Visibility Algorithms in the Plane

We introduce a new algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles. In particular, for a given start points, we build a planar subdivision (ashortest path map) that supports efficient queries for shortest paths froms to any destination pointt. The worst-case time complexity of our algorithm isO(kn log2n), wheren is the number of vertices describing the polygonal obstacles, andk is a parameter we call the “illumination depth” of the obstacle space. Our algorithm usesO(n) space, avoiding the possibly quadratic space complexity of methods that rely on visibility graphs. The quantityk is frequently significantly smaller thann, especially in some of the cases in which the visibility graph has quadratic size. In particular,k is bounded above by the number of different obstacles that touch any shortest path froms.

The problem of determining the Euclidean shortest path between two points in the presence of m simple polygonal obstacles is studied. An O( m2 logn + nlogn ) algorithm is developed, where n is the total number of points in the obstacles. A simple O(E+T) algorithm for determining the visibility graph is also shown, where E is the number of visibility edges and T is the time for triangulating the point set. This is extended to a O(Es + nlogn) algorithm for the shortest path problem where Es is bounded by m2.

The multiple pairs shortest path problem (MPSP) arises in many applications where the shortest paths and distances between only some specific pairs of origin-destination (OD) nodes in a network are desired. The traditional repeated single-source shortest path (SSSP) and all pairs shortest paths (APSP) algorithms often do unnecessary computation to solve the MPSP problem. We propose a new shortest path algorithm to save computational work when solving the MPSP problem. Our method is especially suitable for applications with fixed network topology but changeable arc lengths and desired OD pairs. Preliminary computational experiments demonstrate our algorithm’s superiority on airline network problems over other APSP and SSSP algorithms.

We propose a simple and effective heuristic that allows fast generation of a large set of shortest path alternatives in weighted directed graphs. The heuristic is based on existing deviation path algorithms for exact k shortest paths. It precalculates a backward shortest path tree and thus avoids doing many shortest path computations, but as a result it does not necessarily find the exact set of k shortest paths. Computational results on real-world road networks are reported. Our tests show that the quality of the paths produced by the heuristic is most satisfactory: typically, the kth path found by the heuristic is less than 1% longer than the exact kth shortest path, for values of k up to 10,000. Moreover, the heuristic runs very fast. We also show how the heuristic can be enhanced to an exact k shortest paths algorithm, which performs well in comparison with the existing exact k shortest path algorithms.

Finding shortest distances and paths have always been a crucial area of study. In this article, we have proposed a linear time pre-processing method for optimization of shortest path and distance algorithms, for an undirected weighted graph with non-negative edges. In this approach, certain parts of a graph are grouped together as subgraphs termed as optimized deterministic routing areas (ODRAs). Each ODRA has a unique representative element called optimized proxy. The input graph is reduced by replacing ODRAs with optimized proxies and the new reduced graph is used for the shortest path and distance queries. This is a linear time pre-processing stage which helps in significant decrease in computation time for finding shortest distances and paths in graphs. The lower bound and upper bound of time complexity of the proposed algorithm is O(n/2) and O(n) respectively.

Large numbers of travel traces are collected by vehicles and stored for applications such as optimizing delivery routes, predicting and avoiding traffic, and providing directions. Many of the applications preprocess the travel traces, usually composed of position data, by matching these with links in the underlying road network. This paper addresses the problem of persistent storage of large numbers of vehicle travel traces. We propose two methods for using a routing algorithm and road network to encode a travel trace formed by a sequence of links. An encoded trace, composed of a few links, is useful to store or share and can be decoded into the original travel trace. Considering that drivers tend to proceed from an origin to a destination by using the shortest path or going as straight as possible, the two proposed methods use the following two routing algorithms: a shortest path algorithm; and a following path algorithm, which finds the path that avoids turns. The experimental results for 30 real traces show that a travel trace is encoded into only 5% or 7% of its links on average using the shortest path algorithm or the following path algorithm, respectively.

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Grbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.

A comparison of algorithms for origin–destination matrix generation on real road networks and an approximation approach

We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to n -vertex digraphs of genus O(n 1-ε ) for any ε>0 .

This paper presents six new variants of the polynomially bounded Partitioning Shortest Path (PSP) algorithm for finding the shortest path from one node to all other nodes in a network. Three of these variants, one for negative arc lengths, but without negative cycles, and two for nonnegative arc lengths, augment the PSP algorithm to maintain a property called sharp by Shier and Witzgall. The other three variants augment the PSP algorithm to maintain a property called near-sharp for nonnegative arc lengths. Extensive computational testing is presented on one of the sharp variants for nonnegative arc lengths and two of the near-sharp variants. The empirical results based on 4500 test problems with 90 different configurations and three different network topologies indicate that these new algorithms have excellent computational performance characteristics. Based on total solution times for the 4500 test problems, these new algorithms out-perform all other algorithms tested. In addition, one of the near-sharp algorithms strictly dominates all others on all problem topologies tested.

The motivation of this paper was the development of computer geometry course for students of mathematical specialties. The term “computer geometry” hereafter refers to the mathematical foundations of machine graphics. It is important to emphasize separately that this course should be designed for second-year students and, therefore, they can only be required to have prior knowledge of a standard course in algebra and mathematical analysis. This imposes certain restrictions on the material presented. When studying the thematic literature, it was found out that the de facto standard in modern computer graphics is the use of projective space and homogeneous coordinates. However, the authors faced a methodological problem-the almost complete lack of suitable educational literature in both Russian and English. This paper was written to present the information collected by the authors on this issue.

This paper proposes a parallel shortest path-searching algorithm and implements it on a newly structured parallel reconfigurable processor, DAPDNA-2 (IPFlex Inc). Routing determines the shortest paths from the source to the ultimate destination through intermediate nodes. In open shortest path first (OSPF), Dijkstra's shortest path algorithm, which is the conventional one, finds the shortest paths from the source on a program counter-based processor. The calculation time for Dijkstra's algorithm is O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) when the number of nodes is N. When the network scale is large, calculation time required by Dijkstra's algorithm increases rapidly. It's very difficult to compute Dijkstra's algorithm in parallel because of the need for previous calculation results, so Dijkstra's algorithm is unsuitable for parallel processors. Our proposed scheme finds the shortest paths using a simultaneous multi-path search method. In contrast with Dijkstra's algorithm, several nodes can be determined at one time. Moreover, we partition the network into different groups (network groups) and find the all-node pair's shortest path in each group using a pipeline operation. Networks can be abstracted, and the shortest paths in very large networks can be found easily. The proposed scheme can decrease calculation time from O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) to O(N) using a pipeline operation on DAPDNA-2. Our simulations show that the proposed algorithm uses 99.6% less calculation time than Dijkstra's algorithm. The proposed algorithm can be applied to the very large Internet network designs of the future.

Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in Õ(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04].

Autonomous unmanned aerial vehicles (UAVs) need to dynamically re-plan paths online to avoid newly detected obstacles and no-fly zones. Existing 3D path planning methods are either too computationally intensive for online use or have practical limitations for actual applications. We propose a new method based on visibility graphs that is both computationally efficient for online use and is suitable for actual applications. We consider the 3D space to be composed of many 2D planes that all pass through the current position of the UAV and the destination point. Finding the shortest collision-free path in each plane is a 2D path planning problem which can be solved by using existing visibility graph algorithms. We then collect all the shortest paths generated from each 2D plane and find the shortest path in the whole 3D space. We present the results of the proposed 3D path planning algorithm for two cases to demonstrate that the proposed method is effective.

Labeling methods for partially ordered paths