R. Tamassia. A dynamic data structure for planar graph embedding. In T. Lepisto and A. Salolnaa, editors, Automata, Languages and Prograraraing (Proc. ICALP '88), volulne 317 of Lecture Notes in Computer Science, pages 576 590. Springer-Verlag, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....decade there has been a growing interest in dynamic problems on graphs. In particular, much attention has been devoted to the dynamic maintenance of connected components [14, 16, 43] and higher connectivity [8, 10, 18, 19, 20, 21, 32, 35, 53] transitive closure [29, 30, 31, 37, 47, 55] planarity [7, 8, 46], shortest paths [2, 5, 13, 39, 44] and minimum spanning trees [10, 11, 16] In these problems one would like to answer queries on graphs that are undergoing a sequence of updates, such as insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update ....

....if only one type of update, i.e. either insertions or deletions, is allowed. Partially dynamic problems that deal with insertions only are called incremental. Planarity testing is a basic problem which has inspired an extensive amount of research in graph theory [9, 34, 54] data structures [4, 7, 15, 46], and sequential [28, 38] as well as parallel [33, 42] algorithms. Informally, a graph is planar if it can be embedded onto the plane without edge crossings. The planarity testing problem consists of answering the question whether a given graph is planar and if so of constructing such an embedding ....

[Article contains additional citation context not shown here]

R. Tamassia. A dynamic data structure for planar graph embedding. In Proc. 15th Int. Colloquium on Automata, Languages and Programming, pages 576--590. Lecture Notes in Computer Science 317, Springer-Verlag, Berlin, 1988.

....each update was known. These bounds apply to problems in which insertions need not respect a fixed embedding of the graph; a number of other papers have worked on dynamic graph problems such as minimum spanning forests, connectivity, and planarity testing for graphs with a fixed planar embedding [12, 14, 15, 18, 19, 22, 21, 24, 32, 33]. Finally, our methods apply to static as well as dynamic graph problems. A general certificate construction method from our companion paper, together with the certificates defined here, gives a unified method of testing 3 and 4 edge , and 2 and SPARSIFICATION II: EDGE AND VERTEX CONNECTIVITY ....

R. Tamassia, A dynamic data structure for planar graph embedding, in Proc. 15th Int. Colloq. Automata, Languages, and Programming, Lecture Notes in Computer Science 317, Springer-Verlag, New York, 1988, pp. 576--590.

....and one sink. A spherical st graph is a planar st graph that is embedded in the plane. If in that embedding the source and the sink are on the same face, the graph is a plane st graph. We require st graphs to be acyclic, which agrees with the definition of [18] and disagrees with the one from [16]. Figure 1 shows two spherical st graphs, the left of which is also a plane st graph. The following properties of spherical st graphs can be shown; the last two items may excuse spherical and plane the above definition. 1) Every vertex is on a simple directed path from s to t, called an ....

.... Thetaffi 6 Delta Delta I u v Figure 3. Updates Alternatively, we could also allow all possible insertion and deletion operations and let the data structure decide which updates violate the restrictions. To this end, we could use the planarity testing data structure of Tamassia [16] to decide if u and v are on the same face. The acyclicity condition is of course easily checked using our own data structure: Edge (u; v) induces a cycle if and only if there is a path from v to u. The restriction on the deletion operation is easily checked by maintaining the in and outdegree ....

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Roberto Tamassia, A dynamic data structure for planar graph embedding, Proc. 15th International Colloquium on Automata, Languages, and Programming (ICALP) (T. Lepisto and A. Salomaa, eds.), Lecture Notes in Computer Science, vol. 317, Springer Verlag, Berlin, 1988, pp. 576--590.

....preserve planeness) No reverse reduction is known. Figure 1: An example of transformation for vertex v. 2 Partially Dynamic Graphs Partially dynamic data structures support operations of biconnectivity 2 edge connectivity queries, edges insertions and vertices insertions, intermixed. Tamassia [16, 17] shows a data structure for biconnectivity queries about plane embedded graph that achieves O(log n) amortized time per operation. Westbrook and Tarjan [19] presents algorithms that support biconnectivity or 2 edge connectivity queries in O(ff(m; n) amortized time per operation, where n is the ....

