9 research outputs found
Fast and Compact Exact Distance Oracle for Planar Graphs
For a given a graph, a distance oracle is a data structure that answers
distance queries between pairs of vertices. We introduce an -space
distance oracle which answers exact distance queries in time for
-vertex planar edge-weighted digraphs. All previous distance oracles for
planar graphs with truly subquadratic space i.e., space
for some constant ) either required query time polynomial in
or could only answer approximate distance queries.
Furthermore, we show how to trade-off time and space: for any , we show how to obtain an -space distance oracle that answers
queries in time . This is a polynomial
improvement over the previous planar distance oracles with query
time
Exact Distance Oracles for Planar Graphs with Failing Vertices
We consider exact distance oracles for directed weighted planar graphs in the
presence of failing vertices. Given a source vertex , a target vertex
and a set of failed vertices, such an oracle returns the length of a
shortest -to- path that avoids all vertices in . We propose oracles
that can handle any number of failures. More specifically, for a directed
weighted planar graph with vertices, any constant , and for any , we propose an oracle of size
that answers queries in
time. In particular, we show an
-size, -query-time
oracle for any constant . This matches, up to polylogarithmic factors, the
fastest failure-free distance oracles with nearly linear space. For single
vertex failures (), our -size,
-query-time oracle improves over the previously best
known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for , . For multiple failures, no planarity exploiting
results were previously known
Better Tradeoffs for Exact Distance Oracles in Planar Graphs
We present an -space distance oracle for directed planar graphs
that answers distance queries in time. Our oracle both
significantly simplifies and significantly improves the recent oracle of
Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses
-space and answers queries in time. We achieve this by
designing an elegant and efficient point location data structure for Voronoi
diagrams on planar graphs.
We further show a smooth tradeoff between space and query-time. For any , we show an oracle of size that answers queries in time. This new tradeoff is currently the best (up to
polylogarithmic factors) for the entire range of and improves by polynomial
factors over all the previously known tradeoffs for the range
Shortest Paths in Geometric Intersection Graphs
This thesis studies shortest paths in geometric intersection graphs, which can model, among others, ad-hoc communication and transportation networks. First, we consider two classical problems in the field of algorithms, namely Single-Source Shortest Paths (SSSP) and All-Pairs Shortest Paths (APSP). In SSSP we want to compute the shortest paths from one vertex of a graph to all other vertices, while in APSP we aim to find the shortest path between every pair of vertices. Although there is a vast literature for these problems in many graph classes, the case of geometric intersection graphs has been only partially addressed.
In unweighted unit-disk graphs, we show that we can solve SSSP in linear time, after presorting the disk centers with respect to their coordinates. Furthermore, we give the first (slightly) subquadratic-time APSP algorithm by using our new SSSP result, bit tricks, and a shifted-grid-based decomposition technique.
In unweighted, undirected geometric intersection graphs, we present a simple and general technique that reduces APSP to static, offline intersection detection. Consequently, we give fast APSP algorithms for intersection graphs of arbitrary disks, axis-aligned line segments, arbitrary line segments, d-dimensional axis-aligned boxes, and d-dimensional axis-aligned unit hypercubes. We also provide a near-linear-time SSSP algorithm for intersection graphs of axis-aligned line segments by a reduction to dynamic orthogonal point location.
Then, we study two problems that have received considerable attention lately. The first is that of computing the diameter of a graph, i.e., the longest shortest-path distance between any two vertices. In the second, we want to preprocess a graph into a data structure, called distance oracle, such that the shortest path (or its length) between any two query vertices can be found quickly. Since these problems are often too costly to solve exactly, we study their approximate versions.
Following a long line of research, we employ Voronoi diagrams to compute a (1+epsilon)-approximation of the diameter of an undirected, non-negatively-weighted planar graph in time near linear in the input size and polynomial in 1/epsilon. The previously best solution had exponential dependency on the latter. Using similar techniques, we can also construct the first (1+epsilon)-approximate distance oracles with similar preprocessing time and space and only O(log(1/\epsilon)) query time.
In weighted unit-disk graphs, we present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter and for other related problems, such as the radius and the bichromatic closest pair. To do so, we combine techniques from computational geometry and planar graphs, namely well-separated pair decompositions and shortest-path separators. We also show how to extend our approach to obtain O(1)-query-time (1+epsilon)-approximate distance oracles with near linear preprocessing time and space. Then, we apply these oracles, along with additional ideas, to build a data structure for the (1+epsilon)-approximate All-Pairs Bounded-Leg Shortest Paths (apBLSP) problem in truly subcubic time