Constrained Shortest Paths in Terrains and Graphs

Finding a shortest path is one of the most well-studied optimization problems. In this thesis we focus on shortest paths in geometric and graph theoretic settings subject to different feasibility constraints that arise in practical applications of such paths. One of the most fundamental problems in computational geometry is finding shortest paths in terrains, which has many applications in robotics, computer graphics and Geographic Information Systems (GISs). There are many variants of the problem in which the feasibility of a path is determined by some geometric property of the terrain. One such variant is the shortest descending path (SDP) problem, where the feasible paths are those that always go downhill. We need to compute an SDP, for example, for laying a canal of minimum length from the source of water at the top of a mountain to fields for irrigation purpose, and for skiing down a mountain along a shortest route. The complexity of finding SDPs is open. We give a full characterization of the bend angles of an SDP, showing that they follow a generalized form of Snell's law of refraction of light. We also reduce the SDP problem to the problem of finding an SDP through a given sequence of faces, by adapting the sequence tree approach of Chen and Han for our problem. Our results have two implications. First, we isolate the difficult aspect of SDPs. The difficulty is not in deciding which face sequence to use, but in finding the SDP through a given face sequence. Secondly, our results help us identify some classes of terrains for which the SDP problem is solvable in polynomial time. We give algorithms for two such classes. The difficulty of finding an exact SDP motivates the study of approximation algorithms for the problem. We devise two approximation algorithms for SDPs in general terrains---these are the first two algorithms to handle the SDP problem in such terrains. The algorithms are robust and easy-to-implement. We also give two approximation algorithms for the case when a face sequence is given. The first one solves the problem by formulating it as a convex optimization problem. The second one uses binary search together with our characterization of the bend angles of an SDP to locate an approximate path. We introduce a generalization of the SDP problem, called the shortest gently descending path (SGDP) problem, where a path descends but not too steeply. The additional constraint to disallow a very steep descent makes the paths more realistic in practice. For example, a vehicle cannot follow a too steep descent---this is why a mountain road has hairpin bends. We give two easy-to-implement approximation algorithms for SGDPs, both using the Steiner point approach. Between a pair of points there can be many SGDPs with different number of bends. In practice an SGDP with fewer bends or smaller total turn-angle is preferred. We show using a reduction from 3-SAT that finding an SGDP with a limited number of bends or a limited total turn-angle is hard. The hardness result applies to a generalization of the SGDP problem called the shortest anisotropic path problem, which is a well-studied computational geometry problem with many practical applications (e.g., robot motion planning), yet of unknown complexity. Besides geometric shortest paths, we also study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an easy-to-implement algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The algorithm handles a forbidden path using vertex replication, i.e., replicating vertices and judiciously deleting edges so as to remove the forbidden path but keep all of its subpaths. The main challenge is that vertex replication can result in an exponential number of copies of any forbidden path that overlaps the current one. The algorithm couples vertex replication with the "growth" of a shortest path tree in such a way that the extra copies of forbidden paths produced during vertex replication are immaterial

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.Comment: 37 pages, 9 figure

Quantifiable isovist and graph-based measures for automatic evaluation of different area types in virtual terrain generation

© 2013 IEEE. This article describes a set of proposed measures for characterizing areas within a virtual terrain in terms of their attributes and their relationships with other areas for incorporating game designers\u27 intent in gameplay requirement-based terrain generation. Examples of such gameplay elements include vantage point, strongholds, chokepoints and hidden areas. Our measures are constructed on characteristics of an isovist, that is, the volume of visible space at a local area and the connectivity of areas within the terrain. The calculation of these measures is detailed, in particular we introduce two new ways to accurately and efficiently calculate the 3D isovist volume. Unlike previous research that has mainly focused on aesthetic-based terrain generation, the proposed measures address a gap in gameplay requirement-based terrain generation-the need for a flexible mechanism to automatically parameterise specified areas and their associated relationships, capturing semantic knowledge relating to high level user intent associated with specific gameplay elements within the virtual terrain. We demonstrate applications of using the measures in an evolutionary process to automatically generate terrains that include specific gameplay elements as defined by a game designer. This is significant as this shows that the measures can characterize different gameplay elements and allow gameplay elements consistent with the designers\u27 intents to be generated and positioned in a virtual terrain without the need to specify low-level details at a model or logic level, hence leading to higher productivity and lower cost

Quantifiable isovist and graph-based measures for automatic evaluation of different area types in virtual terrain generation

