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Algorithmic Theory of Random graphs
, 1997
"... The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this pap ..."
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Cited by 23 (1 self)
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The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...
An Empirical Analysis of Algorithms for Constructing a Minimum Spanning Tree
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1991
"... We compare algorithms for the construction of a minimum spanning tree through largescale experimentation on randomly generated graphs of different structures and different densities. In order to extrapolate with confidence, we use graphs with up to 130,000 nodes (sparse) or 750,000 edges (dense). A ..."
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Cited by 20 (1 self)
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We compare algorithms for the construction of a minimum spanning tree through largescale experimentation on randomly generated graphs of different structures and different densities. In order to extrapolate with confidence, we use graphs with up to 130,000 nodes (sparse) or 750,000 edges (dense). Algorithms included in our experiments are Prim's algorithm (implemented with a variety of priority queues), Kruskal's algorithm (using presorting or demand sorting), Cheriton and Tarjan's algorithm, and Fredman and Tarjan 's algorithm. We also ran a large variety of tests to investigate lowlevel implementation decisions for the data structures, as well as to enable us to eliminate the effect of compilers and architectures. Within the range of sizes used, Prim's algorithm, using pairing heaps or sometimes binary heaps, is clearly preferable. While versions of Prim's algorithm using efficient implementations of Fibonacci heaps or rankrelaxed heaps often approach and (on the densest graphs) so...
Finding the k Smallest Spanning Trees
, 1992
"... We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of ..."
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Cited by 18 (2 self)
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We give improved solutions for the problem of generating the k smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes time O(m log #(m, n)+k 2 ); for planar graphs this bound can be improved to O(n + k 2 ). We also show that the k best spanning trees for a set of points in the plane can be computed in time O(min(k 2 n + n log n, k 2 + kn log(n/k))). The k best orthogonal spanning trees in the plane can be found in time O(n log n + kn log log(n/k)+k 2 ).
Checking Mergeable Priority Queues
 In Digest of the 24th Symposium on FaultTolerant Computing
, 1994
"... We present an efficient algorithm which can check the answers given by the fundamental abstract data types priority queues and mergeable priority queues. This is the first lineartime checker for mergeable priority queues. These abstract data types are widely used in routing, scheduling, simulation, ..."
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Cited by 17 (4 self)
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We present an efficient algorithm which can check the answers given by the fundamental abstract data types priority queues and mergeable priority queues. This is the first lineartime checker for mergeable priority queues. These abstract data types are widely used in routing, scheduling, simulation, computational geometry and many other algorithmic domains. We have implemented our answer checker and have performed experiments comparing the speed of our checker to recently benchmarked priority queue and mergeable priority queue implementations, and our checker is substantially faster than the best of these implementations. 1 Introduction This paper concerns the fundamental abstract data types of priority queues (PQs) and mergeable priority queues (MPQs). These abstract data types have been recognized as centrally important from the early days of computeralgorithm design. They appear in seminal algorithm texts such as Knuth's [10] and Aho, Hopcroft and Ullman's [1]. Data structure impl...
A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs
 Algorithmica
, 1992
"... We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a s ..."
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Cited by 16 (0 self)
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We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heapordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a realvalued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimumcost path from r to every...
Sorting Helps for Voronoi Diagrams
 Algorithmica
, 1995
"... It is well known that, using standard models of computation,\Omega\Gamma n log n) time is required to build a Voronoi diagram for n point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. But if the sites are sorted before we start, can the Voronoi diagram ..."
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Cited by 11 (0 self)
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It is well known that, using standard models of computation,\Omega\Gamma n log n) time is required to build a Voronoi diagram for n point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. But if the sites are sorted before we start, can the Voronoi diagram be built any faster? We show that for certain interesting, although nonstandard types of Voronoi diagrams, sorting helps. These nonstandard types of Voronoi diagrams use a convex distance function instead of the standard Euclidean distance. A convex distance function exists for any convex shape, but the distance functions based on polygons (especially triangles) lead to particularly efficient Voronoi diagram algorithms. Specifically, a Voronoi diagram using a convex distance function based on a triangle can be built in O(n log log n) time after initially sorting the n sites twice. Convex distance functions based on other polygons require more initial sorting. 1 Introduction We exam...
