Shortest Paths in Distance-regular Graphs

AbstractWe aim here to introduce a new point of view of the Laplacian of a graph, Γ. With this purpose in mind, we consider L as a kernel on the finite space V(Γ), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(Γ). If Γ is a proper set of a suitable host graph, then the equilibrium problem for Γ can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of Γ. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs

Some applications of the proper and adjacency polynomials in the theory of graph spectra

Given a vertex u\inV of a graph $\Gamma=(V,E)$, the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called $u$-local spectrum of $\Gamma$. These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for te distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, were recently used to study a new concept of distance-regularity for non-regular graphs, and also to give bounds on some distance-related parameters such as the diameter. Here we develop the subject of these polynomials and gave a survey of some known results involving them. For instance, distance-regular graphs are characterized from their spectra and the number of vertices at ``extremal distance'' from each of their vertices. Afterwards, some new applications of both, the proper and adjacency polynomials, are derived, such as bounds for the radius of $\Gamma$ and the weight $k$-excess of a vertex. Given the integers $k,\mu\ge 0$, let $\Gamma_k^{\mu}(u)$ denote the set of vertices which are at distance at least $k$ from a vertex $u\in V$, and there exist exactly $\mu$ (shortest) $k$-paths from $u$ to each each of such vertices. As a main result, an upper bound for the cardinality of $\Gamma_k^{\mu}(u)$ is derived, showing that $|\Gamma_k^{\mu}(u)|$ decreases at least as $O(\mu^{-2})$, and the cases in which the bound is attained are characterized. When these results are particularized to regular graphs with four distinct eigenvalues, we reobtain a result of Van Dam about $3$-class association schemes, and prove some conjectures of Haemers and Van Dam about the number of vertices at distane three from every vertex of a regular graph with four distinct eigenvalues---setting $k=2$ and $\mu=0$---and, more generally, the number of non-adjacent vertices to every vertex $u\in V$, which have $\mu$ common neighbours with it.Peer Reviewe

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms

In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. In 2010, Abraham et al. showed upper bounds on the query time in terms of a graph's highway dimension and diameter for the current fastest routing algorithms, including contraction hierarchies, transit node routing, and hub labeling. In this paper, we show corresponding lower bounds for the same three algorithms. We also show how to improve a result by Milosavljevic which lower bounds the number of shortcuts added in the preprocessing stage for contraction hierarchies. We relax the assumption of an optimal contraction order (which is NP-hard to compute), allowing the result to be applicable to real-world instances. Finally, we give a proof that optimal preprocessing for hub labeling is NP-hard. Hardness of optimal preprocessing is known for most routing algorithms, and was suspected to be true for hub labeling

Shortest-weight paths in random regular graphs

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hat{\alpha}\log n$ edges where $\hat{\alpha}$ is the unique solution of the equation $\alpha \log(\frac{d-2}{d-1}\alpha) - \alpha = \frac{d-3}{d-2}$, for $\alpha > \frac{d-1}{d-2}$.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1112.633