R. Tamassia. A Dynamic Data Structure for Planar graph embedding. ICALP 1988.

....is an st graph if it has exactly one source and one sink. A spherical st graph is a planar st graph that is embedded in the plane. If in that embedding the source and the sink are on the same face, the graph is a plane st graph. Other dynamic problems on planar st graphs are studied in [2] and [41]. Reference [42] contains pointers to a vast number of applications of these graphs within visibility representations, graph drawing and embedding, motion planning, computational geometry, lattice theory, and VLSI design. The definition requires st graphs to be acyclic, which agrees with the ....

....of applications of these graphs within visibility representations, graph drawing and embedding, motion planning, computational geometry, lattice theory, and VLSI design. The definition requires st graphs to be acyclic, which agrees with the definition of [43] and disagrees with the one from [41]. Figure II.1 shows two spherical st graphs, the left of which is also a plane st graph. The following properties of spherical st graphs can be shown; the last two items motivate the somewhat awkward (but standard) terminology of the above definition. 1) Every vertex is on a simple directed ....

[Article contains additional citation context not shown here]

Roberto Tamassia, A dynamic data structure for planar graph embedding, Proc. 15th International Colloquium on Automata, Languages, and Programming (ICALP) (T. Lepisto and A. Salomaa, eds.), Lecture Notes in Computer Science, vol. 317, Springer Verlag, Berlin, 1988, pp. 576--590.

....if it has exactly one source s and one sink t. A spherical st graph is a planar st graph that is embedded in the plane. If in that embedding the source and the sink are on the same face, the graph is a plane st graph. We require st graphs to be acyclic, which agrees with [21] and disagrees with [19]. Figure 1 shows two spherical st graphs, the left of which is also a plane st graph. The following properties of spherical st graphs can be shown; the last two items explain why we used spherical and plane the above definition. 1. Every vertex is on a simple directed path from s to t, called an ....

....v) delete(u, v) q q q q q H HY 6 I u v Other operations. Alternatively, we could allow all insertion and deletion operations and let the data structure decide which updates violate the restrictions. To this end, we could use the planarity testing data structure of Tamassia [19] to decide if u and v are on the same face. The acyclicity condition is of course easily checked using our own data structure: edge (u, v) induces a cycle if and only if there is a path from v to u. The restriction on the deletion operation is easily checked by maintaining the in and outdegree ....

[Article contains additional citation context not shown here]

Roberto Tamassia. A dynamic data structure for planar graph embedding. In Proc. 15th ICALP, volume 317 of Lecture Notes in Computer Science, pages 576--

....in creating good diagrams is designing algorithms to assign a location for each node and a route for each edge; this is the classical graph drawing problem. Since the advent of graphics workstations in the early 1980s, the graph drawing problem has been the subject of a great deal of research [1, 3, 5, 6, 7, 8, 20, 21, 22, 23]. As the amount of information that we want to visualize becomes larger and the relations become more complicated, classical graph drawing methods tend to be inadequate. Even in a small modern file system (say with a 2GB drive on a PC) there are hundreds of nodes and links. Web graphs are much ....

Roberto Tamassia. A dynamic data structure for planar graph embedding. Lecture Notes in Computer Science (Proc. 15th ICALP), 317 (Springer Verlag):576--590, 1988.

....as a function of the output change [17, 40] The main dynamic problems considered on directed graphs include shortest paths and transitive closure. For lack of space, we do not include in this chapter dynamic algorithms for planar graphs, which have received considerable attention in recent years [6, 7, 11, 12, 18, 21, 22, 28, 32, 34, 42, 46, 47, 48, 51], and focus our attention to general undirected graphs only. The remainder of the chapter is organized as follows. In Section 2 we give some preliminary definitions and a little terminology. Dynamic tree problems are considered in Section 3, while in Section 4 we describe partially dynamic ....

