Speeding up shortest path algorithms

Given an arbitrary, non-negatively weighted, directed graph $G=(V,E)$ we present an algorithm that computes all pairs shortest paths in time $\mathcal{O}(m^* n + m \lg n + nT_\psi(m^*, n))$, where $m^*$ is the number of different edges contained in shortest paths and $T_\psi(m^*, n)$ is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial $n$ times application of $\psi$ that runs in $\mathcal{O}(nT_\psi(m,n))$. In our algorithm we use $\psi$ as a black box and hence any improvement on $\psi$ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson's reweighting technique and topological sorting results in an $\mathcal{O}(m^*n + m \lg n)$ all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP.Comment: 10 page

Heuristic estimates in shortest path algorithms

Shortest path problems occupy an important position in Operations Research aswell as in Arti¯cial Intelligence. In this paper we study shortest path algorithms thatexploit heuristic estimates. The well-known algorithms are put into one framework.Besides we present an interesting application of binary numbers in the shortest paththeory.operations research;graph theory;network flows;search problems

Computing shortest paths in 2D and 3D memristive networks

Global optimisation problems in networks often require shortest path length computations to determine the most efficient route. The simplest and most common problem with a shortest path solution is perhaps that of a traditional labyrinth or maze with a single entrance and exit. Many techniques and algorithms have been derived to solve mazes, which often tend to be computationally demanding, especially as the size of maze and number of paths increase. In addition, they are not suitable for performing multiple shortest path computations in mazes with multiple entrance and exit points. Mazes have been proposed to be solved using memristive networks and in this paper we extend the idea to show how networks of memristive elements can be utilised to solve multiple shortest paths in a single network. We also show simulations using memristive circuit elements that demonstrate shortest path computations in both 2D and 3D networks, which could have potential applications in various fields

Open source environment to define constraints in route planning for GIS-T

Route planning for transportation systems is strongly related to shortest path algorithms, an optimization problem extensively studied in the literature. To find the shortest path in a network one usually assigns weights to each branch to represent the difficulty of taking such branch. The weights construct a linear preference function ordering the variety of alternatives from the most to the least attractive.Postprint (published version