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Scans as Primitive Parallel Operations
 IEEE Transactions on Computers
, 1987
"... In most parallel randomaccess machine (PRAM) models, memory references are assumed to take unit time. In practice, and in theory, certain scan operations, also known as prefix computations, can executed in no more time than these parallel memory references. This paper outline an extensive study of ..."
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Cited by 157 (12 self)
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In most parallel randomaccess machine (PRAM) models, memory references are assumed to take unit time. In practice, and in theory, certain scan operations, also known as prefix computations, can executed in no more time than these parallel memory references. This paper outline an extensive study of the effect of including in the PRAM models, such scan operations as unittime primitives. The study concludes that the primitives improve the asymptotic running time of many algorithms by an O(lg n) factor, greatly simplify the description of many algorithms, and are significantly easier to implement than memory references. We therefore argue that the algorithm designer should feel free to use these operations as if they were as cheap as a memory reference. This paper describes five algorithms that clearly illustrate how the scan primitives can be used in algorithm design: a radixsort algorithm, a quicksort algorithm, a minimumspanning tree algorithm, a linedrawing algorithm and a mergi...
Faster Algorithms for Some Geometric Graph Problems in Higher Dimensions
, 1993
"... We show how to apply the wellseparated pair decomposition of a pointset P in ! d to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + ffl times the exa ..."
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Cited by 59 (2 self)
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We show how to apply the wellseparated pair decomposition of a pointset P in ! d to significantly improve known time bounds on several geometric graph problems. We first present an algorithm to find an approximate Euclidean minimum spanning tree of P whose weight is at most 1 + ffl times the exact minimum. We achieve a time complexity of O(n log n + (ffl \Gammad=2 log 1 ffl )n), improving the best known bound of O(ffl \Gammad n log n). We then show how to construct a graph with O(ffl \Gammad+1 n) edges in which the shortest path between any pair of points is within 1 + ffl of the Euclidean distance. Our time complexity is O(n log n+(ffl \Gammad log 1 ffl )n), a significant improvement over the best previous algorithm that produces a graph of this size. Finally, we show how to compute the exact Euclidean minimum spanning tree in time O(T d (n; n) log n), where T d (m; n) is the time to find the bichromatic closest pair between m red points and n blue points. The previo...
Subquadratic Approximation Algorithms For Clustering Problems in High Dimensional Spaces
"... One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and ev ..."
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Cited by 18 (1 self)
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One of the central problems in information retrieval, data mining, computational biology, statistical analysis, computer vision, geographic analysis, pattern recognition, distributed protocols is the question of classification of data according to some clustering rule. Often the data is noisy and even approximate classification is of extreme importance. The difficulty of such classification stems from the fact that usually the data has many incomparable attributes, and often results in the question of clustering problems in high dimensional spaces. Since they require measuring distance between every pair of data points, standard algorithms for computing the exact clustering solutions use quadratic or "nearly quadratic" running time; i.e., O(dn 2\Gammaff(d) ) time where n is the number of data points, d is the dimen Computer Science Department, University of Toronto. Part of this work was done while visiting Bell Communications Research. y Bell Communications Research, MCC1C365...
Reconstructing a Minimum Spanning Tree after Deletion of Any Node
 Algorithmica
, 1999
"... Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper, we consider single node deletions in MSTs. Let G = (V; E) be an undirected graph with n nodes and m edges, and let T be the MST of G. For each node v in V , the node replacement for v is the minim ..."
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Cited by 6 (0 self)
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Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper, we consider single node deletions in MSTs. Let G = (V; E) be an undirected graph with n nodes and m edges, and let T be the MST of G. For each node v in V , the node replacement for v is the minimum weight set of edges R(v) that connect the components of T \Gamma v. We present a sequential algorithm and a parallel algorithm that find R(v) for all V simultaneously. The sequential algorithm takes O(m log n) time, but only O(mff(m; n)) time when the edges of E are presorted by weight. The parallel algorithm takes O(log 2 n) time using m processors on a CREW PRAM. 2 1 INTRODUCTION For communication networks, minimum spanning trees (MSTs) are used for basic network tasks such as broadcast, leader election, and synchronization. Updating the MST after changes in network topology is a fundamental problem. In this paper, we update MSTs after single node deletions. In a graph G with ...