R. Tamassia. A dynamic data structure for planar graph embedding. In Proc. 15th Int. Colloquium on Automata, Languages and Programming, pages 576--590. Lecture Notes in Computer Science 317, Springer-Verlag, Berlin, 1988.

....connectors are shown as thick lines. Notice that each angle of A(T) is a multiple of 5 ffi . 2. Next, for every face f of H(T) join the fans introduced in f , using their connectors, in a cyclic order as shown in Fig. 5. 15(c) For proving that G(T) is triconnected we use the following result of [100]. Two vertices u and v of G(T) are adjacent if there is an edge (u; v) in G(T) Theorem 17 (Tamassia [100] Let G be a biconnected planar graph. Vertices u and v form a separating pair of G if and only if 1. u and v are adjacent and there are at least three faces that contain u and v; or 2. u ....

....every face f of H(T) join the fans introduced in f , using their connectors, in a cyclic order as shown in Fig. 5. 15(c) For proving that G(T) is triconnected we use the following result of [100] Two vertices u and v of G(T) are adjacent if there is an edge (u; v) in G(T) Theorem 17 (Tamassia [100]) Let G be a biconnected planar graph. Vertices u and v form a separating pair of G if and only if 1. u and v are adjacent and there are at least three faces that contain u and v; or 2. u and v are not adjacent and there are at least two faces that contain u and v. Now we are ready to present ....

R. Tamassia. A dynamic data structure for planar graph embedding. In T. Lepisto and A. Salomaa, editors, Automata, Languages and Programming (Proc. 15th ICALP), volume 317 of Lecture Notes in Computer Science, pages 576--590. Springer-Verlag, 1988.

....known beforehand that the edge insertion will preserve the given embedding, then the algorithms are not guaranteed to be correct. However, none of these algorithms can test whether this condition is fulfilled within the given time bounds. Indeed, although there is an O(log n) algorithm by Tamassia [13] to maintain information about the embedding of a plane graph, this algorithm is not able to carry out the full repertoire of updates, since it supports only special cases of deletions. Namely, the algorithm is committed to a precomputed st numbering [4] of the plane graph, and is not able to ....

R. Tamassia, "A Dynamic Data Structure for Planar Graph Embedding", Proc. 15th ICALP , LNCS 317, Springer--Verlag, 1988, 576--590.

.... the corresponding fully dynamic problems [11] Moreover, despite intensive research on dynamic problems on graphs (such as dynamic maintenance of connectivity [7, 8, 10, 11, 14, 20, 22, 29, 30] 2 and 3 connectivity [7, 12, 29, 30] transitive closure [3, 4, 15, 16, 17, 18, 19, 31] planar graphs [6, 7, 19, 25], shortest paths [2, 9, 21, 24, 31] and minimum spanning trees [5, 8, 11, 24] there are very few graphtheoretic problems for which a fully dynamic non trivial algorithm is known. As mentioned in [30] the fully dynamic maintenance of the connected components of a graph differs substantially from ....

R. Tamassia, "A dynamic data structure for planar graph embedding", Proc. 15th Int. Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 317, Springer-Verlag, Berlin, 1988, 576--590.

.... spanning forests [Wes89] shortest paths [Rod68, Che76, GSV78, Fuj81, CC82, Gaz83, EG85, AMSN89, AIMSN90, Ita91] biconnected components [Sac86, WT92, BT90] triconnected components [Ita91, BT90] transitive closure [IK83, Ita86, Ita88, LPv88, YS88, Yel91] planar graphs [Tam88, TP90, BT89, EIT 92, PT88] ffl computational geometry [Ov81, CBT 92] ffl data bases [ABJ89] ffl syntax directed editors and grammars [Rep82, RTD83, Rep88, ACR 87] ffl data flow analysis [Ryd83, RP88, RC86, CR88, Mar89, Bur90, PS89, Zad84, Gho83, KRvM88] ffl code generation and ....

