Graph Traversal Algorithm Research Articles

A graph-theoretic approach to efficiently reason about partially ordered events in the Event Calculus

In this paper, we show how well-known graph-theoretic techniques can be successfully exploited to efficiently reason about partially ordered events in Kowalski and Sergot's Event Calculus and in its skeptical and credulous modal variants. To overcome the computational weakness of the traditional generate-and-test algorithm of (Modal) Event Calculus, we propose two alternative graph-traversal algorithms that operate on the underlying directed acyclic graph of events representing ordering information. The first algorithm pairs breadth-first and depth-first visits of such an event graph in a suitable way, while the second one operates on its transitive closure and reduction. We prove the soundness and completeness of both algorithms, and thoroughly analyze and compare their computational complexity.

Competitive Algorithms for Layered Graph Traversal

A layered graph is a connected graph whose vertices are partitioned into sets L0=s, L1, L2,..., and whose edges, which have nonnegative integral weights, run between consecutive layers. Its width is $\max\{|L_i|\}$. In the on-line layered graph traversal problem, a searcher starts at s in a layered graph of unknown width and tries to reach a target vertex t; however, the vertices in layer i and the edges between layers i-1 and i are only revealed when the searcher reaches layer i-1. We give upper and lower bounds on the competitive ratio of layered graph traversal algorithms. We give a deterministic on-line algorithm which is O(9w)-competitive on width-w graphs and prove that for no w can a deterministic on-line algorithm have a competitive ratio better than 2w-2 on width-w graphs. We prove that for all w, w/2 is a lower bound on the competitive ratio of any randomized on-line layered graph traversal algorithm. For traversing layered graphs consisting of w disjoint paths tied together at a common source, we give a randomized on-line algorithm with a competitive ratio of O(log w) and prove that this is optimal up to a constant factor.

A novel switching architecture for ATM networks

It is known that the flexibility and capacity of asynchronous transfer mode (ATM) networks can meet the bandwidth requirements of multimedia applications. In ATM networks, switching is one of the major bottlenecks of end-to-end communication. We propose using a multiple partitionable circular bus network (MPCBN) as an ATM switch. Connection requests are first transformed into a graph where vertices and edges represent connection requests and conflicts among connection requests, respectively. We then use a graph traversal algorithm to select a maximal set of requests for execution in physically partitioned buses. An approach of using finite projective planes is then used to reduce the number of switch points from O(N/sup 2/) to O(N /spl radic/N), where N is the number of ports of a switch. A performance evaluation for both uniform and bursty data sources shows that the approach of using finite projective planes to reduce the number of switch points results in a small increase of cell loss probability.

Voronoi tessellation of points with integer coordinates: Time-efficient implementation and online edge-list generation

The Voronoi tessellation in the plane can be computed in a particularly time-efficient manner for generators with integer coordinates, such as typically acquired from a raster image. The Voronoi tessellation is constructed line by line during a single scan of the input image, simultaneously generating an edge-list data structure (DCEL) suitable for postprocessing by graph traversal algorithms. In contrast to the generic case, it can be shown that the topology of the grid permits the algorithm to run faster on complex scenes. Consequently, in Computer Vision applications, the computation of the Voronoi tessellation represents an attractive alternative to raster-based techniques in terms of both computational complexity and quality of data structures.

A fast graph traversal algorithm for the computer enumeration of P-V paths of benzenoid graphs

A computer program with O(p.n) time complexity is developed for the combinatorial enumeration of peak to valley augmenting paths of any n-vertex benzenoid graph with p number of peaks, using sets as its primary data structure. Peaks and valleys are recognized by using a breadth-first labeling procedure. The code computes the P-V matrix whose determinant is the number of Kekule structures of the given graph G. this program can handle very large graphs and takes only a few seconds to compute the P-V matrix of the typical graph even on a personal computer. As the code is very efficient and compact it can be used in any program that calculates the Kekule structure count (KSC) as an intermediate result.

Trading Space for Time in Undirected s-t Connectivity

Aleliunas et al. [20th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1979, pp. 218–223] posed the following question: “The reachability problem for undirected graphs can be solved in log space and $O(mn)$ time [m is the number of edges and n is the number of vertices] by a probabilistic algorithm that simulates a random walk, or in linear time and space by a conventional deterministic graph traversal algorithm. Is there a spectrum of time-space trade-offs between these extremes?” This question is answered in the affirmative for sparse graphs by presentation of an algorithm that is faster than the random walk by a factor essentially proportional to the size of its workspace. For denser graphs, this algorithm is faster than the random walk but the speed-up factor is smaller.

