Problem Of Finding Shortest Paths Research Articles

An optimal algorithm for L1 shortest paths in unit-disk graphs

A unit-disk graph G(P) of a set P of points in the plane is a graph with P as its vertex set such that two points of P are connected by an edge if the distance between the two points is at most 1 and the weight of the edge is equal to the distance of the two points. Given P and a source point s∈P, we consider the problem of finding shortest paths in G(P) from s to all other vertices of G(P). In the L2 case where the distance is measured by the L2 metric, the problem has been extensively studied and the current best algorithm runs in O(nlog2⁡n) time, with n=|P|. In this paper, we study the L1 case in which the distance is measured under the L1 metric (and each disk becomes a diamond); we present an O(nlog⁡n) time algorithm, which matches the Ω(nlog⁡n)-time lower bound.

Truncated Tree Based Hybrid Beamforming for Multi-Panel MIMO

Multi-panel MIMO is a practical antenna array structure for massive MIMO, but the optimal hybrid beamforming design has a complexity exponential to the number of antenna panels. In this letter, we propose a truncated tree based hybrid beamforming design with linear complexity for multi-panel MIMO transmission. We convert the hybrid beamforming design problem into the shortest path finding problem that is visualized by tree data structure. To reduce complexity, the tree is truncated by deleting the intermediate paths that are less likely to be the shortest path. By this means, the truncated tree based design reduces exponential time complexity to linear time complexity. Numerical results show that the proposed truncated tree based design far outperforms the heuristic max power gain beamforming design and achieves near optimal performance with a proper truncation ratio.

Shortest Paths in the Plane with Obstacle Violations

We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for $$k \le h$$. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of $$\varTheta (kn)$$ on the size of this map, and show that it can be computed in $$O(k^2n \log n)$$ time, where n is the total number of obstacle vertices.

Shortest Route Optimization for Emergency Service: Case Study in Kumasi, Ghana

It is becoming difficult for emergence services to find the best route especially in Kumasi to any destination in order to save lives in real time. This study deals with the problem of finding shortest paths in traversing some locations within the Kumasi Metropolis in the Ashanti Region of Ghana. Dijkstra‘s Algorithm was selected to determine the shortest distances from any location to any destination within the Kumasi metropolis. The objective of thesis is to use Dijkstra‘s algorithm in constructing the minimum spanning tree considering the dual carriage ways in the road network of Kumasi metropolis within the shortest possible time for emergence services. The distance between 51 locations of the towns with the major roads was measured and a legend and a matrix were formulated. A visual basic program was prepared using Dijkstra‘s algorithm. The distances were used to prepare an input deck for the visual program. The methodology employed included review of relevant literature of the types of Dijkstra‘s algorithm and methods employed in the solution of the Dijkstra‘s algorithm and to develop computer solutions – ArcGIS and VB.net for faster computation of Dijkstra‘s algorithm. The result shows a remarkable reduction in the actual distance as compared with the ordinary routing. These results indicate, clearly the importance of this type of algorithms in the optimization of network flows.  Hence the shortest distance from any area in Kumasi metropolis to another can easily be calculated using this thesis so as to minimize the average loss of lives in case emergences.

The combinatorics of discrete time-trees: theory and open problems

A time-tree is a rooted phylogenetic tree such that all internal nodes are equipped with absolute divergence dates and all leaf nodes are equipped with sampling dates. Such time-trees have become a central object of study in phylogenetics but little is known about the parameter space of such objects. Here we introduce and study a hierarchy of discrete approximations of the space of time-trees from the graph-theoretic and algorithmic point of view. One of the basic and widely used phylogenetic graphs, the mathrm {NNI} graph, is the roughest approximation and bottom level of our hierarchy. More refined approximations discretize the relative timing of evolutionary divergence and sampling dates. We study basic graph-theoretic questions for these graphs, including the size of neighborhoods, diameter upper and lower bounds, and the problem of finding shortest paths. We settle many of these questions by extending the concept of graph grammars introduced by Sleator, Tarjan, and Thurston to our graphs. Although time values greatly increase the number of possible trees, we show that 1-neighborhood sizes remain linear, allowing for efficient local exploration and construction of these graphs. We also obtain upper bounds on the r-neighborhood sizes of these graphs, including a smaller bound than was previously known for mathrm {NNI}. Our results open up a number of possible directions for theoretical investigation of graph-theoretic and algorithmic properties of the time-tree graphs. We discuss the directions that are most valuable for phylogenetic applications and give a list of prominent open problems for those applications. In particular, we conjecture that the split theorem applies to shortest paths in time-tree graphs, a property not shared in the general mathrm {NNI} case.