R. Tamassia. A dynamic data structure for planar graph embedding. In Proceedings of the Fifteenth International Colloquium on Automata, Languages and Programming, pages 576--590, Berlin, 1988. European Association for Theoretical Computer Science, Springer--Verlag. Lecture Notes in Computer Science 317.

....for edge insertions) 2. 6 Planarity Testing In a static environment we can test planarity and compute a planar embedding in optimal O(n) time (see, e.g. 13,42,47,60,72] The dynamic maintenance of a planar embedding under a sequence of edge insertions can be done in time O(log n) per operation [107]. Di Battista and Tamassia [27] have investigated how to test if a new edge can be added to a planar graph G so that G remains planar, and how to add vertices and edges so that planarity is preserved. The semidynamic data structure of [27] for biconnected graphs uses O(n) space and supports ....

....G to sourceleft( 0 ) Then p is contained in face ( 000 ; 0 ) where 000 is the left sibling of 0 . Each of these steps can be implemented in O(log n) time. Therefore, operation Locate is performed in O(log n) time. To implement operation Window we keep the data structure of [107] to to maintain the planar embedding of G. In particular given a face f of G in an upward embedding of series parallel digraph G, we can find two lists of edges and vertices that comprise the left and right boundary of f . Suppose p = x p ; y p ) and q = x q ; y q ) are points with x p x q and ....

[Article contains additional citation context not shown here]

R. Tamassia, "A Dynamic Data Structure for Planar Graph Embedding," Proc. 15th ICALP, LNCS 317(1988), 576--590.

....that s and t are on the boundary of the external face. Planar st graphs were first introduced by Lempel, Even, and Cederbaum [77] in connection with a planarity testing algorithm, and have subsequently been used in a host of applications, dealing with partial orders [68] planar graph embedding [30, 42, 111], graph planarization [91] graph drawing [41, 43] floor planning [123] planar point location [49, 94] visibility [73, 88, 102, 113, 114, 124] motion planning [101] and VLSI layout compaction [123] In this section, we consider a planar st digraph G with V vertices, and recall that G has ....

R. Tamassia. A dynamic data structure for planar graph embedding. In T. Lepisto and A. Salomaa, editors, Automata, Languages and Programming (Proc. 15th ICALP), volume 317 of Lecture Notes in Computer Science, pages 576--590. Springer-Verlag, 1988.

....1 connectivity queries in O(1) time and 1 path queries in O( time. For general graphs and 4, or for ( 1) connected graphs and fixed 4, there are O(t) space data structures that perform connectivity queries in O(1) time, but do not support output sensitive path queries (see [42, 51] for : 2, 14] for : 3, 28] for : 4, and [9] for 4) Table 1 in Appendix A summarizes previous and new results on methods for l path and t connectivity queries. 1.2 Previous Results on Orientations and Orderings of Graphs Orientations and orderings are powerful combinatorial ....

....work related to our combinatorial results. Bipolar orientations and st numberings of biconnected graphs were first defined in conjunc tion with a planarity testing algorithm [18, 33] and were later used for a variety of topological and geometric graph problems, such as embedding (see, e.g. [6, 15, 42]) visibility (see, e.g. 36, 43, 53] drawing (see, e.g. 1, 12, 44] point location (see, e.g. 35, 45] and floorplanning (see, e.g. 30] One of the notable properties of planar bipolar orientations is that they induce a 2 dimensional lattice [31] on the vertices of the graph. See [10] ....

R. Tamassia. A dynamic data structure for planar graph embedding. In T. Lepisto and A. Salolnaa, editors, Automata, Languages and Prograraraing (Proc. ICALP '88), volulne 317 of Lecture Notes in Computer Science, pages 576 590. Springer-Verlag, 1988.

