Shortest Paths In Planar Graphs Research Articles

Polynomial kernels for tracking shortest paths

Given an undirected graph G=(V,E), vertices s,t∈V, and an integer k, Tracking Shortest Paths requires deciding whether there exists a set of k vertices T⊆V such that for any two distinct shortest paths between s and t, say P1 and P2, we have T∩V(P1)≠T∩V(P2). In this paper, we give the first polynomial size kernel for the problem. Specifically we show the existence of a kernel with O(k2) vertices and edges in general graphs and a kernel with O(k) vertices and edges in planar graphs for the Tracking Paths in DAG problem. This problem admits a polynomial parameter transformation to Tracking Shortest Paths, and this implies a kernel with O(k4) vertices and edges for Tracking Shortest Paths in general graphs and a kernel with O(k2) vertices and edges in planar graphs. Based on the above we also give a single exponential algorithm for Tracking Shortest Paths in planar graphs.

An efficient oracle for counting shortest paths in planar graphs

We study the counting shortest paths problem: find the number ns(u,v) of shortest paths from a vertex u to a vertex v in a graph of n vertices. Bezáková and Searns [ISAAC 2018] give an oracle which, given u and v in a planar graph, answers ns(u,v) in O(n) time and O(n1.5) memory space. Applying this oracle directly, it takes O(n) time to answer whether there is a unique shortest path from u to v. We propose an oracle which answers ns(u,v) in O(n) time and O(n1.5) space. Our oracle answers whether there is a unique shortest path from u to v in O(log⁡n) time. Computational studies show that our oracle is faster than the oracle of Bezáková and Searns for large graphs. A key technique to speed up the query time of our oracle is applying Voronoi diagrams on planar graphs, an advanced data structure widely used in shortest distance oracles. Based on Voronoi diagrams on planar graphs, significant theoretical improvements have been made for distance oracles. Our studies confirm that Voronoi diagrams are efficient data structures for distance oracles in practice.

Rerouting shortest paths in planar graphs

A rerouting sequence is a sequence of shortest st-paths such that consecutive paths differ in one vertex. We study the Shortest Path Rerouting Problem, which asks, given two shortest st-paths P and Q in a graph G, whether a rerouting sequence exists from P to Q. This problem is PSPACE-hard in general, but we show that it can be solved in polynomial time if G is planar. In addition, it can be solved in polynomial time when every vertex has at most two neighbors closer to s and at most two neighbors closer to t. To this end, we introduce a dynamic programming method for reconfiguration problems, based on using small encodings of reconfiguration graphs.

Multiple-Source Shortest Paths in Embedded Graphs

Let $G$ be a directed graph with $n$ vertices and nonnegative weights in its directed edges, embedded on a surface of genus $g$, and let $f$ be an arbitrary face of $G$. We describe a randomized algorithm to preprocess the graph in $O(gn \log n)$ time with high probability, so that the shortest-path distance from any vertex on the boundary of $f$ to any other vertex in $G$ can be retrieved in $O(\log n)$ time. Our result directly generalizes the $O(n\log n)$-time algorithm of Klein [Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of $f$. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in $O(g^2 n\log n)$ time with high probability. Our high-probability time bounds hold in the worst case for generic edge weights or with an additional $O(\log n)$ factor for arbitrary edge weights.

Oracles for bounded-length shortest paths in planar graphs

We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = ( V , E ), one can build in O(|V|) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed.Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service.Our results can be easily generalized to other wide classes of graphs---for instance we can take any minor-closed family of graphs.

A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter $\epsilon$ such that $0<\epsilon$ , our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1+ $\epsilon$ factor. The time bounds for both query and update for our algorithm is O( \epsilon -1 n 2/3 log 2 n log D) , where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph.

Faster Shortest-Path Algorithms for Planar Graphs

We give a linear-time algorithm for single-source shortest paths in planar graphs with nonnegative edge-lengths. Our algorithm also yields a linear-time algorithm for maximum flow in a planar graph with the source and sink on the same face. For the case where negative edge-lengths are allowed, we give an algorithm requiringO(n4/3log(nL)) time, whereLis the absolute value of the most negative length. This algorithm can be used to obtain similar bounds for computing a feasible flow in a planar network, for finding a perfect matching in a planar bipartite graph, and for finding a maximum flow in a planar graph when the source and sink are not on the same face. We also give parallel and dynamic versions of these algorithms.

Dynamic algorithms for shortest paths in planar graphs

We propose data structures for maintaining shortest paths in planar graphs in which the weight of an edge is modified. Our data structures allow us to compute, after an update, the shortest-path tree rooted at an arbitrary query node in time O( nlog log n) and to perform an update in O((log n) 3). Our data structure can be applied also to the problem of maintaining the maximum flow problem in an s–t planar network. As far as the all-pairs shortest-path problem is concerned, we are interested in computing the shortest distances between q pairs of nodes. We show how to obtain an o( n 2) algorithm for computing the shortest path between q pairs of nodes whenever q = o( n 2). We also consider the dynamic version of the problem in which we allow the modification of the weight of an edge.

Fast Algorithms for Shortest Paths in Planar Graphs, with Applications

Graph decomposition and data structures techniques are presented that exploit the structure of planar graphs to yield faster algorithms for a number of shortest path and related problems. Improved ...