Colored Non-Crossing Euclidean Steiner Forest

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $k\rho$-approximation algorithm is known, where $\rho \le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered

The Steiner Ratio for the Obstacle-Avoiding Steiner Tree Problem

This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles

On the construction of rectilinear Steiner minimum trees among obstacles.

Rectilinear Steiner minimum tree (RSMT) problem asks for a shortest tree spanning a set of given terminals using only horizontal and vertical lines. Construction of RSMTs is an important problem in VLSI physical design. It is useful for both the detailed and global routing steps, and it is important for congestion, wire length and timing estimations during the floorplanning or placement step. The original RSMT problem assumes no obstacle in the routing region. However, in today’s designs, there can be many routing blockages, like macro cells, IP blocks and pre-routed nets. Therefore, the RSMT problem with blockages has become an important problem in practice and has received a lot of research attentions in the recent years. The RSMT problem has been shown to be NP-complete, and the introduction of obstacles has made this problem even more complicated.In the first part of this thesis, we propose an exact algorithm, called ObSteiner, for the construction of obstacle-avoiding RSMT (OARSMT) in the presence of complex rectilinear obstacles. Our work is developed based on the GeoSteiner approach in which full Steiner trees (FSTs) are first constructed and then combined into a RSMT. We modify and extend the algorithm to allow rectilinear obstacles in the routing region. We prove that by adding virtual terminals to each routing obstacle, the FSTs in the presence of obstacles will follow some very simple structures. A two-phase approach is then developed for the construction of OARSMTs. In the first phase, we generate a set of FSTs. In the second phase, the FSTs generated in the first phase are used to construct an OARSMT. Experimental results show that ObSteiner is able to handle problems with hundreds of terminals in the presence of up to two thousand obstacles, generating an optimal solution in a reasonable amount of time.In the second part of this thesis, we propose the OARSMT problem with slew constraints over obstacles. In modern VLSI designs, obstacles usually block a fraction of metal layers only making it possible to route over the obstacles. However, since buffers cannot be place on top of any obstacle, we should avoid routing long wires over obstacles. Therefore, we impose the slew constraints for the interconnects that are routed over obstacles. To deal with this problem, we analyze the optimal solutions and prove that the internal trees with signal direction over an obstacle will follow some simple structures. Based on this observation, we propose an exact algorithm, called ObSteiner with slew constraints, that is able to find an optimal solution in the extended Hanan grid. Experimental results show that the proposed algorithm is able to reduce nearly 5% routing resources on average in comparison with the OARSMT algorithm and is also very much faster.Huang, Tao.Thesis (Ph.D.)--Chinese University of Hong Kong, 2013.Includes bibliographical references (leaves [137]-144).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- The rectilinear Steiner minimum tree problem --- p.1Chapter 1.2 --- Applications --- p.3Chapter 1.3 --- Obstacle consideration --- p.5Chapter 1.4 --- Thesis outline --- p.6Chapter 1.5 --- Thesis contributions --- p.8Chapter 2 --- Background --- p.11Chapter 2.1 --- RSMT algorithms --- p.11Chapter 2.1.1 --- Heuristics --- p.11Chapter 2.1.2 --- Exact algorithms --- p.20Chapter 2.2 --- OARSMT algorithms --- p.30Chapter 2.2.1 --- Heuristics --- p.30Chapter 2.2.2 --- Exact algorithms --- p.33Chapter 3 --- ObSteiner - an exact OARSMT algorithm --- p.37Chapter 3.1 --- Introduction --- p.38Chapter 3.2 --- Preliminaries --- p.39Chapter 3.2.1 --- OARSMT problem formulation --- p.39Chapter 3.2.2 --- An exact RSMT algorithm --- p.40Chapter 3.3 --- OARSMT decomposition --- p.42Chapter 3.3.1 --- Full Steiner trees among complex obstacles --- p.42Chapter 3.3.2 --- More Theoretical results --- p.59Chapter 3.4 --- OARSMT construction --- p.62Chapter 3.4.1 --- FST generation --- p.62Chapter 3.4.2 --- Pruning of FSTs --- p.66Chapter 3.4.3 --- FST concatenation --- p.71Chapter 3.5 --- Incremental construction --- p.82Chapter 3.6 --- Experiments --- p.83Chapter 4 --- ObSteiner with slew constraints --- p.97Chapter 4.1 --- Introduction --- p.97Chapter 4.2 --- Problem Formulation --- p.100Chapter 4.3 --- Overview of our approach --- p.103Chapter 4.4 --- Internal tree structures in an optimal solution --- p.103Chapter 4.5 --- Algorithm --- p.126Chapter 4.5.1 --- EFST and SCIFST generation --- p.127Chapter 4.5.2 --- Concatenation --- p.129Chapter 4.5.3 --- Incremental construction --- p.131Chapter 4.6 --- Experiments --- p.131Chapter 5 --- Conclusion --- p.135Bibliography --- p.13

