Efficient algorithms for Euclidean shortest path and visibility problems with polygonal obstacles

Labeling methods for partially ordered paths

Shortest path problem is a typical routing optimization problem that is generally involved with a multi-criteria decision-making process. Therefore, the main objective of this paper is to find the shortest path in discrete-time dynamic networks based on bi-criteria of time and reliability by considering the effect of delay times that varies according to different departure time scenarios. Firstly, the well-known single-criterion Dijkstra’s algorithm is extended to fit the conditions of a bi-criteria problem. The solutions obtained from the extended Dijkstra was then compared with a proposed ant colony optimization (ACO) algorithm via a set of multi-objective performance metrics including CPU time, error ratio, spacing and diversity metrics. The analysis was made based on three network scales ranged from small (20-100 nodes), to medium (500-1900 nodes) and large (2000-10000 nodes). The computational results obtained from the analysis suggested that the extended Dijkstra’s algorithm has a higher efficiency in medium and large scaled networks. Furthermore, the comparison of the proposed ACO versus Dijkstra’s algorithm proved the preference of ACO for networks with larger-scaled (nodes over 5000), while, for smaller and medium-scaled networks (nodes 20-2000), the extended Dijkstra’s algorithm has a dominantly better performance in CPU time as compared to proposed ACO.

A finite-time algorithm for shortest path problems with time-varying costs

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).

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 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 shortest path problem (SPP) constitutes one of the most frequently encountered classes of problems in graph theory. It is certainly the most fundamental of components in the fields of transportation and communication networks. Shortest path problems may be encountered directly, possibly as a result of a clever formulation of a problem not at first sight involving shortest paths, or indirectly as a subproblem in the solution of a more complicated optimization problem. This use of shortest path problems as subroutines motivates the search for algorithms with good theoretical bounds on running time. We also seek computer implementations whose empirical performances are rapid in spite of perhaps weak theoretical bounds for the algorithms they implement. As testimony to the importance of shortest path and related problems, a large number of surveys, annotated bibliographies, and reviews have appeared over the past thirty years. Among them are those by Dreyfus (1969), Pierce (1975), Golden and Magnanti (1977), and Gallo and Pallottino (1988).

More than 50 thousand students are studying at Gaziantep University. Most of these students reside around the campus of Gaziantep University, while their hometown is outside Gaziantep. Therefore, they use the intercity bus station in the southeast of Gaziantep very often. In this paper, it is aimed to detect the shortest route between Gaziantep University campus and Gaziantep intercity bus station. Within the aim of the study, the shortest path (SPP) problem was considered. The purpose of the shortest path algorithm is to find the shortest distance between the start and end points in a road network. In this problem, 11 different municipal bus routes operating on the route starting from Gaziantep University campus and ending point Gaziantep bus station were examined and a road network consisting of 46 points was obtained. Then, the distance between each point was calculated with the help of Google Earth and the data was entered to mathematical model developed for the shortest path algorithm. The model was solved by the GAMS 23.1 optimization package and the shortest distance between the two points (10.4 km) was found to be optimal. As a result, a new bus route including 18 destination stops with shorter distances than the existing 11 bus routes has been proposed.

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs (s₁,t₁) and (s₂,t₂), decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an O(mn log n)-time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known O(m⁵n)-time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in O(mn² log n)-time. In addition, we extend our techniques to a more general setting: given two terminal pairs (s₁, t₁) and (s₂, t₂) in a directed graph, find the minimum possible number of vertex intersections between any shortest path from s₁ to t₁ and s₂ to t₂. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an O(m² n³)-time algorithm for directed graphs with positive edge weights, and an O(m+n)-time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same O(m+n)-time. This is somewhat surprising, as there is no known o(mn) time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).

