Optimal Decremental Connectivity in Planar Graphs

Optimal Decremental Connectivity in Planar Graphs∗ Jakub Łącki1 and Piotr Sankowski2 1 University of Warsaw Warsaw, Poland University of Warsaw Warsaw, Poland 2 Abstract We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form ‘Are vertices u and v connected with a path?’ in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in O(n) time. This results improves over previously known O(n log n) time algorithm. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory Keywords and phrases decremental connectivity, planar graphs, dynamic connectivity, algorithms Digital Object Identifier 10.4230/LIPIcs.STACS.2015.608 1 Introduction The dynamic graph connectivity problem consists in maintaining connectivity information about an undirected graph, which is undergoing modifications. Typically, the modifications are additions or removals of edges or vertices. In this paper we focus on the problems in which each modification adds or removes a single edge. These problems have three variants: in the incremental version, edges can only be added to the graph, in the decremental one the edges may only be removed, whereas in the fully dynamic version both edge insertions and deletions are allowed. Graph updates are intermixed with a set of connectivity queries of the form ‘Are vertices u and w in the same connected component?’ We consider the decremental connectivity problem for planar graphs, and show an algorithm that may answer connectivity queries in constant time and process any sequence of edge deletions in O(n) time. The previously known best running time of O(n log n) was obtained by using the fully dynamic algorithm. We assume word-RAM model with standard operations. ∗ Jakub Łącki is a recipient of the Google Europe Fellowship in Graph Algorithms, and this research is supported in part by this Google Fellowship. Piotr Sankowski is partially supported by ERC grant PAAl no. 259515, NCN grant "Efficient planar graph algorithms" and the Foundation for Polish Science. © Jakub Łącki and Piotr Sankowski; licensed under Creative Commons License CC-BY 32nd Symposium on Theoretical Aspects of Computer Science (STACS 2015). Editors: Ernst W. Mayr and Nicolas Ollinger; pp. 608–621 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany J. Łącki and P. Sankowski 1.1 609 Prior work It is easy to see that incremental graph connectivity can be solved using an algorithm for the union-find problem. It follows from the result of Tarjan [16] that a sequence of t edge insertions and t queries can be handled in O(tα(t)) time, where α(t) is the extremely slowly growing inverse Ackermann function. There has been a long line of research considering the fully dynamic connectivity in general graphs [6, 3, 8, 10, 19, 11, 21]. The best currently known algorithms have polylogartithmic update and query time. Thorup [19] has shown a randomized Monte Carlo algorithm with O(log n(log log n)3 ) amortized update and O(log n/ log log log n) query time.1 An algorithm by Wulff-Nilsen [21] handles updates in slightly worse O(log2 n/ log log n) amortized time, but it is deterministic and answers queries in O(log n/ log log n) time. The best algorithm with worst-case update guarantee is a randomized algorithm by Kapron, King and Mountjoy [11], which processes updates in O(log5 n) time and answers queries in O(log n/ log log n) time. However, if we require a deterministic algorithm with worst-case √ running time guarantee, nothing better than a O( n) time algorithm is known [6, 3, 2]. For the decremental variant, Thorup [18] has shown a randomized algorithm, which can process any sequence of edge deletions in O(m log(n2 /m) + n(log n)3 (log log n)2 ) time and answers queries in constant time. Here, m is the initial number of edges in the graph. If m = Θ(n2 ), the update time is O(m), whereas for m = Ω(n(log n log log n)2 ) it is O(m log n). The picture is much simpler in case of planar graphs. Eppstein et. al [5] gave a fully dynamic algorithm which handles updates and queries in O(log n) amortized time, but requires that the graph embedding remains fixed. For the general case (i.e., when the embedding may change) Eppstein et. al [4] gave an algorithm with O(log2 n) worst-case update time and O(log n) query time. In planar graphs, the best known solution for the incremental connectivity problem is the union-find algorithm. However, for the special case when the final resulting planar graph is given upfront, and the edge insertions and queries are given later in a dynamic fashion Gustedt [7] has shown an O(n) time algorithm. On the other hand, for the decremental problem nothing better than a direct application of the fully dynamic algorithm is known. This is different from both general graphs and trees, where the decremental connectivity problems have better solutions than what could be achieved by a simple application of their fully dynamic counterparts. In case of general graphs, the best total update time was O(m log n) [18] (except for very sparse graphs, including planar graphs), compared to O(m log n(log log n)3 ) time for the fully dynamic variant. For trees, only O(n) time is necessary to perform all updates in the decremental scenario [1], while in the fully dynamic case one can use dynamic trees and obtain O(log n) worst case update time. There has also been some progress in obtaining lower bounds for dynamic connectivity problems. Tarjan and La Poutré [17, 15] have shown that incremental connectivity requires Ω(α(n)) time per operation on a pointer machine. Henzinger and Fredman [9] considered the fully dynamic problem and RAM model and obtained a lower bound of Ω(log n/ log log n), which also works for plane graphs. This was improved by Demaine and Pǎtraşcu [14] to a lower bound of Ω(log n) in cell-probe model. The lower bound holds also for plane graphs. 1 Throughout the paper we use n and m to denote, respectively, the number of vertices and the number of edges in the graph. S TA C S 2 0 1 5 610 Optimal Decremental Connectivity in Planar Graphs 1.2 Our results We show an algorithm for the decremental connectivity problem in planar graphs, which processes any sequence of edge deletions in O(n) time and answers queries in constant time. This improves over the previous bound of O(n log n), which can be obtained by applying the fully dynamic algorithm by Eppstein [5], and matches the running time of decremental connectivity on trees [1]. In fact, we present a O(n) time reduction from the decremental connectivity problem to a collection of incremental problems in graphs of total size O(n). These incremental problems have a specific structure: the set of allowed union operations (...truncated)