Heuristic Search on Graphs with Existence Priors for Expensive-to-Evaluate Edges

We address the problem of finding shortest paths in graphs where some edges have a prior probability of existence, and their existence can be verified during planning with time- consuming operations. Our work is motivated by real-world robot motion planning, where edge existence is often expensive to verify (typically involves time-consuming collision-checking between the robot and world models), but edge existence probabilities are readily available. The goal then, is to develop an anytime algorithm that can return good solutions quickly by somehow leveraging the existence probabilities, and continue to return better-quality solutions or provide tighter suboptimality bounds with more time. While our motivation is fast and high-quality motion planning for robots, this work presents two fundamental contributions applicable to generic graphs with probabilistic edges. They are: a) an algorithm for efficiently computing all relevant shortest paths in a graph with probabilistic edges, and as a by-product the expected shortest path cost, and b) an anytime algorithm for evaluating (verifying existence of) edges in a collection of paths, which is optimal in expectation under a chosen distribution of the algorithm interruption time. Finally, we provide a practical approach to integrate a) and b) in the context of robot motion planning and demonstrate significant improvements in success rate and planning time for a 11 degree-of-freedom mobile manipulation planning problem. We also conduct additional evaluations on a 2D grid navigation domain to study our algorithm’s behavior.

Review of A* (A Star) Navigation Mesh Pathfinding as the Alternative of Artificial Intelligent for Ghosts Agent on the Pacman Game

Shortest pathfinding problem has become a populer issue in Game’s Artificial Intelligent (AI). This paper discussed the effective way to optimize the shortest pathfinding problem, namely Navigation Mesh (NavMesh). This method is very interesting because it has a large area of implementation, especially in games world. In this paper, NavMesh was implemented by using A* (A star) algorithm and examined in Unity 3D game engine. A* was an effective algorithm in shortest pathfinding problem because its optimization was made with effective tracing using segmentation line. Pac-Man game was chosen as the example of the shortest pathfinding by using NavMesh in Unity 3D. A* algorithm was implemented on the enemies of Pac-Man (three ghosts),  which path was designed by using NavMesh concept. Thus, the movement of ghosts in catching Pac-Man was the result of this review of the effectiveness of this concept. In further research, this method could be implemented on several optimization programmes, such as Geographic Information System (GIS), robotics, and statistics.

Shortest Path Finding Problem in Stochastic Time-Dependent Road Networks With Stochastic First-In-First-Out Property

As travel times in road networks are dynamic and uncertain, it is difficult and time-consuming to search for the least expected time path in large-scale networks. This paper addresses the problem of finding the least expected time path in stochastic time-dependent (STD) road networks. A stochastic travel speed model is proposed to represent STD link travel times. It is proved that the link travel times in STD networks satisfy the stochastic first-in-first-out (S-FIFO) property. Based on this S-FIFO property, an efficient multicriteria A* algorithm is proposed to exactly determine the least expected time path in STD networks. Computational results using several large-scale road networks show that the proposed algorithm has a significant computational advantage over existing solution algorithms without the S-FIFO property.

Shortest path games

We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.

