Improved Bounds for Shortest Paths in Dense Distance Graphs

We study the problem of computing shortest paths in so-called dense distance graphs, a basic building block for designing efficient planar graph algorithms. Let G be a plane graph with a distinguished set partial{G} of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on partial{G} encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques. Fakcharoenphol and Rao [Fakcharoenphol and Rao, 2006] proposed an efficient implementation of Dijkstra\u27s algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log^2{n}) time per distance clique with b vertices, even though a clique has b^2 edges. Here, n is the total number of vertices of the dense distance graph. The invention of FR-Dijkstra was instrumental in obtaining such results for planar graphs as nearly-linear time algorithms for multiple-source-multiple-sink maximum flow and dynamic distance oracles with sublinear update and query bounds. At the heart of FR-Dijkstra lies a data structure updating distance labels and extracting minimum labeled vertices in O(log^2{n}) amortized time per vertex. We show an improved data structure with O((log^2{n})/(log^2 log n)) amortized bounds. This is the first improvement over the data structure of Fakcharoenphol and Rao in more than 15 years. It yields improved bounds for all problems on planar graphs, for which computing shortest paths in dense distance graphs is currently a bottleneck

Reconstructing Geometric Structures from Combinatorial and Metric Information

In this dissertation, we address three reconstruction problems. First, we address the problem of reconstructing a Delaunay triangulation from a maximal planar graph. A maximal planar graph G is Delaunay realizable if there exists a realization of G as a Delaunay triangulation on the plane. Several classes of graphs with particular graph-theoretic properties are known to be Delaunay realizable. One such class of graphs is outerplanar graph. In this dissertation, we present a new proof that an outerplanar graph is Delaunay realizable. Given a convex polyhedron P and a point s on the surface (the source), the ridge tree or cut locus is a collection of points with multiple shortest paths from s on the surface of P. If we compute the shortest paths from s to all polyhedral vertices of P and cut the surface along these paths, we obtain a planar polygon called the shortest path star (sp-star) unfolding. It is known that for any convex polyhedron and a source point, the ridge tree is contained in the sp-star unfolding polygon [8]. Given a combinatorial structure of a ridge tree, we show how to construct the ridge tree and the sp-star unfolding in which it lies. In this process, we address several problems concerning the existence of sp-star unfoldings on specified source point sets. Finally, we introduce and study a new variant of the sp-star unfolding called (geodesic) star unfolding. In this unfolding, we cut the surface of the convex polyhedron along a set of non-crossing geodesics (not-necessarily the shortest). We study its properties and address its realization problem. Finally, we consider the following problem: given a geodesic star unfolding of some convex polyhedron and a source point, how can we derive the sp-star unfolding of the same polyhedron and the source point? We introduce a new algorithmic operation and perform experiments using that operation on a large number of geodesic star unfolding polygons. Experimental data provides strong evidence that the successive applications of this operation on geodesic star unfoldings will lead us to the sp-star unfolding

Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs

A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of \epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself