K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, Englewood Cliffs, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....query to be a pair of points p; q 2 S. The answer to a path query is a t spanner path from p to q, that is, a path whose length is at most t times the Euclidean distance between p and q. Intuitively, our results may be viewed as one way of generalizing skip lists to higher dimensions (also see [8]) Assume that the points of S are one dimensional. Consider a skip list [10] for the points of S. We can regard this data structure as a directed graph. This graph has an expected number of O(n) edges. For each pair p and q of points, there is a path from p to q having length jp Gamma qj and ....

....graph. This graph has an expected number of O(n) edges. For each pair p and q of points, there is a path from p to q having length jp Gamma qj and containing an expected number of O(log n) edges. In fact, even the expected maximum number of edges on any such path is bounded by O(logn) See [8]. As a result, the skip list is a 1 spanner with expected spanner diameter O(log n) This spanner can be maintained in O(logn) expected time per insertion and deletion. In the first part of this paper, we generalize this idea to the d dimensional case, for any fixed d, by combining the ....

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K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, Englewood Cliffs, 1994.

....a treap where the timestamps are fully ordered and the priorities are heap ordered. Therefore, by the argument in [1] the expected length of this list when there are n active elements is H(n) the nth harmonic number. Furthermore, an application of the Cherno bound on the harmonic numbers (see [5]) demonstrates that the probability that the length of the list will exceed 2c ln n 1 when there are n active elements is less than 2(n=e) c ln(c=e) Thus O(k log n) is both the expected memory usage of priority sample and also a high probability upper bound on the memory usage. 4 ....

K. Mulmuley. An Introduction through Randomized Algorithms. Prentice Hall, 1993.

....1 Introduction Randomized incremental algorithms have become an increasingly popular method for constructing geometric structures such as convex hulls and arrangements. Such algorithms can also be used to maintain structures for dynamic input, in which points are inserted one at a time. Mulmuley [30, 31, 32] and Schwarzkopf [38] generalized this expected case model to fully dynamic geometric algorithms, in which deletions as well as insertions are allowed, and showed that many randomized incremental algorithms can be extended to this fully dynamic model. The resulting model makes only weak ....

....to be practical. Previous studies of average case updates in this model have focused on problems of computing geometric structures: convex hulls, arrangements, and the like. However problems of geometric optimization have been neglected; indeed Mulmuley s text on randomized geometric algorithms [32] does not even mention such basic optimization problems as minimum spanning trees or diameter. In this paper we show that the same model of average case analysis can be used to solve a number of problems of dynamic geometric optimization. We also address some fundamental data structure issues ....

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K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, 1993.

....maintaining bichromatic closest pairs [2, 17] If only insertions are allowed, or if the update sequence is known in advance [16] the minimum spanning tree can instead be maintained in time O(log 2 n) per update. Even more e#ciently, for fully dynamic updates in Mulmuley s expected case model [25, 26, 27, 32], the minimum spanning tree can be maintained in O(log n) expected time per update by using a dynamic Delaunay triangulation algorithm [13, 25, 32] together with a dynamic planar graph minimum spanning tree algorithm [15] In this paper we provide the first dynamic algorithm for the apparently ....

....will be removed and each point will be connected to the newly added point. Hence there would seem to be no hope of an e#cient dynamic maximum spanning tree algorithm, even for insertions only. To avoid this di#culty, we resort to an expected case analysis, in 1 a model popularized by Mulmuley [25, 26, 27] for which the overall point set may be chosen to be a worst case but for which the order of insertions and deletions is randomized. In this model, the maximum spanning tree (like many other geometric graphs) changes by O(1) edges per update. As part of our maximum spanning tree algorithm, we ....

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K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, 1993.

....1 Introduction Randomized incremental algorithms have become an increasingly popular method for constructing geometric structures such as convex hulls and arrangements. Such algorithms can also be used to maintain structures for dynamic input, in which points are inserted one at a time. Mulmuley [22, 23, 24] and Schwarzkopf [29] generalized this expected case model to fully dynamic geometric algorithms, in which deletions as well as insertions are allowed, and showed that many randomized incremental algorithms can be extended to this fully dynamic model. The resulting model makes only weak ....

