D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theor. Comp. Sc., 28:263-- 276, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is that it allows any two paths to be compared at unit cost, without regard to the number of edges in the paths; that is, the computation is assumed to be free. But just as the unbounded word size RAM model can be (theoretically) abused to sort numbers in O(n) time by arithmetic on long numbers [15, 22], path comparison lower bounds can be invalidated by the creation of long paths, which may encode all relevant paths as subpaths. For instance, in our modification of the Karger Koller Phillips construction, we can create a single (non simple) path that includes all the edges of the original ....

D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28(3):263--276, 1984.

....k is the number of re ported answers. In solving problems we will presort the given points. It is well known, that given a set of n keys in U, they can be sorted in O(n u) time using O(n u) storage. However, it is possible to do it more efficiently. Theorem 2. 2: Kirkpatrick and Reisch [11]) Given a set of n keys in U, they can be sorted in O(n(1 loglog,u) time using 0( u) storage. We will now describe a new dynamic data structure, called the interval trie, used to solve the 1 dimensional stabbing problem on a grid, that is, given a set of intervals [ai. bi] with ai and bi ....

D. Kirkpatrick and S. Reisch, Upper Bounds for Sorting Integers on Random Access Machines, Theoretical Computer Science 28 (1984), 263-276

....Our on line approximation algorithm is based on Graham s scan algorithm [12] and can be viewed as a combination of the algorithms by Shamos and by Bentley et al. with the replacement of an exact VEB data structure by an approximate variant. Computation with large words. Kirkpatrick and Reich [17] considered exact sorting with large words, giving upper and lower bounds. Their interest was theoretical, but Lemma 5.1, which in some sense says that maintaining an approximate VEB data structure is equivalent to maintaining an exact counterpart using larger words, suggests that lower bounds on ....

D. Kirkpatrick and S. Reisch. Upper bounds for sort- ing integers on random access machines. Theoretical Computer Science, 28:263 276, 1984.

....solutions to these problems require Theta(log n) time per operation. Striking improvements over comparison based approaches have also been obtained for sorting [3, 5, 16] and computing shortest paths [9, 18, 19, 16] These results are related to an older tradition of RAM algorithms (see e.g. [21, 15]) and follow a direction first taken by [11, 12] One way to view the difference between the old and new algorithms is that the old algorithms have a complexity which increases with w, but are quite fast An experimental study of word level parallelism 194 if w is relatively small 1 . The new ....

....to be simulated on a (virtual) RAM with word size w, for any w 32 specified by the user. The objectives of the simulator and the approach to its implementation are described in Section 2. In Section 3, we describe a simplified implementation of the Kirkpatrick Reisch (KR) sorting algorithm [15]. This algorithm has a running time of O(n(1 logd w log n e) as compared to the running time of radix sort [7] which is O(nd w log n e) Our results indicate, e.g. that for a few million keys and w 256, KR is superior to radix sort on a RAM with word size w. Our results also suggest ....

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theor. Comput. Sci 28 (1984), pp. 263--276.

....[1] achieves T (n; w= logn) O(n log log n) provided that w log n log log n. This can be seen by applying Theorem 1 [1] with parameters k = log n, m = 2 w= log n . Also, using traditional bucket sort, we have T (n; log n) O(n) The range reduction technique by Kirkpatrick and Reisch [6] achieves T (n; b) T (n; b=2) O(n) Department of Computer Science, Lund University, Box 118, S 22100 Lund, Sweden y Max Planck Institut fur Informatik, D 66123 Saarbrucken, Germany Let c = maxflog n; w= logng and consider the following algorithm for sorting n integers in the range ....

D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28:263--276, 1984.

....Our on line approximation algorithm is based on Graham s scan algorithm [12] and can be viewed as a combination of the algorithms by Shamos and by Bentley et al. with the replacement of an exact veb data structure by an approximate variant. Computation with large words. Kirkpatrick and Reich [17] considered exact sorting with large words, giving upper and lower bounds. Their interest was theoretical, but Lemma 5.1, which in some sense says that maintaining an approximate veb data structure is equivalent to maintaining an exact counterpart using larger words, suggests that lower bounds on ....

D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28:263--276, 1984.

....queues for arbitrary integers. In this section we complete the proof of Theorem 1.1. To get from arbitrary integers to short integers we will use a recursive range reduction which can be seen as a simple specialized variant of van Emde Boas data structure [24,25] inspired by the developments in [5,16,17,19]. Let T (n; b) be the time for insert and extract min in a priority queue with up to n b bit integers. We assume that b but not n is known in advance. By Theorem 2.1, T (n; w= logn) O(log log n) We will show that T (n; b) O(1) T (n; b=2) 2.1) Here b is assumed to be a power of 2. For ....

D. Kirkpatrick and S. Reisch, Upper bounds for sorting integers on random access machines, Theor. Comp. Sci. 28 (1984), 263--276.

....on the size. Bucket sorting requires the n input keys to be in the range 0 : n Gamma 1. Radix sorting in k phases, each phase implemented via bucket sorting, can sort n integers in the range 0 : n k Gamma 1 in O(nk) time. A more sophisticated technique, due to Kirkpatrick and Reisch [14], reduces this to O(n log k) but the fact remains that as the size of the integers to be sorted grows to infinity, the cost of the sorting also grows to infinity (or to Theta(n log n) if we switch to a comparison based method at the appropriate point) If we allow intermediate results ....

