R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. Assoc. Comput. Mach. 34 (1) (1987) 200--208.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....narrows down the interval the interval of possible values for # # such that 12 in the whole interval, now two critical points switch positions in the x ory order; So we know that the lower endpoint of this interval must be the Frechet distance #. By utilizing Cole s variant of parametric search [4] we obtain a running time of O(mn log(mn) Theorem 3. Given two curves consisting of m and n pieces, respectively, of smooth algebraic curves of fixed maximum degree we can compute their Frechet distance in O(nm) space and in O(mn log(mn) algebraic operations of bounded degree. Here, an ....

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. Assoc. Comput. Mach. 34 (1987), no. 1, 200--208.

....(e.g. 13] is that there is a point x (not necessarily in P ) of depth at least #n 3#. Such a point is called a centerpoint. The center is the set of all centerpoints. Cole, Sharir and Yap [9] described an O(n(log n) algorithm to construct a centerpoint, and subsequent ideas of Cole [8] could be used to lower the complexity to O(n(log n) Recently, Jadhav and Mukhopadhyay [16] described a linear time algorithm to find a centerpoint. Matousek [19] attacked the harder problem of computing a Tukey median and presented an O(n(log n) algorithm for that task. The algorithm ....

R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34(1):200--208, 1987.

....can count the number of inversions in O(n log n) time sequentially or in O(log n) parallel time using O(n) processors. Plugging these algorithms into the parametric searching paradigm, we obtain an O(n log n) time algorithm for the slope selection problem. 2. 3 Improvements and extensions Cole [76] observed that in certain applications of parametric searching, including the slope selection problem, the running time can be improved to O( P T s )T p ) as follows. Consider a parallel step of the above generic algorithm. Suppose that, instead of invoking the decision procedure O(log n) times ....

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM, 34 (1987), 200--208.

.... yields algorithms that are asymptotically faster than those currently known (e.g. the second and third problems below) by incorporating into our (basic) technique a sub technique that is equivalent to (though much more flexible than) Cole s technique for speeding up parametric searching [17]. We exemplify the technique on three main problems, the slope selection problem, the planar distance selection problem, and the planar two line center problem. For the first problem we develop an O(n log n) solution, which, although suboptimal, is very simple. The other two problems are ....

.... problem is perhaps not ideal, in the sense that (what we regard as) the elegant and simple algorithm that is given below is suboptimal, and runs in time O(n log n) which is, by the way, the same running time yielded by the basic parametric searching method, without the improvements of [17] and of [18] To improve it, one has to develop additional technical tricks that have little to do with the basic method. In a companion paper [25] we do present an alternative O(n log n) algorithm, which is also based on expanders. The solution to the distance selection problem is a much more ....

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R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM 34 (1987), 200--208.

....down the interval the interval of possible values for such that 12 in the whole interval, now two critical points switch positions in the x or y order; So we know that the lower endpoint of this interval must be the Fr echet distance . By utilizing Cole s variant of parametric search [4] we obtain a running time of O(mn log(mn) Theorem 3. Given two curves consisting of m and n pieces, respectively, of smooth algebraic curves of xed maximum degree we can compute their Fr echet distance in O(nm) space and in O(mn log(mn) algebraic operations of bounded degree. Here, an ....

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. Assoc. Comput. Mach. 34 (1987), no. 1, 200-208.

.... the concrete decision; in several cases, sorting can be used instead [3, 8, 11, 13, 16] However, the existing parallel sorting algorithms that have good worst case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormous [2] Cole [9] shows how sorting based parametric search can be optimized even further, but the optimization comes at the expense of making the technique even more complicated than it already is. In this paper we show that Quicksort can be used as the generic algorithm in sorting based parametric search, ....

....algorithm A s . Therefore, a weak model of parallelism, such as the parallel comparison model of Valiant, suffices. In this model, the complexity of an algorithm is only determined by the comparisons being made, and things like communication and synchronization between processes are ignored. Cole [9] shows that in some applications of parametric search, the number of calls to the decision process can be reduced by a log factor, thus improving the running time to O(PT p T p T s T s logP) We will explain his idea and comment on it in Section 4. One of the drawbacks of parametric search ....

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R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34(1):200--208, 1987.

.... (H ) in O(k log n) time using steps 2 and 3 of the 2 d algorithm; these two steps are parallelizable in O(logn) time using O(k ) processors (step 3 requires the AKS sorting network for optimal effect) Thus, can be found by Megiddo s parametric search [40] with Cole s improvement [21], in O(k time. Including the preprocessing, the entire algorithm takes O(n log n k n) expected time. Remark: The problem of finding the smallest circle enclosing all but k points in the plane can be solved by a similar algorithm, because the smallest enclosing circle acts like a ....

R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34:200--208, 1987.

....interest was in obtaining a near linear solution of the 2 center problem. For example, it might be possible to adapt the more efficient technique for off line dynamic maintenance of convex hulls, as described in [8] One might also be able to apply Cole s improved parametric searching technique [3], or to bypass parametric searching altogether by using either randomization (such as in [13] or other geometric techiques (such as in [2, 11] We leave this as an open problem for further research. Some initial ideas towards this goal were suggested to us by Pankaj Agarwal and Matthew Katz, ....

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM 34 (1987), 200--208.

.... Thus, the entire sweep algorithm takes time O(n log n) In the plane, wecansolve this problem in time and space O(n ) using a more complicated algorithm developed by Chazelle and Lee [8] To find minimum circumradius sets, we apply parametric searching with Cole s weighted median strategy [10]. Our sweep algorithm can be parallelized to run in O(log n) steps on O(n ) processors. Thus, the total time is O(n n) in general, and O(n log n) in the plane. The parametric search technique requires the construction of an AKS sorting network [3] with O(n ) inputs, one for each ....

R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34:200--208, 1987.

....RAM model of computation. A centerpoint in R d can be found by solving a set of #(n d ) linear inequalities, using linear programming [2] When d = 2 the situation improves. Cole, Sharir and Yap described an O(n(logn) 5 ) algorithm [5] to construct a centerpoint, and subsequent ideas of Cole [3] could be used to lower the complexity to O(n(log n) 3 ) Recently, Jadhav and Mukhopadhyay [8] described a linear time algorithm to find a Tukey centerpoint. Matousek [9] attacked the harder problem of computing a Tukey median. He first described an O(n(logn) 4 ) algorithm that gives a ....

R. Cole. Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms. J. ACM 34(1), 200-208, (1987).

....is a well known consequence of Helly s Theorem (e.g. 6] that there is a point in R d of Tukey depth at least #n (d 1)#; such a point is called a centerpoint. When d = 2, Cole, Sharir and Yap described an O(n(log n) 5 ) algorithm [5] to construct a centerpoint; ideas presented by Cole in [3] could be used to improve the complexity to O(n(log n) 3 ) Recently, Jadhav and Mukhopadhyay [8] gave a linear time algorithm for this task. Finally, Matousek [11] has described an algorithm that finds a planar Tukey median in time O(n(log n) 5 ) No lower bound for this task has been ....

R. Cole. Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms. J. ACM 34(1), 200-208, (1987).

....cost. A centerpoint in R d can be found by solving a set of #(n d ) linear inequalities, using linear programming (Clarkson et al. 3] When d = 2 the situation improves. Cole, Sharir and Yap [6] described an O(n(log n) 5 ) algorithm to construct a centerpoint, and subsequent ideas of Cole [4] could be used to lower the complexity to O(n(log n) 3 ) Recently, Jadhav and Mukhopadhyay [9] described a linear time algorithm to find a Tukey centerpoint. Matousek [11] attacked the harder problem of computing a Tukey median. Let us first observe that a brute force approach could compute ....

R. Cole. Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms. J. ACM 34(1), 200-208, (1987).

....by asking a constant number of them. Thus, after O(log n) questions, all questions at one level of the sorting network can be answered. Since the network has O(log n) levels, this sums up to O( log n) 2 ) questions. This number can be reduced to O(log n) questions total using Cole s technique [5]. Since each question is answered in O(n log n) the ham sandwich cut is found in O(n(log n) 2 ) and Theorem 1 is proved. 4 Partitioning Lines Let L be a set of n lines in the plane and # A and # B non parallel lines that are not in L, nor parallel to any line in L. Write C = # A # # B . ....

