Improved Bounds for Shortest Paths in Dense Distance Graphs

Improved Bounds for Shortest Paths in Dense Distance Graphs Paweł Gawrychowski Institute of Computer Science, University of Wrocław, Poland Adam Karczmarz1 University of Warsaw, Poland Abstract 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 ∂G of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on ∂G encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques. Fakcharoenphol and Rao [7] proposed an efficient implementation of Dijkstra’s algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log2 n) time per distance clique with b vertices, even though a clique has b2 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 2 minimum labeled vertices in  O(log n) amortized time per vertex. We show an improved data 2 n structure with O loglog amortized bounds. This is the first improvement over the data 2 log n 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. 2012 ACM Subject Classification Theory of computation → Data structures design and analysis Keywords and phrases shortest paths, dense distance graph, planar graph, Monge matrix Digital Object Identifier 10.4230/LIPIcs.ICALP.2018.61 Related Version A full version of this paper is available at [11], https://arxiv.org/abs/1602. 07013. Acknowledgements We thank Piotr Sankowski for helpful discussions. The first author also thanks Oren Weimann and Shay Mozes for discussions about Monge matrices and FR-Dijkstra. 1 Supported by the grants 2014/13/B/ST6/01811 and 2017/24/T/ST6/00036 of the Polish National Science Center. EA TC S © Paweł Gawrychowski and Adam Karczmarz; licensed under Creative Commons License CC-BY 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Editors: Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella; Article No. 61; pp. 61:1–61:15 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 61:2 Improved Bounds for Shortest Paths in Dense Distance Graphs 1 Introduction Finding a truly subquadratic, strongly polynomial algorithm for many of the most basic real-weighted graph problems like the single-source shortest paths or the maximum flow on sparse digraphs seems to be very difficult. However, the situation changes significantly if we restrict ourselves to planar digraphs, which constitute an important class of sparse graphs. In this regime the ultimate goal is to obtain linear or almost linear time complexity. In their breakthrough paper, Fakcharoenphol and Rao gave the first nearly-linear time algorithm for single-source shortest paths in real-weighted planar graphs [7]. Their algorithm 3 had O(n  log n) time complexity. Although their upper bound was eventually improved 2 log n to O n log by Mozes and Wulff-Nilsen [22], the techniques introduced in [7] proved log n very useful in obtaining not only nearly-linear time algorithms for other static planar graph problems, but also first sublinear dynamic algorithms for shortest paths and maximum flows. A major contribution of Fakcharoenphol and Rao was introducing the general concept of a dense distance graph. Let G be a real-weighted plane digraph and let ∂G denote some subset of its vertices, called boundary vertices, such that there exist ` = O(1) faces f1 , . . . , f` of G satisfying ∂G ⊆ V (f1 ) ∪ . . . V (f` ). Such graphs with a topologically nice boundary typically emerge after decomposing a plane graph using a cycle separator. For example, by using a cycle separator of Miller [19], one can decompose any n-vertex triangulated plane graph H into two subgraphs Hin and Hout such that (i) Hin ∪ Hout = H, (ii) Hin and Hout are smaller than H by a constant factor, (iii) the set ∂Hin = ∂Hout = V (Hin ) ∩ V (Hout ) has √ size O( n) and lies both on a single face of Hin and on a single face of Hout . We define a distance clique of G, denoted DC(G), to be a complete graph on ∂G such that the weight of an edge uv is equal to the length of the shortest path from u to v in G. A dense distance graph is a union of possibly many unrelated distance cliques. We note that such a definition of a dense distance graph (also used in [23]) is a bit more general than that of Fakcharoenphol and Rao [7], who defined it only with respect to a recursive decomposition of G using cycle-separators. In fact, subsequently dense distance graphs have been also defined a bit differently with respect to so-called r-divisions [13], and even the two sides of a cycle-separator [15] (i.e., DC(Hin ) ∪ DC(Hout ) in the above example). The definition we assume in this paper captures all these cases. Suppose we are given q distance cliques DC(G1 ), . . . , DC(Gq ) explicitly. Let DDG = Sq Pq 2 i=1 DC(Gi ), V = ∂G1 ∪ . . . ∪ ∂Gq and n = |V |. Clearly, DDG has i=1 |∂Gi | edges in total. Fakcharoenphol and Rao showed how to compute single-source shortest paths in such
P 2 graph DDG with non-negative edge weights in only O |∂G | log n time, i.e., for each i i DC(Gi ) one only needs to spend time nearly-linear in the number of vertices of DC(Gi ), as opposed to its number of edges, i.e., |∂Gi |2 . Their method is often called the FR-Dijkstra, as it follows the overall approach of Dijkstra’s algorithm. Whereas Dijkstra’s algorithm uses a priority queue to maintain its distance labels and extract a non-visited vertex with minimum label, a much more sophisticated data structure is used in FR-Dijkstra. This data structure is capable of relaxing many edges in a single step, by leveraging the fact that certain submatrices of the adjacency matrix of a distance clique are Monge matrices. Applications of Dense Distance Graphs and FR-Dijkstra. Fakcharoenphol and Rao originally employed FR-Dijkstra to construct their dense distance graph recursively and answer distance queries on it. However, the applications of FR-Dijkstra proved much broader and thus it has become an important planar graph primitive used to obtain numerous breakthrough results in recent years. We briefly cover the most important of these results below. P. Gawrychowski and A. Karczmarz 61:3 The dense distance graphs and FR-Dijksta ha (...truncated)