© 2013 IEEE. This article describes a set of proposed measures for characterizing areas within a virtual terrain in terms of their attributes and their relationships with other areas for incorporating game designers\u27 intent in gameplay requirement-based terrain generation. Examples of such gameplay elements include vantage point, strongholds, chokepoints and hidden areas. Our measures are constructed on characteristics of an isovist, that is, the volume of visible space at a local area and the connectivity of areas within the terrain. The calculation of these measures is detailed, in particular we introduce two new ways to accurately and efficiently calculate the 3D isovist volume. Unlike previous research that has mainly focused on aesthetic-based terrain generation, the proposed measures address a gap in gameplay requirement-based terrain generation-the need for a flexible mechanism to automatically parameterise specified areas and their associated relationships, capturing semantic knowledge relating to high level user intent associated with specific gameplay elements within the virtual terrain. We demonstrate applications of using the measures in an evolutionary process to automatically generate terrains that include specific gameplay elements as defined by a game designer. This is significant as this shows that the measures can characterize different gameplay elements and allow gameplay elements consistent with the designers\u27 intents to be generated and positioned in a virtual terrain without the need to specify low-level details at a model or logic level, hence leading to higher productivity and lower cost

In pursuit of linear complexity in discrete and computational geometry

Many computational problems arise naturally from geometric data. In this thesis, we consider three such problems: (i) distance optimization problems over point sets, (ii) computing contour trees over simplicial meshes, and (iii) bounding the expected complexity of weighted Voronoi diagrams. While these topics are broad, here the focus is on identifying structure which implies linear (or near linear) algorithmic and descriptive complexity. The first topic we consider is in geometric optimization. More specifically, we define a large class of distance problems, for which we provide linear time exact or approximate solutions. Roughly speaking, the class of problems facilitate either clustering together close points (i.e. netting) or throwing out outliers (i.e pruning), allowing for successively smaller summaries of the relevant information in the input. A surprising number of classical geometric optimization problems are unified under this framework, including finding the optimal k-center clustering, the kth ranked distance, the kth heaviest edge of the MST, the minimum radius ball enclosing k points, and many others. In several cases we get the first known linear time approximation algorithm for a given problem, where our approximation ratio matches that of previous work. The second topic we investigate is contour trees, a fundamental structure in computational topology. Contour trees give a compact summary of the evolution of level sets on a mesh, and are typically used on massive data sets. Previous algorithms for computing contour trees took Θ(n log n) time and were worst-case optimal. Here we provide an algorithm whose running time lies between Θ(nα(n)) and Θ(n log n), and varies depending on the shape of the tree, where α(n) is the inverse Ackermann function. In particular, this is the first algorithm with O(nα(n)) running time on instances with balanced contour trees. Our algorithmic results are complemented by lower bounds indicating that, up to a factor of α(n), on all instance types our algorithm performs optimally. For the final topic, we consider the descriptive complexity of weighted Voronoi diagrams. Such diagrams have quadratic (or higher) worst-case complexity, however, as was the case for contour trees, here we push beyond worst-case analysis. A new diagram, called the candidate diagram, is introduced, which allows us to bound the complexity of weighted Voronoi diagrams arising from a particular probabilistic input model. Specifically, we assume weights are randomly permuted among fixed Voronoi sites, an assumption which is weaker than the more typical sampled locations assumption. Under this assumption, the expected complexity is shown to be near linear

Tethered Motion Planning for a Rappelling Robot

The Jet Propulsion Laboratory and Caltech developed the Axel rover to investigate and demonstrate the potential for tethered extreme terrain mobility, such as allowing access to science targets on the steep crater walls of other planets. Tether management is a key issue for Axel and other rappelling rovers. Avoiding tether entanglement constrains the robot's valid motions to the set of outgoing and returning path pairs that are homotopic to each other. In the case of a robot on a steep slope, a motion planner must additionally ensure that this ascent-descent path pair is feasible, based on the climbing forces provided by the tether. This feasibility check relies on the taut tether configuration, which is the shortest path in the homotopy class of the ascent-descent path pair. This dissertation presents a novel algorithm for tethered motion planning in extreme terrains, produced by combining shortest-homotopic-path algorithms from the topology and computational geometry communities with traditional graph search methods. The resulting tethered motion planning algorithm searches for this shortest path, checks for feasibility, and then generates waypoints for an ascent-descent path pair in the same homotopy class. I demonstrate the implementation of this algorithm on a Martian crater data set such as might be seen for a typical mission. By searching only for the shortest path, and ordering that search according to a heuristic, this algorithm proceeds more efficiently than previous tethered path-planning algorithms for extreme terrain. Frictional tether-terrain interaction may cause dangerously intermittent and unstable tether obstacles, which can be categorized based on their stability. Force-balance equations from the rope physics literature provide a set of tether and terrain conditions for static equilibrium, which can be used to determine if a given tether configuration will stick to a given surface based on tether tension. By estimating the tension of Axel's tether when driving, I divide potential tether tension obstacles into the following categories: acting as obstacles, acting as non-obstacles, and hazardous intermittent obstacles where it is uncertain whether the tether would slip or stick under normal driving tension variance. This dissertation describes how to modify the obstacle map as the categorization of obstacles fluctuates, and how to alter a motion plan around the dangerous tether friction obstacles. Together, these algorithms and methods form a framework for tethered motion planning on extreme terrain.</p

