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Authenticated Data Structures for Graph and Geometric Searching
 IN CTRSA
, 2001
"... Following in the spirit of data structure and algorithm correctness checking, authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being maintained by a remote host. We present techniques for authenticatin ..."
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Cited by 49 (18 self)
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Following in the spirit of data structure and algorithm correctness checking, authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being maintained by a remote host. We present techniques for authenticating data structures that represent graphs and collection of geometric objects. We use a model where a data structure maintained by a trusted source is mirrored at distributed directories, with the directories answering queries made by users. When a user queries a directory, it receives a cryptographic proof in addition to the answer, where the proof contains statements signed by the source. The user verifies the proof trusting only the statements signed by the source. We show how to efficiently authenticate data structures for fundamental problems on networks, such as path and connectivity queries, and on geometric objects, such as intersection and containment queries.
Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in softwar ..."
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Cited by 34 (4 self)
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Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
Efficient Subtyping Tests with PQEncoding
, 2001
"... Subtyping tests, i.e., determining whether one type is a subtype of another, are a frequent operation during the execution of objectoriented programs. The challenge is in encoding the hierarchy in a small space, while simultaneously making sure that subtyping tests have efficient implementation. We ..."
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Cited by 20 (3 self)
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Subtyping tests, i.e., determining whether one type is a subtype of another, are a frequent operation during the execution of objectoriented programs. The challenge is in encoding the hierarchy in a small space, while simultaneously making sure that subtyping tests have efficient implementation. We present a new scheme for encoding multiple and single inheritance hierarchies, which, in the standardized hierarchies, reduces the footprint of all previously published schemes. The scheme is called PQencoding after PQtrees, a data structure previously used in graph theory for finding the orderings that satisfy a collection of constraints. In particular, we show that in the traditional object layout model, the extra memory requirements for single inheritance hierarchies is zero. In the PQencoding subtyping tests are constant time, and use only two comparisons. Other than PQtrees, PQencoding uses several novel optimization techniques. These techniques are applicable also in improving the performance of other, previously published, encoding schemes.
Drawing SeriesParallel Graphs on a Box
 The University of Lethbridge
, 1997
"... A box is a restricted portion of the threedimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A boxdrawing of a graph is a straightline crossingfree drawing where vertices are located at integer grid points along the four lines. It is known t ..."
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Cited by 17 (3 self)
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A box is a restricted portion of the threedimensional integer grid consisting of four parallel lines of in nite length placed one grid unit apart. A boxdrawing of a graph is a straightline crossingfree drawing where vertices are located at integer grid points along the four lines. It is known that some planar graphs with triconnected components do not admit a boxdrawing. This paper shows that even structurally simpler planar graphs, namely seriesparallel graphs, are not boxdrawable in general. On the positive side, it is proved that every seriesparallel graph whose vertices have maximum degree at most three is boxdrawable. A drawing algorithm is presented that computes a box drawing of a 3planar seriesparallel graph in optimal time and with optimal volume.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Efficient Computation of Causal Behavioural Profiles using Structural Decomposition
, 2010
"... Identification of behavioural contradictions is an important aspect of software engineering, in particular for checking the consistency between a business process model used as system specification and a corresponding workflow model used as implementation. In this paper, we propose causal behaviour ..."
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Cited by 8 (5 self)
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Identification of behavioural contradictions is an important aspect of software engineering, in particular for checking the consistency between a business process model used as system specification and a corresponding workflow model used as implementation. In this paper, we propose causal behavioural profiles as the basis for a consistency notion, which capture essential behavioural information, such as order, exclusiveness, and causality between pairs of activities. Existing notions of behavioural equivalence, such as bisimulation and trace equivalence, might also be applied as consistency notions. Still, they are exponential in computation. Our novel concept of causal behavioural profiles provides a weaker behavioural consistency notion that can be computed efficiently using structural decomposition techniques for sound freechoice workflow systems if unstructured net fragments are acyclic or can be traced back to S or Tnets.
OnLine Convex Planarity Testing
, 1995
"... An important class of planar straightline drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph i ..."
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Cited by 6 (3 self)
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An important class of planar straightline drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph is subject to online insertions of vertices and edges. We present online algorithms for convex planarity testing with the following performance, where t denotes the number of vertices of the graph: convex planarity testing and insertion of vertices take 0(1) worstcase tinhe, insertion of edges takes 0(log n) amortized tinhe, and the space requirement of the data structure is O(n). Furthermore, we give a new combinatorial characterization of convex planar graphs.
Twopage book embedding and clustered graph planarity
, 2009
"... Abstract: A 2page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2page book embedding ..."
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Cited by 3 (0 self)
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Abstract: A 2page book embedding of a graph places the vertices linearly on a spine (a line segment) and the edges on two pages (two half planes sharing the spine) so that each edge is embedded in one of the pages without edge crossings. Testing whether a given graph admits a 2page book embedding is known to be NPcomplete. In this paper, we study the problem of testing whether a given graph admits a 2page book embedding with a fixed edge partition. Based on structural properties of planar graphs, we prove that the problem of testing and finding a 2page book embedding of a graph with a partitioned edge set can be solved in linear time. As an application of our main result, we consider the problem of testing planarity of clustered graphs. The complexity of testing clustered graph planarity is a long standing open problem in Graph Drawing. Recently, polynomial time algorithms have been found for several classes of clustered graphs in which both the structure of the underlying graphs and clustering structure are restricted. However, when the underlying graph is disconnected, the problem remains open. Our book embedding results imply that the clustered planarity problem can be solved in linear time for a certain class of clustered graphs with arbitrary underlying graphs (i.e. possibly disconnected). 1
Efficient authenticated data structures for graph connectivity and geometric search problems. Algorithmica
, 2010
"... Authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being controlled by a remote untrusted host. In this paper we present efficient techniques for authenticating data structures that represent graphs and ..."
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Cited by 3 (1 self)
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Authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being controlled by a remote untrusted host. In this paper we present efficient techniques for authenticating data structures that represent graphs and collections of geometric objects. We use a dataquerying model where a data structure maintained by a trusted source is mirrored at distributed untrusted servers, called responders, with the responders answering queries made by users: when a user queries a responder, along with the answer to the issued query, he receives a cryptographic proof that allows the verification of the answer trusting only a short statement (digest) signed by the source. We introduce the path hash accumulator, a new primitive based on cryptographic hashing for efficiently authenticating various properties of structured data represented as paths, including any decomposable query over sequences of elements. We show how to employ our primitive to authenticate queries about properties of paths in graphs and search queries on multicatalogs. This allows the design of new, efficient authenticated data structures for fundamental problems on networks, such as path and connectivity queries over graphs, and complex queries on twodimensional geometric objects, such as intersection and containment queries.
Monotone Drawings of Graphs
, 2012
"... We study a new standard for visualizing graphs: A monotone drawing is a straightline drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected ..."
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Cited by 2 (1 self)
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We study a new standard for visualizing graphs: A monotone drawing is a straightline drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected planar graphs, we study the interplay between monotonicity, planarity, and convexity, and we outline a number of open problems and future research directions. Submitted: