M. T. Goodrich and E. A. Ramos. Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom., 18:397--420, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and Megiddo [31] gave a randomized algorithm under the CRCW model of computation that runs, with high probability, in O(1) time using O(n) processors. Ajtai and Megiddo [30] gave an O( log log n) time deterministic algorithm using O(n) processors under Valiant s model of computation. Goodrich [138] and Sen [250] gave an O( log log n) time, O(n) work algorithm under the CRCW model; see also [102] 5 Randomized Algorithms for Linear Programming Random sampling has become one of the most powerful and versatile techniques in computational geometry, so it is no surprise that this ....

M. T. Goodrich and E. A. Ramos, Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming, Discrete Comput. Geom., 18 (1997). 397--420.

....induced by (the sorted order of) R is less than n=r. Intuitively, 1 gives even splitting but we will relax that significantly to speed up our algorithm. For approximate splitting, on CRCW model, we use the following result implied by the work of Hagerup and Raman[22] and Goldberg and Zwick[20] (see appendix for details) Lemma 1 For all given integers n 4, and C r n 1=4 , approximate splitting can be done with expansion factor p r in O(log log 2 n) time and O(r n) CRCW processors where C is a sufficiently large constant. For smaller r, we partition the a i2 s into two ....

....expect, the running time of fast parallel algorithms for this problem increases as decreases. Hagerup described an O(log log 3 n) time work optimal CRCW algorithm for the above problem for = 1=poly(log log n) The following improved result result was recently obtained by Goldberg and Zwick[20]. Lemma 3 The approximate compaction problem can be solved on a CRCW PRAM in time t log log n, using n=t processors, with a padding factor of 1=poly(log log n) Suppose we substitute exact compaction by approximate compaction. Then, over the O(d) levels of recursion, there will only be a ....

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M. Goodrich and E. Ramos. Bounded-Independence Derandomization of Geometric Partitioning with Applications to Parallel Fixed-Dimensional Linear Programming, Discrete and Computational Geometry, Vol 18, 1997, 397 -- 420.

....In fact it was even conjectured that the MIS problem does not belong to NC [17] The rst NC algorithm for the MIS problem was presented in [9] but especially Luby s algorithm from [13] received a lot of attention. It was the rst example of the so called derandomization technique [6,2] see [1,3,15,5] for further applications. Roughly speaking the derandomization technique is based on the transformation of a randomized NC algorithm (which is easier to design) into a deterministic NC algorithm by simulating the randomized algorithm in parallel for several possible outcomes of its random ....

M. T. Goodrich and E. A. Ramos. Bounded-independence derandomization of geometric partitioning with applications to parallel xed-dimensional linear programming. GEOMETRY: Discrete & Computational Geometry, 18, 1997.

....Megiddo [31] gave a randomized algorithm under the CRCW model of computation that runs, with high probability, in O(1) time using O(n) processors. Ajtai and Megiddo [30] gave an O( log log n) d ) time deterministic algorithm using O(n) processors under Valiant s model of computation. Goodrich [139] and Sen [252] gave an O( log log n) d 2 ) time, O(n) work algorithm under the CRCW model; see also [103] 5 Randomized Algorithms for Linear Programming Random sampling has become one of the most powerful and versatile techniques in computational geometry, so it is no surprise that this ....

M. T. Goodrich and E. A. Ramos, Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming, Discrete Comput. Geom., 18 (1997). 397--420.

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M. T. Goodrich and E. A. Ramos. Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom., 18:397--420, 1997.

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M. T. Goodrich and E. A. Ramos. Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom., 18:397--420, 1997.

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M. T. Goodrich and E. A. Ramos. Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom., 18:397--420, 1997.

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M. T. Goodrich and E. A. Ramos. Bounded-independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom., 18:397--420, 1997.

.... for the single face problem and leave the other results for a companion paper [9] Finally, we discuss implementations in the CRCW PRAM model, where faster algorithms are possible through the use of approximate counting [5, 29] To take advantage of this, the concept of an approximation is relaxed [35, 33, 34]. We expect that the approach will nd further applications. Contents. The paper consists of two main parts. The rst part consists of Sections 2 to 5, and it deals with the theory and tools from random and deterministic gemetric sampling. The chapters deal respectively with: sampling in con ....

....depends on k. As an implication, in the corollary, deviations can be as high as r instead of logarithmic. This has no e ect as far as the applications where it is sucient to choose c appropriately large. 3.3 Range Spaces A result analogous to Lemma 2. 4 under k wise independence is proved in [34]. However, the guaranteed size is much worst, O(r 2 n ) with = O(1=k) with probabilty at least 1=2. We show here that for the induced range space hH d ; S d 0 i, the size is O(r 2 ) in [34] this size can also be achieved but after further processing while, in contrast, here it is ....

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M. T. Goodrich and E. A. Ramos. Bounded independence derandomization of geometric partitioning with applications to parallel xed-dimensional linear programming. Discrete Comput. Geom. 18 (1997), 397-420.

....25, 7] However, so far these efforts to compute a sample in parallel have resulted only in O( p jSj 1 ffl log m) discrepancies. The situation is similar for other discrepancy like problems, like the lattice approximation problem [27, 23] and some sampling problems in computational geometry [12, 13, 14]. Results. In this paper, we describe NC algorithms (specifically, the algorithms run in time O(log 2 n) using work 1 O(n C ) for some constant C in the EREW PRAM model) that achieve Work performed while at the Max Planck Institut fur Informatik, Germany. y Contact author. The work by ....

