## On optimal and efficient in place merging (2006)

Venue: | SOFSEM 2006. Volume 3831 of Lecture Notes in Computer Science |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Kim06onoptimal,

author = {Pok-son Kim and Arne Kutzner},

title = {On optimal and efficient in place merging},

booktitle = {SOFSEM 2006. Volume 3831 of Lecture Notes in Computer Science},

year = {2006},

pages = {350--359},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. We introduce a new stable in place merging algorithm that needs O(m log ( n +1)) comparisons and O(m+n) assignments. According m to the lower bounds for merging our algorithm is asymptotically optimal regarding the number of comparisons as well as assignments. The stable algorithm is developed in a modular style out of an unstable kernel for which we give a definition in pseudocode. The literature so far describes several similar algorithms but merely as sophisticated theoretical models without any reasoning about their practical value. We report specific benchmarks and show that our algorithm is for almost all input sequences faster than the efficient minimum storage algorithm by Dudzinski and Dydek. The proposed algorithm can be effectively used in practice. 1

### Citations

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Citation Context ...s. An algorithm that meets this lower bound is presented in [7]. (4) Binary Search and Insertion-Sort are two standard algorithms described in almost all introducing literature about algorithms (e.g. =-=[14]-=-). 3 The Core Algorithm We now give the definition of our unstable core algorithm that relies on extra storage of size ⌊ √ m⌋ for local merges. Algorithm 1: Unstable-Core-Merge Let k = ⌊ √ m⌋ and l = ... |

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Citation Context ...put sequences u and v respectively. We will use six other algorithms as subcomponents. We now briefly introduce these algorithms and their complexity (A summary is given in Tab. 1): (1) Hwang and Lin =-=[13]-=- introduced a merging-algorithm that is optimal regarding the number of comparisons as well as assignments. Unfortunately their algorithm is not in-place, it relies on an external buffer of size m whe... |

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Citation Context ...n contained in the C++ Standard Template Libraries (STL) [8]. Significant older works in the area of in place merging are the publications of Pardo [9], Salowe and Steiger [10] and Huang and Langston =-=[11]-=-. All algorithms introduced there are asymptotically optimal regarding the number of assignments, but lack in meeting the lower bound for comparisons. Another class of merging algorithms are the minim... |

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Citation Context ...merge_without_buffer function contained in the C++ Standard Template Libraries (STL) [8]. Significant older works in the area of in place merging are the publications of Pardo [9], Salowe and Steiger =-=[10]-=- and Huang and Langston [11]. All algorithms introduced there are asymptotically optimal regarding the number of assignments, but lack in meeting the lower bound for comparisons. Another class of merg... |

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Citation Context ...shown by Knuth in [1] the lower bound for the number of comparisons is Ω(m log( n m + 1)), where m ≤ n. So far there are three publications about optimal stable in place merging. The work of Symvonis =-=[2]-=- shows how to get an optimal algorithm by combining several given concepts but contains no information about the involved asymptotic constants or implementation aspects. Geffert et. all present in [3]... |

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Citation Context ...wn by the fact that it can compete with the algorithm of Dudzinski and Dydek [7] that is used as foundation of the merge_without_buffer function contained in the C++ Standard Template Libraries (STL) =-=[8]-=-. Significant older works in the area of in place merging are the publications of Pardo [9], Salowe and Steiger [10] and Huang and Langston [11]. All algorithms introduced there are asymptotically opt... |

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Citation Context ...ed as foundation of the merge_without_buffer function contained in the C++ Standard Template Libraries (STL) [8]. Significant older works in the area of in place merging are the publications of Pardo =-=[9]-=-, Salowe and Steiger [10] and Huang and Langston [11]. All algorithms introduced there are asymptotically optimal regarding the number of assignments, but lack in meeting the lower bound for compariso... |

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Citation Context ...in University research grant in 2005.shave some resemblance. They take the algorithm from Mannila and Ukkonen [5] as starting point, rely on the concept of an internal buffer introduced by Kronrod in =-=[6]-=- and develop a stable algorithm out of an unstable one. We will follow this path but with the focus on an improved stable algorithm as well as concrete benchmarking. The proposed stable algorithm can ... |

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Citation Context ...ficant lower bounds for merging. The lower bound for the number of assignments is m + n because every element of the input sequences can change its position in the sorted output. As shown by Knuth in =-=[1]-=- the lower bound for the number of comparisons is Ω(m log( n m + 1)), where m ≤ n. So far there are three publications about optimal stable in place merging. The work of Symvonis [2] shows how to get ... |

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Citation Context ...stable algorithm as well as concrete benchmarking. The proposed stable algorithm can be effectively used in practice as shown by the fact that it can compete with the algorithm of Dudzinski and Dydek =-=[7]-=- that is used as foundation of the merge_without_buffer function contained in the C++ Standard Template Libraries (STL) [8]. Significant older works in the area of in place merging are the publication... |

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Citation Context ...ntation aspects. Geffert et. all present in [3] a rather complex algorithm together with its asymptotic constants, but there are no notes regarding any successful implementation or benchmarking. Chen =-=[4]-=- simplified Geffert’s algorithm for the price of slightly worse asymptotic constants but also without any remarks about a concrete implementation. All three publications ⋆ This work was supported by t... |

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Citation Context ...totically optimal regarding the number of assignments, but lack in meeting the lower bound for comparisons. Another class of merging algorithms are the minimum storage algorithms presented in [7] and =-=[12]-=- which both rely on O(log 2 (m+n)) bits of extra storage. The latter two algorithms are effective in practice and simply structured, but they are not asymptotically optimal regarding the number of ass... |