We give an efficient algorithm for maintaining a minimum spanning forest of a plane graph subject to on-line modifications. The modifications supported include changes in the edge weights, and insertion and deletion of edges and vertices which are consistent with the given embedding. Our algorithm runs in O(log n) time per operation and O(n) space.

Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamically changing graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered. This chapter is intended as an overview of this field. In a typical dynamic graph problem one would like to answer queries on graphs that are undergoing a sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after dynamic changes, rather than having to recompute it from scratch each time. Given their powerful versatility, it is not surprising that dynamic algorithms and dynamic data structures are often more difficult to design and analyze than their static c

Traditionally, interest in parallel computation centered around the speedup provided by parallel algorithms over their sequential counterparts. In this paper, we ask a different type of question: Can parallel computers, due to their speed, do more than simply speed up the solution to a problem? We show that for real-time optimization problems, a parallel computer can obtain a solution that is better than that obtained by a sequential one. Specifically, a sequential and a parallel algorithm are exhibited for the problem of computing the best-possible approximation to the minimum-weight spanning tree of a connected, undirected and weighted graph whose vertices and edges are not all available at the outset, but instead arrive in real time. While the parallel algorithm succeeds in computing the exact minimum-weight spanning tree, the sequential algorithm can only manage to obtain an approximate solution. In the worst case, the ratio of the weight of the solution obtained seque...

We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log 2 n) work per modification. No previous dynamic geometric MST algorithm was known.

Parallel computers can do more than simply speed up sequential computations. They are capable of finding solutions that are far better in quality than those obtained by sequential computers. This fact is demonstrated by analyzing sequential and parallel solutions to numerical problems in a real-time paradigm. In this setting, numerical data required to solve a problem are received as input by a computer system, at regular intervals. The computer must process its inputs as soon as they arrive. It must also produce its outputs at regular intervals, as soon as they are available. We show that for some real-time numerical problems a parallel computer can deliver a solution that is significantly more accurate than when computed by a sequential computer. Similar results were derived recently in the areas of real-time optimization and real-time cryptography. Key words and phrases: Parallelism, real-time computation, numerical analysis. This research was supported by the Natural Sciences a...

The primary purpose of parallel computation is the fast execution of computational tasks that are too slow to perform sequentially. As a consequence, interest in parallel computation to date has naturally focused on the speedup provided by parallel algorithms over their sequential counterparts. The thesis of this paper is that a second equally important motivation for using parallel computers exists. Specifically, the following question is posed: Can parallel computers, thanks to their multiple processors, do more than simply speed up the solution to a problem? We show that within the paradigm of real-time computation, some classes of problems have the property that a solution to a problem in the class, when computed in parallel, is far superior in quality than the best one obtained on a sequential computer. What constitutes a better solution depends on the problem under consideration. Thus, `better' means `closer to optimal' for optimization problems, `more secure' for crypto...

The primary purpose of parallel computation is the fast execution of computational tasks that are too slow to perform sequentially. However, it was shown recently that a second equally important motivation for using parallel computers exists: Within the paradigm of real-time computation, some classes of problems have the property that a solution to a problem in the class computed in parallel is better than the one obtained on a sequential computer. What constitutes a better solution depends on the problem under consideration. Thus, for optimization problems, `better' means `closer to optimal'. The present paper continues this line of inquiry by exploring another class enjoying the aforementioned property, namely, cryptographic problems in a real-time setting. In this class, `better' means `more secure'. A real-time cryptographic problem is presented for which the parallel solution is significantly better than a sequential one.

This paper focuses on the improvement in the quality of computation provided by parallelism. The problem of interest is that of computing the maximum of a nonlinear feedback function in a realtime environment. We show that the solution obtained in parallel is asymptotically better than that computed sequentially. Key words and phrases: Parallelism, real-time computation, nonlinear feedback function, maximization. This research was supported by the Natural Sciences and Engineering Research Council of Canada. 1 1 Introduction The central motivation behind parallelism has always been the speeding up of sequential computations. Recently, another aspect of parallel computation was brought to light. It was shown that under some circumstances it is possible to obtain in parallel solutions to computational problems that are significantly better than any solutions computed sequentially. This phenomenon was demonstrated, in a real-time environment, for problems in combinatorial optimiz...

The problem of finding the most vital edge with respect to a minimum spanning tree of a given connected and weighted graph (with m edges and n vertices) is considered. New sequential and parallel algorithms (3 each) for the problem are proposed, and a lower bound\Omega\Gamma m) is established. We characterize the set of entering edges and show that the cardinality of this set is O(n). We show the connection between most vital edge problem and the minimum spanning tree update problems and exploit this idea in developing one of the proposed sequential algorithms. Two of our sequential algorithms are optimal. One of our parallel algorithms is optimal if the underlying graph is dense, or planar. We also consider a related problem for weighted matroids. Keywords: Data structures, design of algorithms, parallel algorithms, minimum spanning trees, most vital edge, complexity, matroids. 1 INTRODUCTION Networks are ubiquitous in many scientific and technological applications. A few examples ...

We present a general technique for dynamizing a class of problems whose underlying structure is a computation graph embedded in a tree. We associate values, called attributes, with the nodes, paths, and subtrees of our trees. Path attributes form a path attribute system, if they are maintained in constant time under path concatenation. Additionally, attributes form a tree attribute system if the tree attributes of the tail of a path \Pi are determined in constant time from the path attributes of \Pi. We also introduce a new data structure called a linear attribute grammar. An attribute grammar is a tree-based expression where the values a node are calculated from the values at the parent, siblings, and/or the children of . A linear attribute grammar, is an attribute grammar where all dependencies are linear. Our contributions can be summarized as follows. We provide a framework for maintaining attribute systems on trees in a fully dynamic environment. We show that given a ...