A human observer can effortlessly identify visible portions of geometric objects present in the environment. However, computations of visible portions of objects from a viewpoint involving thousands of objects is a time consuming task even for high speed computers. To solve such visibility problems, efficient algorithms have been designed. This book presents some of these visibility algorithms in two dimensions. Specifically, basic algorithms for point visibility, weak visibility, shortest paths, visibility graphs, link paths and visibility queries are all discussed. Several geometric properties are also established through lemmas and theorems. With over 300 figures and hundreds of exercises, this book is ideal for graduate students and researchers in the field of computational geometry. It will also be useful as a reference for researchers working in algorithms, robotics, computer graphics and geometric graph theory, and some algorithms from the book can be used in a first course in computational geometry.

We study the point-to-point shortest path problem with preprocessing. Given an input graph, we preprocess it so as to be able to answer a series of source-to-destination queries efficiently. Our work is motivated by an algorithm of Gutman [ALENEX’04], based on the notion of reach, which measures how important each vertex is with respect to shortest paths. We present a simplified version of his algorithm that does not require explicit lower bounds during queries. We also show how the addition of shortcuts to the graph greatly improves the performance of both preprocessing and queries. Finally, we combine a reach-based algorithm with landmark-based A search to obtain a wide range of space-time trade-offs. For our motivating application, driving directions for road networks, the resulting algorithm is very efficient and practical. The road networks of the USA and Western Europe have roughly 20 million vertices, but on average our algorithm must visit fewer than a thousand to find the distance between two points. Our algorithm also works reasonably well on 2-dimensional grid graphs with random arc weights.

Alternative pathfinding requires finding a set of k alternative paths (including the shortest path) between a given source s and a target t. Intuitively, these paths should be significantly different from each other and meaningful/natural (e.g., must not contain loops or unnecessary detours). While finding alternative paths in road networks has been extensively studied, to the best of our knowledge, we are the first to formally study alternative pathfinding in game maps which are typically represented as Euclidean planes containing polygonal obstacles. First, we adapt the existing techniques designed for road networks to find alternative paths in the game maps. Then, based on our web-based system that visualises alternative paths generated by different approaches, we conduct a user study that shows that the existing road network approaches generate high-quality alternative paths when adapted for the game maps. However, these existing approaches are computationally inefficient especially when compared to the state-of-the-art shortest path algorithms. Motivated by this, we propose novel data structures and exploit these to develop an efficient algorithm to compute high-quality alternative paths. that shows that the existing road network approaches generate high-quality alternative paths in game maps. Our extensive experimental study demonstrates that our proposed algorithm is more than an order of magnitude faster than the existing approaches and returns alternative paths of comparable quality. Furthermore, our algorithm is comparable to a state-of-the-art shortest path algorithm in terms of running time.

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.

We introduce the time-dependent shortest path and vehicle routing problem. In this problem, a set of homogeneous vehicles is used to visit a set of customer locations dispersed over a very large network where the travel times are time-dependent and therefore the shortest path between two locations may change over time. The aim of the problem is to simultaneously determine the sequence in which the customer locations are visited and the arcs traveled on the paths between each pair of consecutively visited customers, such that the total travel time is minimized. We are the first to formally define and solve this fully integrated problem, providing tight bounds to it. We then propose a dynamic time-dependent shortest path algorithm embedded within a simulated annealing metaheuristic to efficiently solve the problem. We also propose a variant of the algorithm where some time-dependent shortest paths are precomputed. We test our formulations and algorithms on a set of real-life instances generated from a dataset of the road network in Québec City, Canada. Our results indicate that the resulting models are too large to be solved even for small instances. However, the obtained bounds show that the developed simulated annealing heuristic performs very well. We also demonstrate that neglecting time-dependent information on traffic leads to imprecise estimation of the traveling time. Moreover, the results show the importance of solving the shortest paths and routing problems simultaneously, as using a set of precomputed shortest paths leads to slightly worse solutions. This work adds new research avenues to city logistics and congestion studies.

Parametric search for the bi-attribute concave shortest path problem