2981830
Characterizing the Complexity of Subgraph Isomorphism for Graphs of Bounded Path-Width
We show that the complexity of the subgraph isomorphism problem on graphs of bounded path-width is inherently dependent on the connectivity of the source and target graphs. In particular, for the problem of determining whether a source graph G of path-width k is a subgraph of a target graph H of path-width k, we present an O(n3) algorithm for G and H both k-connected, for n the sum of the sizes of the graphs, and NP-completeness results for connectivity less than k. In previous polynomial-time algorithms, the degree of the polynomial in the running time was a function of k. In contrast, we show that when neither G nor H is k-connected, the problem becomes NP-complete. The same result also holds if one of the graphs has at least k vertices of unbounded degree. Since bounded path-width graphs are also bounded tree-width graphs, our hardness results immediately extend to this larger class. A further NP-completeness result applies to the situation in which both graphs have tree-width k, but only the target graph is k-connected. This provides a complete characterization of the subgraph isomorphism problem on bounded tree-width graphs, thus answering an open question of Matousek and Thomas.
28349058
A Constructive Proof that Trees Are Well-Quasi-Ordered Under Minors
The recent proof of Robertson and Seymour that graphs are well-quasi-ordered under minors immediately implies that a number of intesting problems have polynomial time algorithms. However, partially because of their non-constructive nature, these proofs do not yield any information about the algorithms. Here we present a constructive proof that trees are well-quasi-ordered under minors. This extends the results of Murty and Russell [MR90] who give a constructive proof of Higman's Lemma. Our proof is based on transforming finite sequences of trees to ordinals. We begin by describing a transformation which carries trees to finite strings of numbers and give an ordering on these strings which preserves the minor ordering on the underlying trees. We show that in our well-quasi-ordering argument, these strings are actually over a finite alphabet. This allows us to conclude the result. We require the well-ordering of the ordinal e0 for our proof.
26737462
The Parallel Complexity of Tree Embedding Problems
Abstract The question of the sequential complexity of various tree embedding problems arises in the recent work of Robertson and Seymour on graph miners; here we consider the parallel complexity of such problems. In particular, we present two CREW PRAM algorithms: an O (log 3 n ) time randomized algorithm for determining whether there is a topological embedding of one rooted tree in another and an O (log 2 n log log n ) time deterministic algorithm for determining whether or not a rooted tree with a degree constraint is a minor of a general rooted tree. These algorithms are two examples of a general technique that can be used for solving other problems on trees.
17058061
Sequential and Parallel Algorithms for Embedding Problems on Classes of Partial k-Trees
We present sequential and parallel algorithms for various embedding problems on bounded degree partial k-trees and k-connected partial k-trees; these include subgraph isomorphism and topological embedding, known to be NP-complete for general partial k-trees. As well as contributing to our understanding of the types of graphs for which these problems are tractable, this paper introduces methods for solving problems on graphs. In particular, we make use of the tree-decomposition of the graph to apply techniques used to solve problems on trees to solve problems on partial k-trees.
209133
Computing planar intertwines
The proof of Wagner's conjecture by N. Robertson and P. Seymour gives a finite description of any family of graphs which is closed under the minor ordering, called the obstructions of the family. Since the intersection and the union of two minor closed graph families are again a minor closed graph family, an interesting question is that of computing the obstructions of the new family given the obstructions for the original two families. It is easy to compute the obstructions of the intersection, but, until very recently, it was an open problem to compute the obstructions of the union. It is shown that if the original families are planar, then the obstructions of the union are no larger than n to the O(n/sup 2/) power, where n is the size of the largest obstruction of the original family.<<ETX>>
204999323
Acyclic Directed Hypercubes may have Exponential Diameter
Abstract The diameter of a directed graph is the maximum of the lengths of the shortest paths between all pairs of vertices (if such paths exist). We prove that for every n , the edges of the n -dimensional hypercube can be acyclically oriented such that the diameter of the cube is F n + 1 , the ( n + 1)st Fibonacci number.
18656965
Integrating GIS and Imagery Through XML-Based Information Mediation
In this paper, we investigate a mediation-based approach to integrating information from geospatial information sources such as GIS's and databases of geo-referenced imagery. This approach is based on the use of XML to represent the information at the sources, by exporting XML Document Type Definitions (DTD's) associated with each source, and employing XML-based query languages to query integrated views of information at the mediator. Our goal is to enable users to issue a single query to the mediator, and provide the capability to return a result that seamlessly combines information from multiple sources. This paper describes issues in designing such a mediated system for geospatial data. Our notion of mediation for geospatial data is the "logical integration" of data achieved by creating correspondences between related spatial information, similar to the way mediation is done in "standard" databases. Thus, this notion of mediation does not depend on the development of spatial algorithms that operate on images or vector data and achieve "physical integration" through techniques such as image conflation. However, the mediator architecture does allow for the integration of physical integration techniques. The paper discusses issues related to wrapping of GIS and other geospatial sources and issues in querying such sources using an XML-based query language.
14459246
On optimal embeddings into incomplete hypercubes
The authors show the embeddings of various types of n-node incomplete binary trees into n-node or (n+1)-node composite hypercubes with dilation of at most 2. They also present lower bound proofs showing optimality of the dilation. They characterize the class of incomplete binary trees which are subgraphs of composite hypercubes. They present dilation 1 embedding of a two-dimensional n-node mesh, where one dimension is a power of two, into its optimal n-node composite hypercube. When neither dimension is a power of two, it is shown that a dilation 1 embedding is not possible; thereby characterizing the class of two-dimensional meshes that can be embedded into composite hypercubes with dilation 1. All two-dimensional meshes are shown to be embeddable with dilation 1 if expansion greater than 1 but less than 2 is allowed. The authors also consider two types of incomplete meshes and their embeddings into their optimal composite hypercubes.<<ETX>>
14215965
On the Covering Steiner Problem
The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems. An instance of the Covering Steiner problem consists of an undirected graph with edge-costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. When all requirements are equal to 1, this is the Group Steiner problem.
2668248
A Data Model for Querying Wavelet Features in Image Databases
Multimedia databases deal with storage and retrieval of complex descriptors of image contents, called features. Traditional techniques consider features as “black boxes,” often represented as vectors for which the only operation defined is distance computation. This “modus describendi” resulted in the widespread utilization of similarity queries, which rest solely on the computation of distances between feature vectors. The capacity to express queries more complex than simple similarity, however, rests on the possibility of describing and manipulating the structure of the features. In order to achieve this, features should be described as complex data types inside the database, and opportune operators for manipulating these data types should be defined. In this paper we try to demonstrate the efficacy and feasibility of such a program by modeling a widely used image feature: the wavelet transform. We represent features using a complex data type derived from arrays, and propose a basic algebra from which operators can be derived to manipulate and query wavelet features.
207081206
Bounding the Size of Planar Intertwines
The proof of Wagner's conjecture by Robertson and Seymour gives a finite description of any family of graphs which is closed under the minor ordering. This description is a finite set of minimal graphs not in the family; these graphs are called the obstructions of the family. Since the intersection and union of two minor closed graph families is again a minor closed graph family, an interesting question regards computing the obstructions of the new family given the obstructions for the original two families. It is easy to compute the obstructions of the intersection, but nontrivial to compute those of the union. In this paper, we show that if the original families are planar then the planar obstructions of the union are no larger than nO(n2), where n is the size of the largest obstruction of the original families.