....practical. Previous studies of average case updates in this model have focused on problems of computing geometric structures: convex hulls, arrangements, and the like. However problems of geometric optimization have been neglected; indeed Mulmuley s recent text on randomized geometric algorithms [24] does not even mention such basic optimization problems as minimum spanning trees or diameter. In this paper we show that the same model of average case analysis can be used to solve a number of problems of dynamic geometric optimization. We also address some fundamental data structure issues ....

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K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, 1993.

....on the structure and representation of B. For example B can be given as an implicit equation describing its surface, in which case numerical root finders can be used. Often B is a polyhedron, and finding its intersection with a line is a widely studied topic in computational geometry [PS85, Mul94] After finding a new formula for the potential VB (P ) in a point, we will address the problem of computing the electrostatic potential energy of a system of homogeneusly charged 1 bodies. The classical expression for this quantity is U(B 1 ; B 2 ) Z ae 1 ae 2 jQ 1 Gamma Q 2 j dQ 1 dQ ....

Ketan Mulmuley. Computational Geometry. An Introduction Through Randomized Algorithms. Prentice Hall, 1994. 18

....using the Endpoint Dominance Problem The O(sort(N) solution to EPD has several almost immediate consequences as discussed in [GIS] As already mentioned it leads to an O(sort(N) algorithm for segment sorting. Similarly it leads to an algorithm for trapezoid decomposition of a set of segments [92, 108] with the same I O bound, as the core of this problem precisely is to find for each segment endpoint the segment immediately above it. The ability to compute a trapezoid decomposition of a simple polygon also leads to an O(sort(N) polygon triangulation algorithm using a slightly modified version ....

K. Mulmuley. Computational Geometry. An introduction through randomized algorithms. Prentice-Hall, 1994.

....(17) By running some of the earlier developed procedures twice simultaneously for the real part p e (q) and for the imaginary part p o (q) we obtain a set of points in the complex plane. Then we compute their convex hull what can be done in optimal time using O( log ) operations, e.g. [24, 26], where denotes the number of points, and check whether the origin of the complex plane is contained in this convex hull. If it is outside and if there exists a stable member, the family of polynomials is robustly stable. Otherwise (if the origin is inside the convex hull) we perform an inclusion ....

....the explicit reference to D and write ConvB for Convfb I : I 2 Sg. In [27] it is shown that the convex hull property also holds for univariate polynomials having complex coefficients. By (10) and (11) it is easy to see that Convfp(x) x 2 Dg ConvB: By any standard convex hull algorithm, cf. [24, 26], we can check whether the origin belongs to ConvB. If it is inside we apply the sweep procedure to get a better approximation of the value set. An approach based merely on the sweep procedure and a convex hull algorithm is only sufficient to verify robust stability. However, to show that a family ....

K. Mulmuley, Computational Geometry, An Introduction through Randomized Algorithms. EnglewoodCliffs: Prentice Hall, 1994.

.... and polytope duality [81, 63] and mirror symmetry [39] algebraic combinatorics a la Stanley [Stanley] 44, 82, 83, 84, 85] Various proofs for the UBT: for Eulerian complexes with many vertices [65] for polytopes using shellability [73] a simple dual form using linear objective functions, [187], for spheres using the Cohen Macaulay property [79] using shellability and the Cohen Macaulay property [64] using shellability and a strong form of an extremal theorem of Bollob as [36] for manifolds, using relations between face numbers and Betti numbers of Buchsbaum rings [74] a strong form ....

....type algorithm (even randomized) will already be a major achievement; see [173] for a special case. 10.6 Some links and references Linear programming [190] average case behavior of the simplex method [171] randomized pivot rules [172, 173, 189, 175, 180, 188, 185] computational geometry [187] algorithmic applications of random walks [183, 186] the diameter problem for polyhedra [179] 42 11 Applications and Expectations Judging from the conference on Vision in Mathematics , mathematicians have a strong desire to interact and in uence other sciences, as well as technology, ....

K. Mulmuley, Computational Geometry, An Introduction Through Randomized Algorithms, Prentice-Hall, Englewoods Clis, 1994.

....in the face F . These considerations have far reaching applications on the understanding of the combinatorial structure of simple polytopes. We refer the reader to Ziegler s book [32] for historical notes and for references to the original papers. Our presentation is also quite close to that in [26]. We hope that the theory of h numbers described below will reflect back on linear programming but this is left to be seen. Degrees and h numbers Let P be a simple d polytope and let OE be a generic linear objective function. For a vertex v of P define the degree of v denoted by deg(v) to be the ....

