Lower Bounds for Parallel Algebraic Decision Trees, Complexity of Convex Hulls and Related Problems
15 Dec 1994-pp 193-204
TL;DR: It is shown that any parallel algorithm in the fixed degree algebraic decision tree model that answers membership queries in W ⊑ R n using p processors, requires Ω(¦W¦/n log(p/n) rounds where ¦w¦ is the number of connected components of W.
Abstract: We show that any parallel algorithm in the fixed degree algebraic decision tree model that answers membership queries in W ⊑ R n using p processors, requires Ω(¦W¦/n log(p/n)) rounds where ¦W¦ is the number of connected components of W. We further prove a similar result for the average case complexity. We give applications of this result to various fundamental problems in computational geometry like convex-hull construction and trapezoidal decomposition and also present algorithms with matching upper bounds.
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01 May 1996
TL;DR: A fast randomized algorithm for planar hulls that runs in expected time O(log H . log log n) and does optimal O(rz log H) work where n and ET are the input and output sizes respectively and a very simple O (log nlog H) time optimal deterministic algorithm which is an improvement for small outputs is described.
Abstract: optimal sublogarithmic algorithms for small outputs Neelima Gupta and Sandeep Sen Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi 110016, India. {neelima,ssen}@cse.iitcl.ernet.in In this paper we focus on the problem of designing very fast parallel algorithms for the convex hull problem in two and three dimensions in the arbitrary CRCW model whose running times are output-size sensitive. We present a fast randomized algorithm for planar hulls that runs in expected time O(log H . log log n) and does optimal O(rz log H) work where n and ET are the input and output sizes respectively. For log II = Q(log log n), we can achieve the optimal running time of O(log II) for planar hulls while simultaneously keeping the work optimal. In three dimensions, our algorithm runs in expected time O(log log2 n logH) with optimal O(rz log H) work for all I/. Hence, for O(logO(l ) n) size outputs, our algorithms in two and three dimensions achieve poly(log log n) running time and optimal O(n log log n) work. The previously known output-sensitive workoptimal algorithms for convex hulls have running times Q (log n) (expected) and Q(log3 n) in two and three dimensions respectively. Our algorithms assume no input distribution and the running times hold with high probability. We also describe a very simple O (log n log H) time optimal deterministic algorithm for planar hulls which is an improvement for small outputs. For larger output-sizes, a running time of O(log n log log n)) can be achieved. Permission to make digitahmd copies of ail or part of thk material for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commemial advantage, the copyright notice, the title of the publication and ita date appear, and notice is given that copyright is by permission of the ACM, Jnc. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires specific permission andlor fee. Computational Geometry’96, Philadelphw PA, USA @1996 ACM 0-89791-804-5/96/05. .$3.50
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References
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06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.
1,163 citations
"Lower Bounds for Parallel Algebraic..." refers background in this paper
...A sample is 'good' if the maximum sub-problem size is less than O(nl-~logn) and the sum of the sub-problem sizes is less than gn for some constant 6. From the probabilistic bounds proved in [ 7 , 15], it is known that the first condition for a 'good' sample holds with high probability....
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TL;DR: A parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small.
Abstract: We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and $O(\log n)$ time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and $O(\log n)$ time. The constant in the running time is still moderate, though not as small.
847 citations
"Lower Bounds for Parallel Algebraic..." refers methods in this paper
...A general approach for deterministic PRAM algorithms was pioneered by Aggarwal et al. [1] and subsequently improved upon by Atallah, Cole and Goodrich [3] by extending the techniques of Cole [ 8 ]....
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01 Dec 1983
TL;DR: All the apparently known lower bounds for linear decision trees are extended to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20].
Abstract: A topological method is given for obtaining lower bounds for the height of algebraic computation trees, and algebraic decision trees. Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear lower bounds for algebraic computations. Applying the method to decision trees we extend all the apparently known lower bounds for linear decision trees to bounded degree algebraic decision trees, thus answering the open questions raised by Steele and Yao [20]. We also show how this new method can be used to establish lower bounds on the complexity of constructions with ruler and compass in plane Euclidean geometry.
584 citations
"Lower Bounds for Parallel Algebraic..." refers background in this paper
...For this, we will first prove a worst case bound along the lines of Ben-Or [ 6 ] and subsequently extend it to the average case....
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TL;DR: In this paper, the authors present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.
Abstract: We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.
311 citations
TL;DR: It is shown that any algorithm in the quadratic decision-tree model must make cn log n tests for some input.
Abstract: Given a set S of n distinct points {($x_i$,$y_i$) | 0 $\leq$ i > n}, the convex hull problem is to determine the vertices of the convex hull H(S). All the known algorithms for solving this problem have a worst-case running time of cn log n or higher, and employ only quadratic tests, i.e., tests of the form f($x_0$, $y_0$, $x_1$, $y_1$,...,$x_{n-1}$, $y_{n-1}$): 0 with f being any polynomial of degree not exceeding 2. In this paper, we show that any algorithm in the quadratic decision-tree model must make cn log n tests for some input.
169 citations
"Lower Bounds for Parallel Algebraic..." refers background in this paper
...Yao [ 18 ] had proved that in the sequential context, the identification of the convex hull vertices was no easier than sorting, but our simple argument shows that such is not the case with parallel algorithms....
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