.... for some range spaces that here we just call linearizable, and for r n ffl , some ffl 0 depending on the range space, the construction can be performed in time O(n log r) In parallel, however, only size O(r 2 ffi ) has been achieved using k wise independence probability spaces [12, 13, 14]. There is a close relation to the DP. In fact, when the random sample R is of size jX j=2, the low discrepancy and approximation properties are equivalent. From the definition, it is clear that the same approach used for the DP can be used to compute an approximation of optimal size in parallel. ....

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M.T. Goodrich and E.A. Ramos. Bounded independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. To appear in Discrete and Comput. Geom.

....with the monotonicity property, see Section 2) such derandomization is not known. Finally, we discuss implementations in the CRCW PRAM model, where faster algorithms are possible through the use of approximate counting [5, 32] To take advantage of this, the concept of an approximation is relaxed [38, 36, 37]. The derandomization tools that our algorithms use have been well developed in the case of a sequential model of computation [23, 48, 50, 53] For parallel models, a corresponding adaptation of the algorithms has been partly performed [36, 37] Here, we give a complete review of those tools, ....

.... this, the concept of an approximation is relaxed [38, 36, 37] The derandomization tools that our algorithms use have been well developed in the case of a sequential model of computation [23, 48, 50, 53] For parallel models, a corresponding adaptation of the algorithms has been partly performed [36, 37]. Here, we give a complete review of those tools, giving in some cases algorithms simpler than those in [36, 37] and providing new e cient implementations of algorithms by Matou sek [50] for constructing approximations. Contents. The paper consists of two main parts. The rst part consists of ....

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M. T. Goodrich and E. A. Ramos. Bounded independence derandomization of geometric partitioning with applications to parallel xed-dimensional linear programming. Discrete Comput. Geom. 18 (1997), 397-420.

....by successive refinement using the previous algorithm. Our algorithms rely essentially on the CRCW PRAM implementation by Goldberg and Zwick [20] of the approximate counting circuits of Ajtai [2] Also, we take advantage of previous work on random sampling in the context of computational geometry [38, 15, 32, 33, 21, 22, 24], like the concept of semi splitters and constant time computation of good samples and good splitters. bf Contents. In Section 2 we review results concerning approximate counting and its applications which are needed in our algorithms. In Section 3 we describe our pad sorting algorithm. The ....

....on t, with probability at least 1 2 the following holds: i) If s = Dr 1 2=t then S is a (Dr 2=t ; r) splitter for X. ii) If s = D 0 r 2 4=t then S is a (1; r) sample for X. iii) If s = r then S is a (t=2 Gamma 4; D 00 ; r) semi splitter for X. Remark. The more general results in [21, 22, 24] only claim sizes rn ffi and r 2 n ffi in (i) and (ii) of the theorem. For our purposes, in (i) and (ii) it will be sufficient to use t = 2; but in (iii) we need at least t = 12 to obtain a (2; Delta; Delta) semi splitter (and we actually need an even higher exponent in Lemma 5) 4.4 ....

[Article contains additional citation context not shown here]

M. T. Goodrich and E. A. Ramos. Bounded independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom., 18 (1997) 397--420.

.... 0 ffl 1, such that for r n ffl , the construction can be performed in O(n log r) time [29] This extends to a wide class of range spaces, called linearizable, that can be embedded into a linear range space in IR l , for some l (see e.g. 1, 28] In NC however, the best known algorithms [18, 19, 20] only guarantee (1=r) appoximations of larger size, namely O(r 2 ffi ) The ffi factor appears once again because of the use of k wise independent probability spaces, with k proportional to 1=ffi. There is a close relation with the discrepancy problem. In fact, when R is a (1=2) sample, the low ....

....of the good behavior of approximations under partitioning and iteration [27] the running times of the algorithms can be improved as stated in the following theorem, with only a constant factor loss in the size. See App. E for an outline of the proof. The results for the CRCW PRAM model in [19, 20] can be similarly improved. Theorem 5.1 Let (X; R) be a range space with VC exponent bounded by d and X of size n. Then there is a constant c = c(d) such that a (1=r) approximation of size O(r 2 log r) for (X; R) can be computed deterministically in the EREW PRAM model in O(log n log 2 r) ....

M.T. Goodrich and E.A. Ramos. Bounded independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. Discrete Comput. Geom. 18 (1997), 397--420.

....envelopes of 2 d algebraic functions with a site having a simply connected face) These results also parallelize. The output sensitive results were not known previously even under randomization. An additional tool that we use is the fast construction of approximations and cuttings in parallel [24, 25, 27]. Even in the particular case of planar Euclidean Voronoi diagrams, we can improve previous work in [16] by obtaining a deterministic worst case optimal work CRCW algorithm that runs in time O(log n log log n) this can also be obtained using the original divide and conquer 1 d lower envelope ....

.... PRAM the construction can be performed in times O(log log n) and O(log log n log r) respectively (the output is padded, or add time O(logn) for not padded output) See [9, 33, 24, 4] Only the CRCW construction has not been noted before, it uses the recent fast construction of approximation in [25, 27]. Briefly, for later reference, a cutting is computed by first (i) obtaining a (1=2r) approximation A for (Q; Q E ) then (ii) computing a (1=2r) cutting T for (Y; A) and finally (iii) computing the conflict lists of T in Q. It is important to note that for the fast O(n log r) construction, for ....

M.T. Goodrich and E.A. Ramos. Bounded independence derandomization of geometric partitioning with applications to parallel fixed-dimensional linear programming. To appear in Discrete and Computational Geometry.

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