....equalities among face numbers of simple d polytopes. The cyclic polytopes The cyclic d polytope with n vertices C(d; n) is the convex hull of n distinct point on the moment curve x(t) t; t 2 ; t d ) ae R d . This is a remarkable class of polytopes and the reader should consult [10, 26, 32] for their properties. C (d; n) will denote a polar polytope to C(d; n) For the definition of polarity see [10, 26, 32] C (d; n) is a simple d polytope with n facets. The upper bound theorem Motzkin conjectured that the maximal number of vertices (and more generally of k dimensional faces) ....

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K. Mulmuley, Computational Geometry, An Introduction Through Randomized Algorithms, Prentice-Hall, Englewoods Cliffs, 1994.

....factor of the graph. Ruppert and Seidel [71] improved this trade off using some stronger definition of the graph and they generalized it for higher dimensions. Arya et al.: 8] studied the problem of maintaining the spanner property of a graph under a sequence of random insertion and deletions [64] of points of the original point set. They obtained a polylogarithmic update time using a graph based data structure. 2.1 The two dimensional graph In this section we present the definition and the proof of the spanner property of the graph done by Ruppert and Seidel [71] Then we study ....

K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, Englewood Cliffs, 1994.

....in (x; y; z) then also oe is a polynomial in (x 0 ; y 0 ; z 0 ) thus, it is possible to compute symbolically oe, oe and R oe 2 . 3. Compute the partition of the plane induced by the projected edges; to this purpose a variety of algorithms exist in computational geometry literature [36, 37]. The work required at this step is O( n k) log n) where n is the number of edges and k the number of intersections between projected edges, using a method of Bentley and Ottman [38] 4. The global integral K(u) can be obtained by summing the quantities K f (u) R s2f wB (s; u) ds, where f ....

Ketan Mulmuley. Computational Geometry. An Introduction Through Randomized Algorithms. Prentice Hall, 1994.

....contains the origin. By expanding p e (x) and p o (x) simultaneously into their Bernstein forms, we obtain a set of points ( b (e) I (U) b (o) I (U) in the plane, denoted by b I (U) Then we compute its convex hull, which can be done in optimal time using O( log ) operations, see, e.g. [27], where denotes the number of points. Then we check whether the origin of the plane is contained in the convex hull, since Conv P(U) Conv B(U) holds true. If the origin is outside, the family of polynomials is robustly stable. Otherwise an inclusion test given in [21] is performed. If it fails, ....

K. Mulmuley, Computational Geometry, An Introduction through Randomized Algorithms, Englewood-Cliffs: Prentice Hall,1994.

.... g(x) x 2 Ug contains the origin. By expanding h(x) and g(x) simultaneously into their Bernstein forms we obtain a set of points (b I ( h; U) b I ( g; U) in the plane, denoted by b I (U) Then we compute their convex hull which can be done in optimal time using O( log ) operations, e.g.[19], 20] where denotes the number of points. As in [17] it can be shown that Conv P(U) ConvB(U) holds true. By any standard convex hull algorithm, cf. 19] 20] we can check whether the origin belongs to Conv B(U) If it is outside, the family of polynomials (1) is robustly stable. Otherwise an ....

....( g; U) in the plane, denoted by b I (U) Then we compute their convex hull which can be done in optimal time using O( log ) operations, e.g. 19] 20] where denotes the number of points. As in [17] it can be shown that Conv P(U) ConvB(U) holds true. By any standard convex hull algorithm, cf. [19], 20] we can check whether the origin belongs to Conv B(U) If it is outside, the family of polynomials (1) is robustly stable. Otherwise an inclusion test given in [1] is performed. If it fails, i.e. it can not be verified that the origin is in the set P(U ) the sweep procedure is applied ....

K. Mulmuley, Computational Geometry, An Introduction through Randomized Algorithms, Englewood-Cliffs: Prentice Hall,1994.

....only occur at certain prespecified times the algorithms are not fully dynamic. A number of other papers have considered dynamic computational geometry problems under an average case model that assumes that among a given set of points each point is equally likely to be inserted or deleted next [10, 19, 20, 21, 24]. However we are interested here in worst case bounds. We are particularly interested in the dynamic geometric MST problem. If only insertions are allowed, it is not hard to maintain the MST in time O(log 2 n) per update. The same bound has recently been achieved for offline updates consisting ....

K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, 1993.