A Border Line-Based Pruning Scheme for Shortest Path Computations

With the progress of IT and mobile positioning technologies, various types of location-based services (LBS) have been proposed and implemented. Finding a shortest path between two nodes is one of the most fundamental tasks in many LBS related applications. So far, there have been many research efforts on the shortest path finding problem. For instance, A * algorithm estimates neighboring nodes using a heuristic function and selects minimum cost node as the closest one to the destination. Pruning method, which is known to outperform the A * algorithm, improves its routing performance by avoiding unnecessary exploration in the search space. For pruning, shortest paths for all node pairs in a map need to be pre-computed, from which a shortest path container is generated for each edge. The container for an edge consists of all the destination nodes whose shortest path passes through the edge and possibly some unnecessary nodes. These containers are used during routing to prune unnecessary node visits. However, this method shows poor performance as the number of unnecessary nodes included in the container increases. In this paper, we focus on this problem and propose a new border line-based pruning scheme for path routing which can reduce the number of unnecessary node visits significantly. Through extensive experiments on randomly-generated, various complexity of maps, we empirically find out optimal number of border lines for clipping containers and compare its performance with other methods.

Dynamic shortest path problems with time-varying costs

This paper concerns the problem of finding shortest paths from one node to all other nodes in networks for which arc costs can vary with time, each arc has a transit time, and parking with a corresponding time-varying cost is allowed at the nodes. The transit times can also take negative values. A general labeling method, as well as several implementations, are presented for finding shortest paths and detecting negative cycles under the assumption that arc traversal costs are piecewise linear and node parking costs are piecewise constant.

Inferring branching pathways in genome-scale metabolic networks

BackgroundA central problem in computational metabolic modelling is how to find biochemically plausible pathways between metabolites in a metabolic network. Two general, complementary frameworks have been utilized to find metabolic pathways: constraint-based modelling and graph-theoretical path finding approaches. In constraint-based modelling, one aims to find pathways where metabolites are balanced in a pseudo steady-state. Constraint-based methods, such as elementary flux mode analysis, have typically a high computational cost stemming from a large number of steady-state pathways in a typical metabolic network. On the other hand, graph-theoretical approaches avoid the computational complexity of constraint-based methods by solving a simpler problem of finding shortest paths. However, while scaling well with network size, graph-theoretic methods generally tend to return more false positive pathways than constraint-based methods.ResultsIn this paper, we introduce a computational method, ReTrace, for finding biochemically relevant, branching metabolic pathways in an atom-level representation of metabolic networks. The method finds compact pathways which transfer a high fraction of atoms from source to target metabolites by considering combinations of linear shortest paths. In contrast to current steady-state pathway analysis methods, our method scales up well and is able to operate on genome-scale models. Further, we show that the pathways produced are biochemically meaningful by an example involving the biosynthesis of inosine 5'-monophosphate (IMP). In particular, the method is able to avoid typical problems associated with graph-theoretic approaches such as the need to define side metabolites or pathways not carrying any net carbon flux appearing in results. Finally, we discuss an application involving reconstruction of amino acid pathways of a recently sequenced organism demonstrating how measurement data can be easily incorporated into ReTrace analysis. ReTrace is licensed under GPL and is freely available for academic use at http://www.cs.helsinki.fi/group/sysfys/software/retrace/.ConclusionReTrace is a useful method in metabolic path finding tasks, combining some of the best aspects in constraint-based and graph-theoretic methods. It finds use in a multitude of tasks ranging from metabolic engineering to metabolic reconstruction of recently sequenced organisms.

Hamiltonian oracles

Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of finding shortest paths on a manifold. The study of these shortest paths leads to lower bounds of the original unitary oracle problem. A number of example Hamiltonian oracles are studied in this paper, including oracle interrogation and the problem of computing the XOR of the hidden bits. Both of these problems are related to the study of geodesics on spheres with nonround metrics. For the case of two hidden bits a complete description of the geodesics is given. For $n$ hidden bits a simple lower bound is proven that shows the problems require a query time proportional to $n$, even in the continuum limit. Finally, the problem of continuous Grover search is reexamined leading to a modest improvement to the protocol of Farhi and Gutmann.