.... n) if we switch to a comparison based method at the appropriate point) If we allow intermediate results containing many more bits than the input numbers, we can actually sort integers in linear time independently of their size, as demonstrated by Paul and Simon [18] and Kirkpatrick and Reisch [14]. But again, from a practical point of view, this is not what we want, since a real machine is unlikely to have unit time instructions for operating on integers containing a huge number of bits. Instead, if the input numbers are w bit integers, we would like all intermediate results computed by a ....

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoret. Computer Science, 28, pp. 263--276, 1984.

....work in linear time, bucket sorting requires its input keys to come from a range of linear size. Radix sorting in k phases, each phase implemented via bucket sorting, can sort n integers in the range 0 : n k Gamma 1 in O(nk) time. A more sophisticated technique, due to Kirkpatrick and Reisch [24], reduces this to O(n log k) for arbitrary k 2, but the fact remains that as the size of the integers to be sorted grows to infinity, the cost of the sorting also grows to infinity (or to Theta(n log n) if we switch to a comparison based method at the appropriate point) If we allow ....

.... method at the appropriate point) If we allow intermediate results containing many more bits than the input numbers, while maintaining a unit cost assumption, we can actually sort integers in linear time independently of their size, as demonstrated by Paul and Simon [29] and Kirkpatrick and Reisch [24]. But again, from a practical point of view, this is not what we want, since a real machine is unlikely to have unit time instructions for operating on integers containing a huge number of bits. Instead, if the input numbers are w bit integers, we would like all intermediate results computed by a ....

[Article contains additional citation context not shown here]

D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoret. Computer Science, 28, pp. 263--276, 1984.

....range 1 : m is Theta(log(n m) bits, i.e. that unit time operations on integers of size (n m) O(1) are provided, but that all integers manipulated must be of size (n m) O(1) A number of papers have explored the implications of allowing a larger word length. Paul and Simon (1980) and Kirkpatrick and Reisch (1984) demonstrated that with no restrictions on the word length at all, arbitrary integers can be sorted in linear sequential time. Hagerup and Shen (1990) showed that in fact a word length of about O(n log n log m) bits suffices to sort n integers in the range 1 : m in O(n) sequential time or in ....

....refer to this process as stabilizing the original algorithm. Given an element x of a totally ordered domain U and a finite sequence L of elements of U , the rank of x in L is the number of elements in L no larger than x. Our basic sorting algorithm is nonconservative, following the terminology of Kirkpatrick and Reisch (1984), i.e. the word length allowed for sorting n integers in the range 1 : m is not limited to O(log(n m) bits. More precisely, we will use a word length of Theta(kl log n) bits, where k is a power of 2 with 2 k n and l is an integer with l dlog(m k)e 2. This word length enables us to ....

Kirkpatrick, D., and Reisch, S. (1984), Upper bounds for sorting integers on random access machines, Theoret. Comput. Sci. 28, 263--276.

.... preprocessing and sorting, we get the following results: ffl In the simplest case, when bucketing is used as a subroutine, the complexity is the same as that achieved by Paige and Tarjan [16] Theta n B log n 1 : ffl Using the sorting algorithm by Kirkpatrick and Reisch [12] we get two cases. Assuming that w = Omega Gamma B) the cost is Theta n log B log n 2 : For input from a stationary ergodic process, the complexity is Theta n log 1 H 1 : In general, making no assumption on the size of w, the cost is O n B w 1 ....

....process we get the bound Theta i n H j : 5) According to the trivial lower bound above, this algorithm is optimal if the word length is Theta(log n) 5.1. 2 Applying the algorithm by Kirkpatrick and Reisch Under the assumption that b w, the sorting algorithm by Kirkpatrick and Reisch [12] sorts n integers in the range [0; 2 b Gamma 1] in Theta (n log(b= log n 2) time. The algorithm uses Theta(2 b=2 ) extra space, but this can be reduced to O(n) with universal hashing [3, 5] Using this sorting algorithm as a subroutine we get the time complexity Theta n B b ....

D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28:263--276, 1984.

.... [AHU74] Because of the central role that the sorting problem plays in computer science, numerous papers are devoted to study opportunities for improving this time bound to o(n log n) One approach is to consider idealized (and non standard) versions of the RAM model; as, for instance, in [KR84] and [PS80] where very large words are assumed. The practicality of such an assumption is unclear. Another approach is to focus on instances of the sorting problem, where the input consists of integers drawn from a restricted interval [1: m] For m = O(n) the known Bucket Sort algorithm ....

....of his result is an O(n log log m) time and O(m 1=c ) space algorithm for sorting, where c 0 is a constant. Johnson recognizes the problem with the space requirements of the algorithm, and writes that the algorithm is not practical and only of theoretical interest. Kirkpatrick and Reisch [KR84] presented an algorithm, based on a range reduction technique, that has the same complexity bounds as Johnson s algorithm. They state that the algorithm is of little practical value due to both large constants, that are hidden in the asymptotic bounds, and storage requirements. The following open ....

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28:263--276, 1984.

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theor. Comp. Sc., 28:263-- 276, 1984.

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science 28, pp. 263276 (1984).

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science 28, 263-276(1984).

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D. Kirkpatrick and S. Reisch, Upper bounds for sorting integers on random access machines, Theoretical Computer Science 28, 263-276(1984).

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D. Kirkpatrick and S. Reisch, Upper bounds for sorting integers on random access machines, Theoretical Computer Science 28, pp. 263-276(1984).

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28(3):263--276, 1984.

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D. Kirkpatrick and S. Reisch. Upper bounds for sorting integers on random access machines. Theoretical Computer Science, 28(3):263-276, 1984.

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Kirkpatrick,D., and Reisch,S., "Upper Bounds for Sorting Integers on Random Access Machines, " Theoretical Computer Science, vol28, 1984, pp.263-276.

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