Generalized Shortest Path Kernel on Graphs

We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known families, by the number of clusters they contain. We verify empirically that the generalized shortest path kernel outperforms the original shortest path kernel on a number of datasets. We give a theoretical analysis for explaining our experimental results. In particular, we estimate distributions of the expected feature vectors for the shortest path kernel and the generalized shortest path kernel, and we show some evidence explaining why our graph kernel outperforms the shortest path kernel for our graph classification problem.Comment: Short version presented at Discovery Science 2015 in Banf

Shortest Paths Avoiding Forbidden Subpaths

In this paper we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P. Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception-avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x in X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's algorithm incrementally after replicating vertices when an exception is discovered.Comment: 12 pages, 2 figures. Fixed a few typos, rephrased a few sentences, and used the STACS styl

Computing a rectilinear shortest path amid splinegons in plane

We reduce the problem of computing a rectilinear shortest path between two given points s and t in the splinegonal domain \calS to the problem of computing a rectilinear shortest path between two points in the polygonal domain. As part of this, we define a polygonal domain \calP from \calS and transform a rectilinear shortest path computed in \calP to a path between s and t amid splinegon obstacles in \calS. When \calS comprises of h pairwise disjoint splinegons with a total of n vertices, excluding the time to compute a rectilinear shortest path amid polygons in \calP, our reduction algorithm takes O(n + h \lg{n}) time. For the special case of \calS comprising of concave-in splinegons, we have devised another algorithm in which the reduction procedure does not rely on the structures used in the algorithm to compute a rectilinear shortest path in polygonal domain. As part of these, we have characterized few of the properties of rectilinear shortest paths amid splinegons which could be of independent interest

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