Control for Localization and Visibility Maintenance of an Independent Agent using Robotic Teams

Given a non-cooperative agent, we seek to formulate a control strategy to enable a team of robots to localize and track the agent in a complex but known environment while maintaining a continuously optimized line-of-sight communication chain to a fixed base station. We focus on two aspects of the problem. First, we investigate the estimation of the agent\u27s location by using nonlinear sensing modalities, in particular that of range-only sensing, and formulate a control strategy based on improving this estimation using one or more robots working to independently gather information. Second, we develop methods to plan and sequence robot deployments that will establish and maintain line-of-sight chains for communication between the independent agent and the fixed base station using a minimum number of robots. These methods will lead to feedback control laws that can realize this plan and ensure proper navigation and collision avoidance

Efficient Mission Planning for Robot Networks in Communication Constrained Environments

Many robotic systems are remotely operated nowadays that require uninterrupted connection and safe mission planning. Such systems are commonly found in military drones, search and rescue operations, mining robotics, agriculture, and environmental monitoring. Different robotic systems may employ disparate communication modalities such as radio network, visible light communication, satellite, infrared, Wi-Fi. However, in an autonomous mission where the robots are expected to be interconnected, communication constrained environment frequently arises due to the out of range problem or unavailability of the signal. Furthermore, several automated projects (building construction, assembly line) do not guarantee uninterrupted communication, and a safe project plan is required that optimizes collision risks, cost, and duration. In this thesis, we propose four pronged approaches to alleviate some of these issues: 1) Communication aware world mapping; 2) Communication preserving using the Line-of-Sight (LoS); 3) Communication aware safe planning; and 4) Multi-Objective motion planning for navigation. First, we focus on developing a communication aware world map that integrates traditional world models with the planning of multi-robot placement. Our proposed communication map selects the optimal placement of a chain of intermediate relay vehicles in order to maximize communication quality to a remote unit. We also vi propose an algorithm to build a min-Arborescence tree when there are multiple remote units to be served. Second, in communication denied environments, we use Line-of-Sight (LoS) to establish communication between mobile robots, control their movements and relay information to other autonomous units. We formulate and study the complexity of a multi-robot relay network positioning problem and propose approximation algorithms that restore visibility based connectivity through the relocation of one or more robots. Third, we develop a framework to quantify the safety score of a fully automated robotic mission where the coexistence of human and robot may pose a collision risk. A number of alternate mission plans are analyzed using motion planning algorithms to select the safest one. Finally, an efficient multi-objective optimization based path planning for the robots is developed to deal with several Pareto optimal cost attributes

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

Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours

In this paper, we study problems of connecting classes of points via noncrossing structures. Given a set of colored terminal points, we want to find a graph for each color that connects all terminals of its color with the restriction that no two graphs cross each other. We consider these problems both on the Euclidean plane and in planar graphs. On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored traveling salesman problem as well as for the red-blue-green separation problem (in which we want to separate terminals of three colors with two noncrossing polygons of minimum length), both on the Euclidean plane. This improves the work of Arora and Chang (ICALP 2003) who gave a slower PTAS for the simpler red-blue separation problem. For the case of unweighted plane graphs, we also show a PTAS for the two-colored traveling salesman problem. All these results are based on our new patching procedure that might be of independent interest. On the negative side, we show that the problem of connecting terminal pairs with noncrossing paths is NP-hard on the Euclidean plane, and that the problem of finding two noncrossing spanning trees is NP-hard in plane graphs.Comment: 36 pages, 15 figures (colored