An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n2) time sequential algorithm and anO(n2/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.

AN OPTIMAL ALGORITHM FOR SOLVING ALL-PAIRS SHORTEST PATHS ON TRAPEZOID GRAPHS

International Journal of Computational Engineering ScienceVol. 03, No. 02, pp. 103-116 (2002) No AccessAN OPTIMAL ALGORITHM FOR SOLVING ALL-PAIRS SHORTEST PATHS ON TRAPEZOID GRAPHSSUKUMAR MONDAL, MADHUMANGAL PAL, and TAPAN K. PALSUKUMAR MONDALDepartment of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 721 102, West Bengal, India Search for more papers by this author , MADHUMANGAL PALDepartment of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 721 102, West Bengal, India Search for more papers by this author , and TAPAN K. PALDepartment of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 721 102, West Bengal, India Search for more papers by this author https://doi.org/10.1142/S1465876302000575Cited by:14 Next AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractThe shortest-paths problem is an important problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The Shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one node to another often gives the best way to route message between the nodes. This paper presents an O(n2) time algorithm for solving all pairs shortest path problems on trapezoid graphs which are extensions of interval graphs and permutation graphs. The space complexity of this algorithm is of O(n2). This problem has been solved by constructing n breadth-first search (BFS) trees with each of the n vertices as root. As the lower bound of time complexity for computing the all pairs shortest paths is known to be of O(n2), this proposed algorithm is optimal.Keywords:Design of algorithmsanalysis of algorithmsbreadth-first searchshortest-pathstrapezoid graph FiguresReferencesRelatedDetailsCited By 14Computation of diameter, radius and center of permutation graphsShaoli Nandi, Sukumar Mondal, and Sambhu Charan Barman6 December 2021 | Discrete Mathematics, Algorithms and Applications, Vol. 14, No. 08An optimal algorithm to find minimum k-hop connected dominating set of permutation graphsAmita Samanta Adhya, Sukumar Mondal, and Sambhu Charan Barman25 March 2020 | Asian-European Journal of Mathematics, Vol. 14, No. 04An Introduction to Intersection GraphsMadhumangal Pal1 Jan 2020Computation of Inverse 1-Center Location Problem on the Weighted Trapezoid GraphsBiswanath Jana, Sukumar Mondal and Madhumangal Pal1 May 2019 | Missouri Journal of Mathematical Sciences, Vol. 31, No. 1An optimal algorithm to find minimum k-hop dominating set of interval graphsSambhu Charan Barman, Madhumangal Pal, and Sukumar Mondal24 April 2019 | Discrete Mathematics, Algorithms and Applications, Vol. 11, No. 02Surjective L (2, 1)-labeling of cycles and circular-arc graphsSk. Amanathulla and Madhumangal Pal27 Jul 2018 | Journal of Intelligent & Fuzzy Systems, Vol. 35, No. 1L(0, 1)-Labelling of Trapezoid GraphsSatyabrata Paul, Madhumangal Pal and Anita Pal8 June 2017 | International Journal of Applied and Computational Mathematics, Vol. 3, No. S1Scheduling Algorithm to Select Optimal Programme Slots in Television Channels: A Graph Theoretic ApproachMadhumangal Pal and Anita Pal20 July 2016 | International Journal of Applied and Computational Mathematics, Vol. 3, No. 3Unrestricted and complete Breadth-First Search of trapezoid graphs in timeChristophe Crespelle and Philippe Gambette1 Jun 2010 | Information Processing Letters, Vol. 110, No. 12-13A linear time algorithm to construct a tree 4-spanner on trapezoid graphsSambhu Charan Barman, Sukumar Mondal and Madhumangal Pal1 Mar 2010 | International Journal of Computer Mathematics, Vol. 87, No. 4Counting the number of vertex covers in a trapezoid graphMin-Sheng Lin and Yung-Jui Chen1 Oct 2009 | Information Processing Letters, Vol. 109, No. 21-22An efficient algorithm to find next-to-shortest path on permutation graphsSambhu Charan Barman, Sukumar Mondal and Madhumangal Pal11 December 2008 | Journal of Applied Mathematics and Computing, Vol. 31, No. 1-2Efficient algorithms for the minimum connected domination on trapezoid graphsYin-Te Tsai, Yaw-Ling Lin and F.R. Hsu1 Jun 2007 | Information Sciences, Vol. 177, No. 12An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphsAnita Saha, Madhumangal Pal and Tapan K. Pal1 Mar 2005 | Journal of Applied Mathematics and Computing, Vol. 17, No. 1-2 Recommended Vol. 03, No. 02 Metrics History KeywordsDesign of algorithmsanalysis of algorithmsbreadth-first searchshortest-pathstrapezoid graphPDF download

