Journal of Computer and System Sciences SS1352 journal of computer and system sciences 52, 327 (1996) article no. 0002 View metadata, citation and similar papers at Separatorcore.ac.uk Based Sparsification brought to you by CORE provided by Elsevier - Publisher Connector I. Planarity Testing and Minimum Spanning Trees* David Eppstein- Department of Information and Computer Science, University of California, Irvine, California 92717 Zvi Galil Computer Science Department, Columbia University, New York, New York 10027; and Computer Science Department, Tel-Aviv University, Tel-Aviv, Israel Giuseppe F. Italiano Dipartimento di Informatica e Sistemistica, UniversitaÁ di Roma ``La Sapienza'', Rome, Italy and Thomas H. Spencer Department of Mathematics and Computer Science, University of Nebraska at Omaha, Omaha, Nebraska Received October 11, 1993 1. INTRODUCTION We describe algorithms and data structures for maintaining a dynamic planar graph subject to edge insertions and edge deletions Sparsification was introduced in [2] as a technique for that preserve planarity but that can change the embedding. We give a fully dynamic planarity testing algorithm that maintains a graph subject designing fully dynamic graph algorithms, in which edges to edge insertions and deletions and that allows queries that test may be inserted into and deleted from a graph while some whether the graph is currently planar, or whether a potential new edge graph property must be maintained. This technique is based 1Â2 would violate planarity, in O(n ) amortized time per update or query. upon a combination of graph decomposition and edge We give fully dynamic algorithms for maintaining the connected com- elimination. Let G be a graph with m edges and n vertices: we ponents, the best swap and the minimum spanning forest of a planar graph in O(log n) worst-case time per insertion and O(log2 n) per partition the edges of G into a collection of sparse subgraphs deletion. Finally, we give fully dynamic algorithms for maintaining (i.e., subgraphs with O(n) edges), and summarize the relevant the 2-edge-connected components of a planar graph in O(log n) information for each subgraph in an even sparser certificate. 2 amortized time per insertion and O(log n) per deletion. All of the data We merge certificates in pairs, producing larger subgraphs structures, except for the one that answers planarity queries, handle only insertions that keep the graph planar. All our algorithms improve which we make sparse by again applying the certificate reduc- previous bounds. The improvements are based upon a new type of tion. The result is a balanced binary tree in which each node sparsification combined with several properties of separators in planar is represented by a sparse certificate. Each edge insertion or graphs. ] 1996 Academic Press, Inc. deletion causes changes in log(mÂn) tree nodes, but each such change occurs in a subgraph with O(n) edges, reduced from the m edges in the original graph. This reduces a time * Portions of this paper were presented at the ``25th Annual ACM Symp. bound of T(m, n)toO(T(O(n), n) log(mÂn)). Using a more on Theory of Computing, San Diego, CA, 1993'' [4] - Supported by NSF grant CCR-9258355. sophisticated approach (described in [3]), we can eliminate Work supported in part by NSF Grant CCR-90-14605 and CISE the logarithmic factor from this bound. This reduces the time Institutional Infrastructure Grant CCR-90-24735. bounds for many dynamic graph problems, including vertex On leave from IBM T. J. Watson Research Center, Yorktown Heights, and edge connectivity, and minimum spanning forests, to NY 10598. Work supported in part by ESPRIT BRA ALCOM II under exactly match the bounds known for sparse graphs. contract no. 7141 and by the Italian MURST Project ``Algoritmi, Modelli di Calcolo e Strutture Informative''. While sparsification has many applications in algorithms Research partially supported by the University Committee on Research, for arbitrary graphs, it seemed unlikely that it could be used University of Nebraska at Omaha. to speed up algorithms for families of graphs that are 3 0022-0000Â96 12.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. File: 571J 135201 . By:CV . Date:23:01:00 . Time:11:44 LOP8M. V8.0. Page 01:01 Codes: 6767 Signs: 4092 . Length: 60 pic 11 pts, 257 mm 4 EPPSTEIN ET AL. already sparse, such as planar graphs. However, algorithms previous bound was that for general graphs, O(n1Â2)[3].In for planar graphs are especially important, as these graphs the easier case where the embedding of the graph is fixed, it arise frequently in applications. We call a planar graph that was known how to solve this problem in time O(log n) per is committed to an embedding plane, and we use the term update [6, 9]. planar when changes in the embedding are allowed. We call v Third, we maintain the best swap [8] in a planar an edge insertion that preserves the embedding embed- graph, in O(log n) worst-case time per insertion or edge ding-preserving, whereas we call it planarity-preserving if it weight decrease, and O(log2 n) per deletion or edge weight keeps the graph planar (even though its embedding can increase. In the easier case of embedding-preserving inser- change); finally, we call an edge insertion arbitrary if it is not tions (i.e., when the embedding of the graph is fixed), it was known to preserve planarity. Previous work on dynamic known how to maintain the best swap in O(log3 n) time [8]. graph algorithms has used ad hoc techniques to solve a Our algorithm improves this bound as well. Frederickson number of problems such as minimum spanning forests, [8] uses a dynamic best swap algorithm to compute the k 2-edge-connectivity and planarity testing for plane graphs smallest spanning trees of a graph; with our new techniques, (with embedding-preserving insertions) [68, 12, 15, 18], this can be done in O(n+k log2 n) time for any planar graph. and 2- and 3-vertex-connectivity, and planarity under The best previous bound was O(n+k log3 n) [8]. planarity-preserving insertions or deletions on planar v Fourth, we maintain the 2-edge-connected components graphs [10, 11]. of a planar graph in O(log n) amortized time per planarity- In this paper we provide a new, general technique for preserving insertion or query and O(log2 n) per deletion. The dynamic planar graph problems, including those mentioned best previous bound was that for general graphs, O(n1Â2)[3]. above; in all these problems, we deal with either arbitrary or In the easier case where the embedding of the graph is fixed, planarity-preserving insertions and therefore allow changes it was known how to solve 2-edge-connectivity n O(log2 n) of the embedding. The new ideas behind this technique are time per update [15]. the following. We expand the notion of a certificate, to a definition for graphs in which a subset of the vertices are In the companion paper [5] we use the same technique denoted as interesting; these compressed certificates may to maintain the 3-edge-connected, 4-edge-connected, reduce the size of the graph by removing uninteresting ver- 2-vertex-connected, and 3-vertex-connected components tices. Using this notion, we define a type of sparsification of a planar graph in O(n1Â2) amortized time per operation. based on separators, small sets of vertices the removal of We believe that our minimum spanning forest, 2-edge- which splits the graph into roughly equal size components. connectivity and connectivity bounds are particularly Recursively finding separators in these components gives a interesting. Indeed, so far there was a huge gap in the time separator tree which we also use as our sparsification tree; complexity of fully dynamic algorithms for plane and for the interesting vertices in each certificate will be those planar graphs (such as, for instance, fully dynamic connec- vertices used in separators at higher levels of the tree. We tivity or minimum spanning forest). For plane graphs we introduce the notion of a balanced separator tree, which also typically knew bounds like O(polylog n)(O(log n) in our partitions the interesting vertices evenly in the tree: we show example), while for arbitrary updates on planar graphs the how to compute such a tree in linear time, and how to main- bounds were much higher (O(n1Â2) in our example). Even tain it dynamically. We believe that balanced separator trees worse, because of sparsification the latter bounds were often are interesting on their own and may find applications in exactly the bounds known for general graphs. In other other areas. Using this technique, we achieve the following words, while we knew how to exploit the properties of plane results. graphs, so far we could not exploit any property of planar graphs. Our new technique for the first time breaks this v First, we maintain a (not necessarily planar) graph barrier and shows how to efficiently exploit properties of subject to arbitrary edge insertions and deletions and planar graphs in a fully dynamic scenario. allow queries that test whether the graph is currently The remainder of the paper consists of the following sec- planar, or whether a potential new edge would violate 1Â2 tions. Section 2 defines precisely the compressed certificates planarity, in O(n ) amortized time per update or query. we need for speeding up algorithms on planar graphs, while The previous best known fully dynamic algorithm only Section 3 uses these certificates to develop our sparsifica- allowed planarity-preserving insertions and took time 2Â3 tion technique.