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

An Almost Optimal Edit Distance Oracle

We consider the problem of preprocessing two strings S and T, of lengths m and n, respectively, in order to be able to efficiently answer the following queries: Given positions i,j in S and positions a,b in T, return the optimal alignment score of S[i..j] and T[a..b]. Let N = mn. We present an oracle with preprocessing time N^{1+o(1)} and space N^{1+o(1)} that answers queries in log^{2+o(1)}N time. In other words, we show that we can efficiently query for the alignment score of every pair of substrings after preprocessing the input for almost the same time it takes to compute just the alignment of S and T. Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS). The best previously known oracle with construction time and size ?(N) has slow ?(?N) query time [Sakai, TCS 2019], and the one with size N^{1+o(1)} and query time log^{2+o(1)}N (using a planar graph distance oracle) has slow ?(N^{3/2}) construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a ? N factor

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

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