....1 connectivity queries in O(1) time and 1 path queries in O( time. For general graphs and k 4, or for (k Gamma 1) connected graphs and fixed k 4, there are O(n) space data structures that perform k connectivity queries in O(1) time, but do not support output sensitive k path queries (see [42, 51] for k = 2, 14] for k = 3, 28] for k = 4, and [9] for k 4) Table 1 in Appendix A summarizes previous and new results on methods for k path and k connectivity queries. 1.2 Previous Results on Orientations and Orderings of Graphs Orientations and orderings are powerful combinatorial ....

....work related to our combinatorial results. Bipolar orientations and st numberings of biconnected graphs were first defined in conjunction with a planarity testing algorithm [18, 33] and were later used for a variety of topological and geometric graph problems, such as embedding (see, e.g. [6, 15, 42]) visibility (see, e.g. 36, 43, 53] drawing (see, e.g. 1, 12, 44] point location (see, e.g. 35, 45] and floorplanning (see, e.g. 30] One of the notable properties of planar bipolar orientations is that they induce a 2 dimensional lattice [31] on the vertices of the graph. See [10] ....

R. Tamassia. A dynamic data structure for planar graph embedding. In T. Lepisto and A. Salomaa, editors, Automata, Languages and Programming (Proc. ICALP '88), volume 317 of Lecture Notes in Computer Science, pages 576--590. Springer-Verlag, 1988.

....21, 34, 23] and shortest paths [1, 8, 25, 28, 34] Dynamic planar graphs arise in communication networks, graphics, and VLSI design, and they occur in algorithms that build planar subdivisions such as Voronoi diagrams. Algorithms have been proposed for maintaining the embedding of a planar graph [29] and for incremental planarity testing [2, 3] The dynamic minimum spanning tree problem has been considered by Spira and Pan [28] Chin and Houck [7] Frederickson [10] and Gabow and Stallmann [11] Frederickson gives an algorithm based on topology trees that runs in O( p m) time per ....

.... insert edge(d; e) which inserts an edge between orig(d) and orig(e) dividing the face to the left of d and e, can be implemented by x =make edge followed by splice(d; x) and splice(e; sym(x) We can similarly implement other standard operations such as delete edge, expand, and contract (see [29]) Let G denote the planar multigraph induced by the vertices and edges of a collection of subdivisions. Each subdivision induces a connected component of G. We may use make edge and splice to generate any multigraph G not containing isolated vertices. New vertices are always created by make ....

[Article contains additional citation context not shown here]

R. Tamassia. A dynamic data structure for planar graph embedding. In Proc. 15th Int. Conf. on Automata, Languages, and Programming, (ICALP 1988), Lecture Notes in Computer Science, vol. 317, pages 576-- 590. Springer-Verlag, Berlin, 1988.

....G to sourceleft( 0 ) Then p is contained in face ( 000 ; 0 ) where 000 is the left sibling of 0 . Each of these steps can be implemented in O(log n) time. Therefore, operation Locate is performed in O(log n) time. To implement operation Window we keep the data structure of [42] to to maintain the planar embedding of G. In particular given a face f of G in an upward embedding of seriesparallel digraph G, we can find two lists of edges and vertices that comprise the left and right boundary of f . Suppose p = x p ; y p ) and q = x q ; y q ) are points with x p x q and ....

....planar graphs. Planar st digraphs, which include series parallel digraphs as a special case, were first introduced by Lempel, Even, and Cederbaum [27] in connection with a planarity testing algorithm, and they have subsequently been used in several applications, including planar graph embedding [4,8,42], graph drawing [7,9] and planar point location [16, 21,32,48] A planar st digraph is a planar acyclic directed graph with exactly one source vertex s and exactly one sink vertex t, which is embedded in the plane such that s and t are on the boundary of the external face. The following ....

[Article contains additional citation context not shown here]

R. Tamassia, "A Dynamic Data Structure for Planar Graph Embedding," Proc. 15th ICALP, LNCS 317 (1988), 576--590.

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