R. Cole. Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms. J. ACM, 34(1):200--208, 1987.

....ed variant of the algorithm of [5] for the slope selection problem. Speci cally, this algorithm applies parametric searching to the inversion counting algorithm. The simplest version runs in O(n log 3 n) time, which is reduced to O(n log 2 n) time using an enhancement technique due to Cole [4]. Both of these approaches are applicable in our case too. The nal improvement in the algorithm of [5] which reduces its complexity to O(n log n) does not seem to be applicable in our case, and has no e ect anyway on the overall asymptotic bound on the running time of the algorithm. 2 The ....

....parallel step. The cost of a single parallel step is O ( log(m n) m n log n) log(mn) and since there are O(log n) parallel steps, the overall cost of this step is O( m n log n) log n log 2 (mn) We can improve this further, by a factor of O(log n) by using the technique of Cole [4]. This technique is applicable when the parallel execution can be simulated on a network with bounded fan out, and this property holds for our algorithm, which is just a collection of binary searches. At this stage, we have limited the range for r further, and for any r in the new range, we ....

R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34(1):200-208, 1987.

....lel version of the test algorithm. In our case what is needed is a parallel median selection algorithm. Typically all comparisons with A# done in a single step of the parallel algorithm can be tested by binary search with O(log n) calls to the (sequential) test algorithm. A technique of Cole [4] further speeds up this idea, so that for our problem the resulting time bound should be O(n log n) However we find this unsatisfactory for two reasons. First, the time is still greater than we want. Second, and more importantly, the resulting algorithm is extremely complicated and not easily ....

R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 31 (1984) 200--208.

....execution is the desired r i . Concerning the running time of this generic simulation, the first two stages require time O(n log 2 n) this time is dominated by O(log n) calls to Calc Depth with specific radii) The third step takes O(n log 3 n) time, but using a standard trick due to Cole [6], this can be reduced to O(n log 2 n) Repeating this procedure for each p i , we obtain r as the minimum of all the r i s, at a total cost of O(n 2 log 2 n) We will improve this naive bound in subsequent sections of the paper. The Improved Algorithms 8 3 The Improved Algorithms The ....

....cost of the generic simulation is O(nk log 3 n) Note that, since the generic simulation follows the execution of the oracle at r init , we are guaranteed that the generic execution does not require more than 18nk n processors. This can be improved to O(nk log 2 n) using Cole s trick [6]. This amounts to executing only a constant number of binary search steps at each stage of the algorithm, thereby resolving only some fixed large fraction of the number of comparisons. Nodes whose comparisons were resolved can proceed to the next level while the other nodes are stuck and have to ....

[Article contains additional citation context not shown here]

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM, 34:200--208, 1987.

....interest was in obtaining a near linear solution of the 2 center problem. For example, it might be possible to adapt the more efficient technique for off line dynamic maintenance of convex hulls, as described in [8] One might also be able to apply Cole s improved parametric searching technique [3], or to bypass parametric searching altogether by using either randomization (such as in [13] or other geometric techiques (such as in [2, 11] We leave this as an open problem for further research. Some initial ideas towards this goal were suggested to us by Pankaj Agarwal and Matthew Katz, ....

R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. ACM 34 (1987), 200--208.

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R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, J. Assoc. Comput. Mach. 34 (1) (1987) 200--208.

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R. Cole, Slowing down sorting networks to obtain faster sorting algorithms, Journal of the Association for Computing Machinery 34 (1987), 200-208.

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R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34:200--208, 1987. 8

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R. Cole, \Slowing down sorting networks to obtain faster sorting algorithms," Journal of the ACM, 34, 200-208, 1987.

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R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34:200--208, 1987.

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R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. ACM, 34:200--208, 1987.

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R. Cole. Slowing Down Sorting Networks to Obtain Faster Sorting Algorithms. J. ACM 34(1), 200-208, (1987).

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R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. J. Assoc. Comput. Mach., 34(1):200-208, 1987.

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