Modeling Autonomous Agents In Military Simulations

Simulation is an important tool for prediction and assessment of the behavior of complex systems and situations. The importance of simulation has increased tremendously during the last few decades, mainly because the rapid pace of development in the field of electronics has turned the computer from a costly and obscure piece of equipment to a cheap ubiquitous tool which is now an integral part of our daily lives. While such technological improvements make it easier to analyze well-understood deterministic systems, increase in speed and storage capacity alone are not enough when simulating situations where human beings and their behavior are an integral part of the system being studied. The problem with simulation of intelligent entities is that intelligence is still not well understood and it seems that the field of Artificial Intelligence (AI) has a long way to go before we get computers to think like humans. Behavior-based agent modeling has been proposed in mid-80\u27s as one of the alternatives to the classical AI approach. While used mainly for the control of specialized robotic vehicles with very specific sensory capabilities and limited intelligence, we believe that a behavior-based approach to modeling generic autonomous agents in complex environments can provide promising results. To this end, we are investigating a behavior-based model for controlling groups of collaborating and competing agents in a geographic terrain. In this thesis, we are focusing on scenarios of military nature, where agents can move within the environment and adversaries can eliminate each other through use of weapons. Different aspects of agent behavior like navigation to a goal or staying in group formation, are implemented by distinct behavior modules and the final observed behavior for each agent is an emergent property of the combination of simple behaviors and their interaction with the environment. Our experiments show that while such an approach is quite efficient in terms of computational power, it has some major drawbacks. One of the problems is that reactive behavior-based navigation algorithms are not well suited for environments with complex mobility constraints where they tend to perform much worse than proper path planning. This problem represents an important research question, especially when it is considered that most of the modern military conflicts and operations occur in urban environments. One of the contributions of this thesis is a novel approach to reactive navigation where goals and terrain information are fused based on the idea of transforming a terrain with obstacles into a virtual obstacle-free terrain. Experimental results show that our approach can successfully combine the low run-time computational complexity of reactive methods with the high success rates of classical path planning. Another interesting research problem is how to deal with the unpredictable nature of emergent behavior. It is not uncommon to have situations where an outcome diverges significantly from the intended behavior of the agents due to highly complex nonlinear interactions with other agents or the environment itself. Chances of devising a formal way to predict and avoid such abnormalities are slim at best, mostly because such complex systems tend to be be chaotic in nature. Instead, we focus on detection of deviations through tracking group behavior which is a key component of the total situation awareness capability required by modern technology-oriented and network-centric warfare. We have designed a simple and efficient clustering algorithm for tracking of groups of agent suitable for both spatial and behavioral domain. We also show how to detect certain events of interest based on a temporal analysis of the evolution of discovered clusters

Survey of semi-regular multiresolution models for interactive terrain rendering

Rendering high quality digital terrains at interactive rates requires carefully crafted algorithms and data structures able to balance the competing requirements of realism and frame rates, while taking into account the memory and speed limitations of the underlying graphics platform. In this survey, we analyze multiresolution approaches that exploit a certain semi-regularity of the data. These approaches have produced some of the most efficient systems to date. After providing a short background and motivation for the methods, we focus on illustrating models based on tiled blocks and nested regular grids, quadtrees and triangle bin-trees triangulations, as well as cluster-based approaches. We then discuss LOD error metrics and system-level data management aspects of interactive terrain visualization, including dynamic scene management, out-of-core data organization and compression, as well as numerical accurac

Hybrid Device-to-Device and Device-to-Vehicle Networks for Energy-Efficient Emergency Communications

Recovering postdisaster communications has become a major challenge for search and rescue. Device-to-device (D2D) and device-to-vehicle (D2V) networks have drawn attention. However, due to the limited D2D coverage and onboard energy, establishing a hybrid D2D and D2V network is promising. In this article, we jointly establish, optimize, and fuse D2D and D2V networks to support energy-efficient emergency communications. First, we establish a D2D network by optimally dividing ground devices (GDs) into multiple clusters and identifying temporary data caching centers (TDCCs) from GDs in clusters. Accordingly, emergency data returned from GDs is cached in TDCCs. Second, given the distribution of TDCCs, unmanned aerial vehicles (UAVs) are dispatched to fetch data from TDCCs. Therefore, we establish a UAV-assisted D2V network through path planning and network configuration optimization. Specifically, optimal path planning is implemented using cascaded waypoint and motion planning and optimal network configurations are determined by multiobjective optimization. Consequently, the best tradeoff between emergency response time and energy consumption is achieved, subject to a given set of constraints on signal-to-interference-plus-noise ratios, the number of UAVs, transmit power, and energy. Simulation results show that our proposed approach outperforms benchmark schemes in terms of energy efficiency, contributing to large-scale postdisaster emergency response.Comment: 12 page