On the Circuit Cover Problem for Mixed Graphs

The circuit cover problem for mixed graphs (those containing edges and/or arcs) is defined as follows. Given a mixed graph M with a nonnegative integer weight function p on its edges and arcs, decide whether (M, p) has a circuit cover, that is, a list of circuits in M such that every edge (arc) e is contained in exactly p(e) circuits of the list. In the special case when M is a directed graph (contains only arcs), the problem is easy, but when M is an undirected graph not many results are known. For general mixed graphs this problem was shown to be NP-complete by Arkin and Papadimitriou in 1986. We prove that this problem remains NP-complete for planar mixed graphs. Furthermore, we present a good characterization for the existence of a circuit cover when M is series-parallel (a similar result holds for the fractional version). We also describe a polynomial algorithm to find such a circuit cover, when it exists. This is an ellipsoid-based algorithm whose separation problem is the minimum circuit problem on series-parallel mixed graphs, which we show to be polynomially solvable. Results on two well-known combinatorial problems, the problem of detecting negative circuits and the problem of finding shortest paths, are also presented. We prove that both problems are NP-hard for planar mixed graphs.

Optimizedk-shortest-paths algorithm for facility restoration

The problem of finding shortest paths arises in many contexts; testing restoration algorithms and developing design packages for large telecommunications networks are two cases where the simple task of finding sets of restoration paths can consume up to 95 per cent of the execution time of an application program. This paper presents experimental studies of several well-known shortest-paths algorithms adapted to the task of finding the k-successively-shortest link-disjoint replacement paths for restoration in a telecommunications network with n nodes. The implementations range in complexity from O(kn2) when based on Dijkstra's original method, through several improvements to an efficient implementation of O(kn[v+longn]) complexity, and finally to an O(kn) implementation for the special case of edge-sparse graphs with small integer edge weights. Here v is the maximum degree of a node in the network. Several alternatives were tested during the course of these studies, particularly with a view to minimizing the number of heap updates, These alternatives are possible because we are searching for several paths between a given pair of nodes, rather than just one path between one or more pairs of nodes. Two fairly straightforward changes yield a decrease in execution time, whereas a more complex heap management strategy consumes as much time in the added code as it releases from the main routine. Experimental results confirm the theoretical complexity of q k n log n) and demonstrate a speed-up of nearly an order of magnitude over the simpler O(kn2) implementation in the largest networks tested. The optimized implementation is recommended for planning and operational applications of k-shortest paths rerouting for telecommunications network restoration and restorable network design. If hop counts or small integer link weights can be used to measure distances, then the qkn) implementation is recommended, as typical telecommunications networks are edge-sparse.

Parallel algorithms for the single source shortest path problem

We present several parallel algorithms for the problem of finding shortest paths from a specified vertex (called the source) to all others in a weighted directed graph. The algorithms are for machines ranging from array processors, multipleinstruction multiple-data stream machines to a special network of processors. These algorithms have been designed by “parallelizing” two classic sequential algorithms — one due to Dijkstra (1959), the other due to Moore (1957). Our interest is not only in obtaining speeded-up parallel versions of the algorithms but also in exploring the design principles, the commonality of correctness proofs of the different versions, and the subjective complexity of explaining and understanding these versions.