A model search procedure for hierarchial models

Large data sets cross-classified according to multiple factors are available in epidemiology and other disciplines. Their analysis often calls for finding a small set of best hierarchical models to serve as a basis for further analysis. This selection can be based on some well defined model optimality criterion. Fitting all possible models to find a best set is usually not feasible for as few as five factors (7581 possible models). We note that the set of hierarchical models and their relationships can be represented by a graph and develop an algorithm to generate it efficiently. We further develop a graph traversal algorithm that requires fitting of only fraction of all models to find exactly a best subset of the models. The algorithm classifies as many models as possible on the basis of each fit. A data structure implementing the graph of model nodes keeps track of the information required by the model search algorithm.

PCBN: a high-performance partitionable circular bus network for distributed systems

The authors present a dynamically partitionable circular bus network (PCBN) and efficient algorithms for maximizing its utilization. In their approach, a distributed network is transformed into a graph, in which a vertex represents a communication request and an edge denotes the conflict between a pair of communication requests. A graph traversal algorithm is applied to the graph to identify some maximal independent sets of vertices. The communication requests corresponding to the vertices of a maximum independent set call proceed in parallel. By computing the expected size of the maximal independent sets of a graph, the improvement ratio of the network can be obtained. The network control and synchronization techniques of PCBN are described in detail. The idling problem in the execution of nonconflicting requests is also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Three-dimensional unstructured multigrid for the Euler equations

The three dimensional Euler equations are solved on unstructured tetrahedral meshes using a multigrid strategy. The driving algorithm consists of an explicit vertex-based finite element scheme, which employs an edge-based data structure to assemble the residuals. The multigrid approach employs a sequence of independently generated coarse and fine meshes to accelerate the convergence to steady-state of the fine grid solution. Variables, residuals and corrections are passed back and forth between the various grids of the sequence using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using an efficient graph traversal algorithm. The preprocessing operation is shown to require a negligible fraction of the CPU time required by the overall solution procedure, while gains in overall solution efficiencies greater than an order of magnitude are demonstrated on meshes containing up to 350,000 vertices. Solutions using globally regenerated fine meshes as well as adaptively refined meshes are given.

Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm

An efficient dynamic graph traversal algorithm is used to identify nonconflicting requests and to allocate network resources in a dynamically partitionable bus network (DPBN). In centralized network control a special processor receives from the control computer of a partitionable bus network an adjacency matrix which indicates conflicts among requests. It applies the dynamic graph traversal algorithm and returns the identified nonconflicting requests to the control computer. The control computer then physically partitions the network into a number of subnetworks for processing the nonconflicting requests in parallel. In distributed control, each station determines conflicts and sets the switches. The results of performance evaluation show a 40% decrease of network delay as compared with a fully utilized, but unpartitioned local area network.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Filling regions in binary raster images

Filling regions in raster images is the term given to the problem of extracting a connected region (that contains some preselected seed pixel) and filling it with some color . A connected region is formally defined as the collection of all pixels that are in the transitive closure of a pixel-connectivity operator that is applied to the seed. For example, the 4-pixel-connectivity operator selects all pixels that are (spatially) 4-connected to the pixel operand, and have the same color. This problem can be solved relatively easily if the fill color is distinguishable so that every pixel, once colored (i.e. flagged OLD) will not be considered again. In binary images, the fill “color” is usually a binary pattern and therefore, a “colored” pixel may still have its previous (e.g., black or white) color. This fact makes the problem non-trivial. In this paper, a region to be filled is represented as a connected directed a-cyclic planar graph in which nodes are regular regions (defined below) that are easy to handle. An arc connects a regular region to its neighbor below which shares a common horizontal sub-boundary. Based on this abstract representation of a region, a formulation of the filling problem is developed as a variant of graph traversing. The difficulties imposed by filling with a binary pattern, and the avoidance of an explicit description of the graph are explored and a solution is presented. This solution utilizes the frame buffer (that is used to display the image) for improved efficiency of a graph traversal algorithm. This method turns out to be similar to [Lieberman-78], that is shown here to be incorrect. The complexity of the new algorithm is 0(N*L+ N*Log N), compared to 0(N*L*Log N) there, where N is the number of nodes in the graph and L is the average composite degree of each node. A proof of correctness for the new algorithm is given too.