Showing papers on "Graph (abstract data type) published in 1981"

Graph decomposition: A new key to coding theorems

Abstract: A new and simple method is proposed for finding good encoders both for channels and for sources with side information. This method relies on the continuous version of a graph decomposition result of Lovasz. The presently known best exponential error bounds for both problems follow in a unified manner with an improvement on the source coding bound. The previous bounds for universal codes of the authors and Marton are also improved.

Fault Diagnosis in a Boolean n Cube Array of Microprocessors

Abstract: Fault- tolerant characteristics of a Boolean n cube array of microprocessors are analyzed. Connectivity properties of the network graph are used to show that n processor or link failures are required to isolate a processor. For processor failures the network is shown to be n (one step) diagnosable. A testing algorithm is presented which can diagnose up to n processor failures.

A complexity measure based on nesting level

Abstract: For the past several years an accepted method of determining the complexity of computer programs has involved developing a direct graph, G=(V,E) which represents the flow of control of the program. The directed graph, G consists of a set of nodes, V which represent "blocks" of groups of code, and a set E, of edges which corresponds to the flow of control among the various nodes. The graph is usually restricted to having one initial node which is always executed first. In addition, each block has two properties:

Scheduling in Sports

Abstract: The purpose of this paper is to present the problem of scheduling the games of a hockey or football league. It is shown how a graph theoretical model may be used and how some constraints related to the alternating pattern of home-games and away-games can be handled. Finally some other requirements occurring in practice are also discussed and introduced into the model.

Sample function properties of multi-parameter stable processes

Abstract: The almost sure behaviour of the sample functions of the vectorvalued N-parameter Wiener process and its stable analogues is investigated, especially their small fluctuations. In particular, laws of the simple and of the iterated logarithm are established for the supremum of the local time increments or sojourn times. These results give precise information about the minimum oscillation of the sample functions. In addition, the Hausdorff measure problem for the graph and the range of these processes is solved.

Linear Algorithms for Finding the Jordan Center and Path Center of a Tree

Abstract: This paper contains linear algorithms for finding the Jordan center, the weighted Jordan center and the path center of a tree. All of these algorithms take advantage of an efficient data structure for representing a tree called a canonical recursive representation. As a result, the first two algorithms are different from and faster than similar algorithms in the literature. This paper also defines a new parameter of a graph called the path center, and shows that the path center of a tree T consists of a unique subpath of T.

Drawing planar graphs

Abstract: Given a planar graph, we wish to draw it in the plane so that no edges cross. We might be given a particular planarisation (a specification giving the faces of the desired drawing) instead of merely the graph, but this is not required. Given a planarisation and a choice of outermost face, all drawings are in a sense equivalent; indeed, when the drawing is performed on the surface of a sphere instead of on the plane, even the choice of outermost face is irrelevant. To make the problem meaningful, we must introduce further constraints. Two general forms of constraints are: (1) absolute restrictions on the details of the drawing, that is, disallowing or requiring certain features, and (2) weighted restrictions, that is, additional objectives to be met to whatever extent is possible. We first examine the general problem, and look at various constraints that have been found to yield useful drawings in some applications. We then examine in more detail one particular set of constraints, developing a fast algorithm for producing drawings meeting those constraints, and proving some theorems relating to the overall complexity of the problem. Finally, we look at what results are known regarding other variations on the general problem.

On domination related concepts in Graph Theory

Abstract: A survey of recent results on domination and related concepts in graph theory is presented.

Dynamic Programming as Graph Searching: An Algebraic Approach

Abstract: Finding the solution of a dynamic programming problem m the form of polyadlc funcUonal equatmns is shown to be equivalent to searching a mmmaal cost path in an AND/OR graph with monotone cost functions The proof is given in an algebraic framework and is based on a commutaUvity result between solutton and mterpretauon of a symbohc system This approach Is simdar to the one used by some authors to prove the eqmvalence between the operaUonal and denotatmnal semantics of programming languages

New layouts for the shuffle-exchange graph(Extended Abstract)

Abstract: In this extended abstract, we present several new layouts for the shuffle-exchange graph, including one which requires only 0(n2/log2n) area. The optimal layout is described and analyzed in section 3. The analysis is heavily dependent on several combinatorial results which we state in section 2 and prove in the appendix. The other layouts are described in section 4. Although these layouts are not asymptotically optimal (most require 0(n2/log3/2n) area), the theory behind their development is interesting and may eventually lead to good practical layouts as well as other asymptotically optimal layouts.

