A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system Gamma, there is a sphere S that intersects at most O(k(1/d)n(1...
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A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system Gamma, there is a sphere S that intersects at most O(k(1/d)n(1-1/d)) balls of Gamma and divides the remainder of Gamma into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1-1/(d+2))n balls. This bound of O(k(1/d)n(1-1/d)) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every k-nearest neighbor graph's of n points in d dimensions has a separator of size O (k(1/d)n(1-1/d)). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.
Consider the class of discrete time, general state space Markov chains which satisfy a ''uniform ergodicity under sampling'' condition. There are many ways to quantify the notion of ''mixing ti...
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Consider the class of discrete time, general state space Markov chains which satisfy a ''uniform ergodicity under sampling'' condition. There are many ways to quantify the notion of ''mixing time'', i.e., time to approach stationarity from a worst initial state. We prove results asserting equivalence (up to universal constants) of different quantifications of mixing time. This work combines three areas of Markov theory which are rarely connected: the potential-theoretical characterization of optimal stopping times, the theory of stability and convergence to stationarity for general-state chains, and the theory surrounding mixing times for finite-state chains. (C) 1997 Elsevier Science B.V.
Harmonic update is a randomized on-line algorithm which, given a random rn-set of vertices U(m) subset of or equal to {-1, 1}(n) in the n-dimensional cube, generates a random vertex w is an element of {-1, 1}(n) as a ...
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Harmonic update is a randomized on-line algorithm which, given a random rn-set of vertices U(m) subset of or equal to {-1, 1}(n) in the n-dimensional cube, generates a random vertex w is an element of {-1, 1}(n) as a putative solution to the system of linear inequalities: Sigma(i=1)(n) w(i)u(i) greater than or equal to 0 for each u is an element of U(m). Using tools from large deviation multivariate normal approximation and Poisson approximation, we show that root n/root log n is a threshold function for the property that the vertex w generated by harmonic update has positive inner product with each vertex in U(m). More explicitly, let P(n, rn) denote the probability that Sigma(i=1)(n) w(i)u(i) greater than or equal to 0 for each u is an element of U(m). Then, as n --> infinity, P(n, m) --> 0 or 1 according to whether m = m(n) varies with n such that >> root n/root log n or m << root n/root log n, respectively. The analysis also exposes the fine structure of the threshold function.
For a connected, undirected and weighted graph G = (V, E), the problem of finding the k most vital edges of G with respect to minimum spanning tree is to find k edges in G whose removal will cause greatest weight incr...
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ISBN:
(纸本)0780337263
For a connected, undirected and weighted graph G = (V, E), the problem of finding the k most vital edges of G with respect to minimum spanning tree is to find k edges in G whose removal will cause greatest weight increase in the minimum spanning tree of the remaining graph. This problem is known to be NP-hard for arbitrary k. In this paper, we first describe a simple exact algorithm for this problem, based on the approach of edge replacement in the minimum spanning tree of G. Next we present polynomial-time randomized algorithms that produce optimal and approximate solutions to this problem. For /V/ = n and /E/ = m, our algorithm producing optimal solution has a time complexity of O(mn) with probability of success at least E-k2/2(m-n-1 - 2 log(c) k/k = 4), c = 1 + 1/2(k/2), and the algorithm producing approximate solution runs in time O(mn + nk(2) log k) and yields results within factor 2 to the optimal one. Finally we show that both of our randomized algorithms can be easily parallelized. On a CREW PRAM the first algorithm runs in O(n) time using n(2) processors, and the second algorithm runs in O(log(2) n) time using mn/log n processors.
We introduce two probabilistic algorithms to determine the motion parameters of a planar shape without knowing a priori the point-to-point correspondences. If the target is limited to rigid objects, an Euclidean trans...
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We introduce two probabilistic algorithms to determine the motion parameters of a planar shape without knowing a priori the point-to-point correspondences. If the target is limited to rigid objects, an Euclidean transformation can be expressed as a linear equation with six parameters, i.e. two translational parameters and four rotational parameters (the axis of rotation and the rotational speed about the axis). These parameters can be determined by applying the randomized Hough transform. One remarkable feature of our algorithms is that the calculations of the translation and rotation parameters are performed by using points randomly selected from two image frames that are acquired at different times. The estimation of rotation parameters is done using one of two approaches, which we call the triangle search and the polygon search algorithms respectively. Both methods focus on the intersection points of a boundary of the 2D shape and the circles whose centers are located at the shape's centroid and whose radii are generated randomly. The triangle search algorithm randomly selects three different intersection points in each image, such that they form congruent triangles, and then estimates the rotation parameter using these two triangles. However, the polygon search algorithm employs all the intersection points in each image, i.e. all the intersection points in the two image frames form two polygons, and then estimates the rotation parameter with aid of the vertices of these two polygons.
In this paper, we propose a randomized algorithm to estimate the motion parameters of a planar shape without knowing a priori the point-to-point correspondences. By randomly searching points on two shapes measured at ...
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In this paper, we propose a randomized algorithm to estimate the motion parameters of a planar shape without knowing a priori the point-to-point correspondences. By randomly searching points on two shapes measured at different times, we determine the centroids, after which the algorithm proceeds to determine the rotation by randomly searching points on each shape that form congruent polygons.
This note is a report of testing a straightforward generalization of the randomized 3-coloring algorithm of Petford and Welsh (1989) on the decision problems of 4- and 10-coloring. We observe similar behavior, namely ...
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This note is a report of testing a straightforward generalization of the randomized 3-coloring algorithm of Petford and Welsh (1989) on the decision problems of 4- and 10-coloring. We observe similar behavior, namely the existence of critical regions. Experimentally, the average time complexity for large n again seems to grow slowly, although in some cases the number of transitions needed is prohibitively high for practical applications.
This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph's minimum cut form an extremely small fraction of the graph's edges. U...
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This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph's minimum cut form an extremely small fraction of the graph's edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n(2)log(3)n) time, a significant improvement over the previous (O) over tilde(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in RNC with n(2) processors;this gives the first proof that the minimum cut problem can be solved in RNC. The algorithm does more than find a single minimum cut;it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of cu of the minimum cut's in expected (O) over tilde(n(2 alpha)) time, or in RNC with n(2 alpha) processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected (O) over tilde(n(2(r-1))) time, or in RNC with n(2(r-1)) processors. The ''trace'' of the algorithm's execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the more standard cactus representation for minimum cuts.
The so-called Maximum Clique Problem is one of the most famous NP-complete problems for which it is difficult to find a solution. Given an indirected graph, we present here a polynomial-time randomized algorithm RaCLI...
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The so-called Maximum Clique Problem is one of the most famous NP-complete problems for which it is difficult to find a solution. Given an indirected graph, we present here a polynomial-time randomized algorithm RaCLIQUE for finding a near-maximum clique. While the basic idea of the algorithm comes from Boltzmann machines, it employs no simulated annealing at all and hence it is simple to control its execution. We have confirmed in experiments for several random and nonrandom graphs with up to 400 nodes that very good solutions can be found efficiently compared with the other conventional algorithms.
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