Clustering in Massive Data Sets
 Handbook of massive data sets
, 1999
"... We review the time and storage costs of search and clustering algorithms. We exemplify these, based on casestudies in astronomy, information retrieval, visual user interfaces, chemical databases, and other areas. Sections 2 to 6 relate to nearest neighbor searching, an elemental form of clustering, ..."
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Cited by 11 (0 self)
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We review the time and storage costs of search and clustering algorithms. We exemplify these, based on casestudies in astronomy, information retrieval, visual user interfaces, chemical databases, and other areas. Sections 2 to 6 relate to nearest neighbor searching, an elemental form of clustering, and a basis for clustering algorithms to follow. Sections 7 to 11 review a number of families of clustering algorithm. Sections 12 to 14 relate to visual or image representations of data sets, from which a number of interesting algorithmic developments arise.
A New Evolutionary Approach to the Degree Constrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 2000
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a well studied NPhard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the dMST, either using heuristic or evolutionary app ..."
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Cited by 11 (0 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a well studied NPhard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the dMST, either using heuristic or evolutionary approaches. The basis of this encoding is a spanningtree construction algorithm which we call the Randomised Primal Method (RPM), based on the wellknown Prim's algorithm [6], and an extension [4] which we call `dPrim's'. We describe a novel encoding for spanning trees, which involves using the RPM to interpret lists of potential edges to include in the growing tree. We also describe a random graph generator which produces particularly challenging dMST problems. On these and other problems, we find that an evolutionary algorithm (EA) using the RPM encoding outperforms the previous best published technique from the operations research literature, and also outperforms simulated...
Symbolic Analysis of Large Analog Integrated Circuits By Approximation During Expression Generation
, 1994
"... A novel algorithm is presented that generates approximate symbolic expressions for smallsignal characteristics of large analog integrated circuits. The method is based upon the approximation of an expression while it is being computed. The CPU time and memory requirements are reduced drastically wi ..."
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Cited by 9 (1 self)
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A novel algorithm is presented that generates approximate symbolic expressions for smallsignal characteristics of large analog integrated circuits. The method is based upon the approximation of an expression while it is being computed. The CPU time and memory requirements are reduced drastically with regard to previous approaches, as only those terms are calculated which will remain in the final expression. As a consequence, the maximum circuit size amenable to symbolic analysis has largely increased. The simplification procedure explicitly takes into account variation ranges of the symbolic parameters to avoid inaccuracies of conventional approaches which use a single value. The new approach is also able to take into account mismatches between the symbolic parameters. INTRODUCTION Symbolic circuit analysis refers to the calculation of network functions H(s,x) in the form: (1) where x T ={x 1 , x 2 , . . . x Q } is the vector of circuit parameters which remain as symbols, and the...
Parallel MarkerBased Image Segmentation with Watershed Transformation
, 1998
"... The parallel watershed transformation used in grayscale image segmentation is here augmented to perform with the aid of a priori supplied image cues called markers. The reason for introducing markers is to calibrate a resilient algorithm to oversegmentation. In a hybrid fashion, pixels are first cl ..."
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Cited by 7 (1 self)
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The parallel watershed transformation used in grayscale image segmentation is here augmented to perform with the aid of a priori supplied image cues called markers. The reason for introducing markers is to calibrate a resilient algorithm to oversegmentation. In a hybrid fashion, pixels are first clustered based on spatial proximity and graylevel homogeneity with the watershed transformation. Boundarybased region merging is then effected to condense nonmarked regions into marked catchment basins. The agglomeration strategy works with a weighted neighborhood graph representation of the oversegmented image. The throughput of a parallel Boru # vkalike minimum spanning forest (MSF) operator, applied on the considered graph, embodies the desired image partition, reasoning that all regions in a tree fuse into a homogeneous area containing a unique marker. Two figures of merit of the parallel algorithm are worth mentioning: the local detection of the catchment basins conforming to the watershed principle (which strongly depends on the history of the regions ' growth) and the parallel computation of the Boru # vkalike MSF which merges, at the same time, partial regions, produced by the local labeling, and nonmarked regions to marked basins. Both modules are designed with great concurrency, locality, and reduced software engineering cost, emerging into a scalable algorithm.