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Asymptotically Efficient inPlace Merging
 Theoretical Computer Science
"... Two lineartime algorithms for inplace merging are presented. Both algorithms perform at most m(t+1)+n=2 t +o(m) comparisons, where m and n are the sizes of the input sequences, m n, and t = blog 2 (n=m)c. The first algorithm is for unstable merging and it carries out no more than 3(n+m)+o(m) el ..."
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Cited by 14 (3 self)
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Two lineartime algorithms for inplace merging are presented. Both algorithms perform at most m(t+1)+n=2 t +o(m) comparisons, where m and n are the sizes of the input sequences, m n, and t = blog 2 (n=m)c. The first algorithm is for unstable merging and it carries out no more than 3(n+m)+o(m) element moves. The second algorithm is for stable merging and it accomplishes at most 5n+12m+o(m) moves. Key words: Inplace algorithms, merging, sorting ? A preliminary and weaker version of this work appeared in Proceedings of the 20th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 969, SpringerVerlag, Berlin/Heidelberg (1995), 211220. 1 Supported by the Slovak Grant Agency for Science under contract 1/4376/97 (Project "Combinational Structures and Complexity of Algorithms"). 2 Partially supported by the Danish Natural Science Research Council under contracts 9400952 (Project "Computational Algorithmics") and 9701414 (Project "Experimental Algorithmics"). Preprint submitted to Elsevier Preprint December 19, 1995 1
Practical InPlace Mergesort
, 1996
"... Two inplace variants of the classical mergesort algorithm are analysed in detail. The first, straightforward variant performs at most N log 2 N + O(N ) comparisons and 3N log 2 N + O(N ) moves to sort N elements. The second, more advanced variant requires at most N log 2 N + O(N ) comparisons and " ..."
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Cited by 10 (3 self)
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Two inplace variants of the classical mergesort algorithm are analysed in detail. The first, straightforward variant performs at most N log 2 N + O(N ) comparisons and 3N log 2 N + O(N ) moves to sort N elements. The second, more advanced variant requires at most N log 2 N + O(N ) comparisons and "N log 2 N moves, for any fixed " ? 0 and any N ? N ("). In theory, the second one is superior to advanced versions of heapsort. In practice, due to the overhead in the index manipulation, our fastest inplace mergesort behaves still about 50 per cent slower than the bottomup heapsort. However, our implementations are practical compared to mergesort algorithms based on inplace merging. Key words: sorting, mergesort, inplace algorithms CR Classification: F.2.2 1.
Radix sorting with no extra space
 In Proceedings of the 15th European Symposium on Algorithms
, 2007
"... It is well known that n integers in the range [1, n c] can be sorted in O(n) time in the RAM model using radix sorting. More generally, integers in any range [1, U] can be sorted in O(n √ log log n) time [5]. However, these algorithms use O(n) words of extra memory. Is this necessary? We present a s ..."
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Cited by 7 (0 self)
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It is well known that n integers in the range [1, n c] can be sorted in O(n) time in the RAM model using radix sorting. More generally, integers in any range [1, U] can be sorted in O(n √ log log n) time [5]. However, these algorithms use O(n) words of extra memory. Is this necessary? We present a simple, stable, integer sorting algorithm for words of size O(log n), which works in O(n) time and uses only O(1) words of extra memory on a RAM model. This is the integer sorting case most useful in practice. We extend this result with same bounds to the case when the keys are readonly, which is of theoretical interest. Another interesting question is the case of arbitrary c. Here we present a blackbox transformation from any RAM sorting algorithm to a sorting algorithm which uses only O(1) extra space and has the same running time. This settles the complexity of inplace sorting in terms of the complexity of sorting. 1
InSitu, Stable Merging by way of the Perfect Shuffle.
, 1999
"... We introduce a novel approach to the classical problem of insitu, stable merging, where "insitu" means the use of no more than O(log 2 n) bits of extra memory for lists of size n. Shufflemerge reduces the merging problem to the problem of realising the "perfect shuffle" permutation, that is, the ..."
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Cited by 1 (0 self)
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We introduce a novel approach to the classical problem of insitu, stable merging, where "insitu" means the use of no more than O(log 2 n) bits of extra memory for lists of size n. Shufflemerge reduces the merging problem to the problem of realising the "perfect shuffle" permutation, that is, the exact interleaving of two, equal length lists. The algorithm is recursive, using a logarithmic number of variables, and so does not use absolutely minimum storage, i.e., a fixed number of variables. A simple method of realising the perfect shuffle uses one extra bit per element, and so is not insitu. We show that the perfect shuffle can be attained using absolutely minimum storage and in linear time, at the expense of doubling the number of moves, relative to the simple method. We note that there is a worst case for Shufflemerge requiring time\Omega\Gamma n log n), where n is the sum of the lengths of the input lists. We also present an analysis of a variant of Shufflemerge which uses a ...
On the Competitiveness of Linear Search
 In Proceedings of the 8th Annual European Symposium on Algorithms
, 2000
"... We reexamine offline techniques for linear search. Under a reasonable model of computation, a method is given to perform offline linear search in amortized cost proportional to the entropy of the request sequence. It follows that no online technique can have an amortized cost of that which one ..."
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We reexamine offline techniques for linear search. Under a reasonable model of computation, a method is given to perform offline linear search in amortized cost proportional to the entropy of the request sequence. It follows that no online technique can have an amortized cost of that which one could obtain if given the request sequence in advance, i.e., there is no competitive linear search algorithm. 1 Introduction This is a paper about the competitiveness of algorithms. That is, the extent to which an algorithm can receive and immediately process a sequence of queries almost as well as could be done if all queries were given in advance, and a more global scheme could be developed for their processing. Online algorithms and competitiveness have been a major focus in the theory of query processing over the past 10 or 15 years ([1],[2],[6],[7]). Borodin and El Yani [2], in particular, give an excellent treatment of the topic. At the heart of proving anything about "optimal" perf...
Supervised by
, 2011
"... The discrete geometry is to classical geometry what the language is to thought, i.e. an imperfect means to represent the reality. It took centuries for the language to evolve in a way almost capable to faithfully describe our though. tel00596947, version 1 30 May 2011 Today the discrete geometry t ..."
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The discrete geometry is to classical geometry what the language is to thought, i.e. an imperfect means to represent the reality. It took centuries for the language to evolve in a way almost capable to faithfully describe our though. tel00596947, version 1 30 May 2011 Today the discrete geometry tries to do the same thing with the continuous geometry. The continuous geometry is a mathematical model that cannot be correctly or exactly reproduced in the real world and in computer science. A simple example is the famous number π. The theoretical mathematic model supposes an exact value of this number, however, the representation of a circle on the ground with a rope or on a sheet of paper by a compass can only give an approximated value of π, whatever the diameter of the circle, the size of the rope or the precision of the compass. In computer science, for any approximation of π used during computations, results will always be an approximation. Today, one of the biggest challenge in computer science is to nd new methods so that computers can represent reality as faithfully as possible. Regarding geometry, we strongly believe that these methods belong to the discrete geometry. tel00596947, version 1 30 May 2011