....using the Endpoint Dominance Problem The O(sort(N) solution to EPD has several almost immediate consequences as discussed in [GIS] As already mentioned it leads to an O(sort(N) algorithm for segment sorting. Similarly it leads to an algorithm for trapezoid decomposition of a set of segments [92, 108] with the same I O bound, as the core of this problem precisely is to find for each segment endpoint the segment immediately above it. The ability to compute a trapezoid decomposition of a simple polygon also leads to an O(sort(N) polygon triangulation algorithm using a slightly modified version ....

K. Mulmuley. Computational Geometry. An introduction through randomized algorithms. Prentice-Hall, 1994.

.... [10] An introduction to probability theory can be found in the book by Cormen, Leiserson and Rivest [2] The standard book on this topic is Feller [3] Comprehensive overviews of randomized algorithms and data structures can be found in the books by Motwani and Raghavan [5] and by Mulmuley [6]. ....

K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, Englewood Clis, 1994.

....and tools from the area of computational geometry that we will be using throughout the report. For the remainder of this section we consider the vector space to be R d , and the distance metric to be the Euclidean norm L 2 . The presentation in this section follows closely the presentation in [70]. 3.1 Hyperplanes and half spaces We define a hyperplane h in R d as the set of points in R d that satisfy a linear equality of the form a 1 x 1 a 2 x 2 : a d x d = a 0 , where x j s denote the coordinates in R d , the coefficients a j s are arbitrary real numbers, and a 1 ; ....

....space, and with O(n log n) preprocessing time, we have a data structure with O(log n) search time. Dobkin and Lipton [33] proposed the following natural generalization of the binary search to two dimensions. Given the point set P , create the Voronoi diagram of P . Using the plane sweep algorithm [70] this takes O(n log n) time. Then sort all the vertices of the Voronoi diagram according to their x coordinate. Any two consecutive x coordinates define a slab of the two dimensional space. This slab is intersected by a set of edges of the Voronoi diagram which do not intersect each other within ....

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K. Mulmuley. Computational Geometry. An Introduction Through Randomized Algorithms. Prentice Hall, 1994.

....finish with some remarks on the practical realization of the randomized algorithm and its deterministic counterpart, after having derandomized it. 1 Introduction Until now randomized algorithms were almost exclusively used for problems from computational geometry, see e.g. 2] 7] 10] 11] or [12]. But they also prove valuable for the classical linear or quadratic programming problems, as was observed by [1] 4] 6] 9] and [14] The following paper gives an example of such a randomized algorithm from [4] and its derandomization in [3] both put into a rather general framework. The ....

K.Mulmuley: Computational Geometry, An Introduction through Randomized Algorithms. Prentice Hall, London/Sydney/Toronto u.a., 1994.

....to combine the Theta graph with skip lists [100] to get a t spanner having O(n) edges and O(log n) spanner diameter, both with high probability. They also show how to maintain this spanner efficiently under insertions and deletions of points, in the model of random updates, as defined in Mulmuley [95]. Although the out degree of any vertex in the Theta graph is bounded by a constant, the maximum in degree can be as large as n Gamma 1: Take n Gamma 1 points on a circle, and let the n th point be its center. In the corresponding Theta graph, each point on the circle has an edge towards the ....

K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, Englewood Cliffs, 1994.

....for at most two INSERT and at most two SPLIT operations, each taking O(log n) time. The update of D with P takes O(jP j) time. Since P jP j over the execution of the algorithm is O(n) the total time required by step 1 is O(n log n) We recall the following well known fact (e.g. see [12], page 77) Lemma 3 For any planar graph with n vertices, one can build a point location data structure of O(n) size in O(n log n) time, guaranteeing O(log n) query time. The point location is performed for b n points, so the total time of step 2 is also O(n log n) The complexity of the last ....

K. Mulmuley, Computational Geometry { An Introduction through Randomized Algorithms, Prentice Hall, Englewood Clis, 1994.

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K. Mulmuley. Computational Geometry, an Introduction through Randomized Algorithms. Prentice Hall, Englewood Cliffs, 1994.

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K. Mulmuley, Computational Geometry, An Introduction Through Randomized Algorithms, Prentice Hall, 1994.

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K. Mulmuley. Computational Geometry. An Introduction Through Randomized Algorithms. Prentice Hall, 1994.

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Ketan Mulmuley. Computational Geometry. An Introduction Through Randomized Algorithms. Prentice Hall, 1994.

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