Towards a Formal Treatment of VLSI Arrays

Abstract: This paper presents a formalism for describing the behavior of computational networks at the algorithmic level. It establishes a direct correspondence between the mathematical expressions defining a function and the computational networks which compute that function. By formally manipulating the symbolic expressions that define a function, it is possible to obtain different networks that compute the function. From this mathematical description of a network, one can directly determine certain important characteristics of computational networks, such as computational rate, performance and communication requirements. The use of this formalism for design and verification is demonstrated on computational networks for Finite Impulse Response (FIR) filters, matrix operations, and the Discrete Fourier Transform (DFT). The progression of computations can often be modeled by wave fronts in an illuminating way. The formalism supports this model. A computational network can be viewed in an abstract form that can be represented as a graph. The duality between the graph representation and the mathematical expressions is briefly introduced.

A priori based techniques for determining visibility priority for 3-d scenes

Abstract: Currently, simulation of real time motion of 3-D images requires expensive special purpose hardware and consequently is limited chiefly to training simulators. Only a restricted number of types of scenes can be generated, the primary limiting factor being the difficulty of solving the "hidden surface problem". Simulators provide an environment in which it has been acceptable to restrict simulated movement primarily to the trainee's viewing position. As the chief modeling technique is to approximate objects with closed polyhedra, restricting the database to a static condition geometrically allows for "offline" preprocessing to be performed which is able to capitalize on the time invariant geometric relations of the polygons comprizing the polyhedra. Such preprocessing produces a data structure which enables a faster solution to the hidden surface problem than methods which involve no preprocessing. However, methods based on preprocessing that have been developed heretofore are not automated, which greatly restricts the displayable objects. Presented is a formalization of the concepts of a priority ordering and of partitioning a collection of sets of points by a hyperplane. These form a basis from which it is possible to develop two methods which allow for a complete automation of the preprocessing and which is capable of handling any arbitrary scene. One approach involves creation of a directed graph of the relation of "potential visibility obstruction". It is known that a primary limitation to the previous approach is the presence of cycles in this graph which consequently prevent an assignment of a priority ordering. This graph is reduced to its strongly connected components, and a technique is presented for obtaining a priority ordering for a given viewing position which is based on a solution for a single simple cycle. The second approach entails the construction of a "binary space partitioning tree". This data structure is a generalization of multidimensional binary search trees (k-d trees). The method involves recursively partitioning space by the planes of the polygons comprizing the visible database and the attendent construction of a binary space partitioning tree which corresponds to this partitioning. It is then possible to obtain a priority ordering on (parts of) the polygons by an inorder-like traversal of the tree, where polygons are represented by the nodes of the tree. The order of the traversal is dependent upon the current viewing position.

Summary AN ALGORITHM FOR PARTITIONING THE NODES OF A GRAPH

Abstract: We present a heuristic algorithm for partitioning the nodes of a graph into a given number of subsets in such a way that the number of edges connecting the various subsets is a minimum. The sizes of the subsets must be specified in advance.

Some new problems and results in Graph Theory and other branches of Combinatorial Mathematics

Abstract: Recently I published several papers on finite and infinite combinatorial problems. I will try to make the overlap with this paper as small as possible ; as a result I have to omit some of my most interesting problems, but first of all I give some references to my older papers where these questions have been discussed P. Erdős, Old and new problems in combinatorial analysis and graph theory, Secondference). This paper contains extensive references to my previous papers. P. Erdős, Combinatorial problems which I would most like to see solved, will soon appear in the new Hungarian periodical Combinatorics. For applications of probabilistic methods to combinatorial analysis see our book, First I discuss problems connected with Ramsey's theorem and its generalisations, here I of course can not avoid overlap with previous papers. r(nl,. . .,nk) is the smallest integer for which if one colors the edges of K(r(nl,. . .,nk)) by k colors (K(t)

Sensitivity analysis of optimal matchings

Abstract: After stating the weighted, perfect matching problem and briefly describing Edmonds' algorithm, certain postoptimality procedures are described. These procedures aid in reoptimizing related matching problems in which a few edge weights are altered. Regardless of the actual implementation of the matching algorithm used, when changing a single edge weight, the postoptimality procedures are on the order of cardinality (N) more efficient than solving the modified problem “from scratch,” where N is the node set of the underlying graph.

Computational comparison of two methods for finding the shortest complete cycle or circuit in a graph

Abstract: — Two methodsforfinding the shortest complete cycle or circuit in a graph are compared. Thefirst method which is well knowntransforms the problem into a travelling salesman problem. Under the second approach^ the problem isformulated directly as an integer linear program and then solved by relaxing most of Us constraints. The results show the superiority of the second method.

Scaling transformation of multiple-feedback filters

Abstract: The scaling of signal levels in multiple-feedback filters is presented in a systematic way, as an operation on a cutset of the corresponding signal-flow graph. This operation leads to an interesting summed sensitivity invariant.