so far are a combinatorial 3-approximation algorithm in time O(n log n), and an LP-based 2-approximation algorithm. We present a new combinatorial 2-approximation for this problem in time O(n log n)

Exploiting Problem Structure in Pathfinding

With a given map and a start and a goal position on the graph, a pathfinding algorithm typically searches on this graph from the start node and exploring its neighbour nodes until reaching the goal. It is closely related to the shortest path problem. A* is one of the best and most popular heuristic-guided algorithms used in pathfinding for video games. The algorithm always picks the node with the smallest f value and process this node. The f value is the sum of two parameters g (the actual cost from the start node to the current node) and h (estimated cost from the current node to the goal). At each step of the algorithm, the node with lowest f will be removed from an open list and its neighbour nodes with their f values would be updated in this list. The main cost of this algorithm is the frequent insertion and deleteMin operations of the open list. Typically, implementation of A* uses a priority queue or min-heap to implement the open list, which takes O(log n) for the operations in the worst case. But this is still expensive when using the algorithm in a large and complicated map with numerous nodes. We came up with a new data structure called multi-stack heap for the open list based on the 2D grid map and Manhattan distance, which only costs O(1) for insertion and deleteMin. It is very efficient especially when we have a considerable number of nodes to explore. Additionally, traditional A* requires checking whether the open list contains a duplicated of the being inserted node before every insertion, which takes O(n). We proposed a new implementation method based on admissible and consistent heuristic called “Check From Closed List”, it can reduce the time of this process to O(1)

Tight Lower Bounds for Greedy Routing in Higher-Dimensional Small-World Grids

We consider Kleinberg's celebrated small world graph model (Kleinberg, 2000), in which a D-dimensional grid {0,...,n-1}^D is augmented with a constant number of additional unidirectional edges leaving each node. These long range edges are determined at random according to a probability distribution (the augmenting distribution), which is the same for each node. Kleinberg suggested using the inverse D-th power distribution, in which node v is the long range contact of node u with a probability proportional to ||u-v||^(-D). He showed that such an augmenting distribution allows to route a message efficiently in the resulting random graph: The greedy algorithm, where in each intermediate node the message travels over a link that brings the message closest to the target w.r.t. the Manhattan distance, finds a path of expected length O(log^2 n) between any two nodes. In this paper we prove that greedy routing does not perform asymptotically better for any uniform and isotropic augmenting distribution, i.e., the probability that node u has a particular long range contact v is independent of the labels of u and v and only a function of ||u-v||. In order to obtain the result, we introduce a novel proof technique: We define a budget game, in which a token travels over a game board, while the player manages a "probability budget". In each round, the player bets part of her remaining probability budget on step sizes. A step size is chosen at random according to a probability distribution of the player's bet. The token then makes progress as determined by the chosen step size, while some of the player's bet is removed from her probability budget. We prove a tight lower bound for such a budget game, and then obtain a lower bound for greedy routing in the D-dimensional grid by a reduction

Bidirected minimum Manhattan network problem

In the bidirected minimum Manhattan network problem, given a set T of n terminals in the plane, we need to construct a network N(T) of minimum total length with the property that the edges of N(T) are axis-parallel and oriented in a such a way that every ordered pair of terminals is connected in N(T) by a directed Manhattan path. In this paper, we present a polynomial factor 2 approximation algorithm for the bidirected minimum Manhattan network problem.Comment: 14 pages, 16 figure

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 ℝ^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 ∈ 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 4ω^2 approximations, where ω ≤ 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 ℝ^d identifies a closest point x on the unit sphere S^{d-1}. We are interested in computing an ε-approximation y ∈ ℚ^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 ℝ^2 and ℝ^3. However, we show how to construct a rational point with denominators of at most 10(d-1)/ε^2 for any given ε ∈ (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 vielfältige Lösungsstrategieen aber auch Hindernisse. Diese Arbeit betrachtet drei Probleme im Gebiet der geometrischen Netzwerk Planung, des geometrischen Data Minings und der sphärischen Geometrie. Geometrische Netzwerk Planung: Im d-dimensionalen Generalized Minimum Manhattan Network Problem (d-GMMN) möchte man ein günstigstes geradliniges Netzwerk finden, welches jedes der gegebenen n Punktepaare aus ℝ^d mit einem kürzesten 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 für d>2 und eine O(log n) Approximation für 2-GMMN. Durch diese Arbeit geben wir etwas mehr Einblick, ob das Problem eine Approximation mit konstantem Faktor in polynomieller Zeit zulässt. Wir entwickeln zwei neue Algorithmen. Ersterer nutzt die `Skalendiversität' und hat eine O(D) Approximationsgüte für 2-GMMN. Hierbei ist D ein Maß für die in Eingaben auftretende `Skalen'. D ∈ O(log n), aber potentiell deutlichen kleiner für 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 Approximationsgüten auf Instanzen von 2-GMMN verhalten sich gut. Dieser Algorithmus kann für 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. Darüber hinaus zeigen wir, dass Lösungen des Primal-Dual Algorithmus 4ω^2 Approximationen sind, wobei ω ≤ n die maximale Überlappung einer Probleminstanz bezeichnet. Daher müssen potentielle Beweise, die konstante Approximationen für 2-GMMN ausschließen möchten, Instanzen mit vielen unterschiedlichen Skalen und nicht konstanter Überlappung konstruieren. Geometrisches Map Matching für heterogene Daten: Für eine gegebene Sequenz von Positionsmessungen ist das Ziel des geometrischen Map Matchings eine Sequenz von Bewegungen entlang Kanten eines räumlich eingebetteten Graphen zu finden, welche eine `gute Erklärung' für die Messungen ist. Das Problem wird anspruchsvoll da reale Messungen, wie beispielsweise Traces oder Graphen des OpenStreetMap Projekts, keine homogene Datenqualität aufweisen. Graphdetails und -fehler variieren in Gebieten und jeder Trace hat wechselndes Rauschen und Messgenauigkeiten. Zu formalisieren, was eine `gute Erklärung' ist, wird dadurch schwer. Wir stellen einen neuen Map Matching Ansatz vor, welcher sich lokal der Datenqualität 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 Datenqualität erzielbar sind ohne vorher Parameter kalibrieren zu müssen. Rationale Punkte auf Einheitssphären: Jeder, von Null verschiedene, Punkt in ℝ^d identifiziert einen nächsten Punkt x auf der Einheitssphäre S^{d-1}. Wir suchen eine ε-Approximation y ∈ ℚ^d für x zu berechnen, welche exakt auf S^{d-1} ist und niedrige Bit-Größe hat. Wir wiederholen untere Schranken an rationale Approximationen und liefern explizite, sphärische Instanzen. Wir beweisen, dass Floating-Point Zahlen nur triviale Lösungen zur Sphären-Gleichung in ℝ^2 und ℝ^3 liefern können. Jedoch zeigen wir die Konstruktion eines rationalen Punktes mit Nennern die maximal 10(d-1)/ε^2 sind für gegebene ε ∈ (0, 1/8], was ein bekanntes Resultat verbessert. Darüber hinaus profitiert die Methode von Algorithmen für simultane Diophantische Approximationen. Unsere quell-offene Implementierung und die Experimente demonstrieren die Praktikabilität unseres Ansatzes für sehr große, durch geometrische Längen- und Breitengrade referenzierte, Datensätze

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

A Simulation Framework for Fast Design Space Exploration of Unmanned Air System Traffic Management Policies

The number of daily small Unmanned Aircraft Systems (sUAS) operations in uncontrolled low altitude airspace is expected to reach into the millions. UAS Traffic Management (UTM) is an emerging concept aiming at the safe and efficient management of such very dense traffic, but few studies are addressing the policies to accommodate such demand and the required ground infrastructure in suburban or urban environments. Searching for the optimal air traffic management policy is a combinatorial optimization problem with intractable complexity when the number of sUAS and the constraints increases. As the demands on the airspace increase and traffic patterns get complicated, it is difficult to forecast the potential low altitude airspace hotspots and the corresponding ground resource requirements. This work presents a Multi-agent Air Traffic and Resource Usage Simulation (MATRUS) framework that aims for fast evaluation of different air traffic management policies and the relationship between policy, environment and resulting traffic patterns. It can also be used as a tool to decide the resource distribution and launch site location in the planning of a next-generation smart city. As a case study, detailed comparisons are provided for the sUAS flight time, conflict ratio, cellular communication resource usage, for a managed (centrally coordinated) and unmanaged (free flight) traffic scenario.Comment: The Integrated Communications Navigation and Surveillance (ICNS) Conference in 201