QuickXsort: A Fast Sorting Scheme in Theory and Practice

Edelkamp, Stefan; Weiß, Armin; Wild, Sebastian

doi:10.1007/s00453-019-00634-0

Cited by 7 publications

(18 citation statements)

References 39 publications

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“…However, the equation from Theorem 2 is hard to work with, so we approximate it with the binomial distribution p elements less compared to the worst case. Combining that with the bounds from [4] we obtain Theorem 3. The complete proof is given in Appendix B.3.…”

Section: Improved Upper Bounds For Mergeinsertionmentioning

confidence: 72%

“…However, since we are looking at the On the Average Case of MergeInsertion 25 average case we need to consider that in some cases b i can be inserted using just k i − 1 comparisons. This is reflected by α m and β k , the first of which has already been studied by [4].…”

Section: Florian Stober and Armin Weißmentioning

confidence: 96%

“…The worst case is near x = 0.6 (i. e., n roughly a power of two) where c(x) is just slightly larger than 1.4005. The proof of Theorem 3 analyzes the insertion of one batch of elements more carefully than in [4]. The exact probability that b t k−1 +i is inserted into j elements is given by Theorem 2.…”

Section: Improved Upper Bounds For Mergeinsertionmentioning

confidence: 99%

“…We calculate the number of comparisons required in the average case in a similar fashion to [4]. Recall Eq.…”

Section: Florian Stober and Armin Weißmentioning

confidence: 99%

“…To estimate the cost of an insertion we use the formula T InsAvg (m) = ⌈log m⌉+ 1 − 2 ⌈log m⌉ m by [4]. Technically this formula is only correct if the probability of an element being inserted is the same for each position.…”

Section: Florian Stober and Armin Weißmentioning

confidence: 99%

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On the Average Case of MergeInsertion

Stober

Weiß

2019

Lecture Notes in Computer Science

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MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of n log n − 1.4005n + o(n) comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for n up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to the recent combination with (1,2)-Insertionsort by Iwama and Teruyama.

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Section: Improved Upper Bounds For Mergeinsertionmentioning

confidence: 72%

Section: Florian Stober and Armin Weißmentioning

confidence: 96%

Section: Improved Upper Bounds For Mergeinsertionmentioning

confidence: 99%

“…We calculate the number of comparisons required in the average case in a similar fashion to [4]. Recall Eq.…”

Section: Florian Stober and Armin Weißmentioning

confidence: 99%

Section: Florian Stober and Armin Weißmentioning

confidence: 99%

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On the Average Case of MergeInsertion

Stober

Weiß

2019

Lecture Notes in Computer Science

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On the Average Case of MergeInsertion

Stober

Weiß

2020

Theory Comput Syst

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MergeInsertion, also known as the Ford-Johnson algorithm, is a sorting algorithm which, up to today, for many input sizes achieves the best known upper bound on the number of comparisons. Indeed, it gets extremely close to the information-theoretic lower bound. While the worst-case behavior is well understood, only little is known about the average case. This work takes a closer look at the average case behavior. In particular, we establish an upper bound of $n \log n - 1.4005n + o(n)$ n log n − 1.4005 n + o ( n ) comparisons. We also give an exact description of the probability distribution of the length of the chain a given element is inserted into and use it to approximate the average number of comparisons numerically. Moreover, we compute the exact average number of comparisons for n up to 148. Furthermore, we experimentally explore the impact of different decision trees for binary insertion. To conclude, we conduct experiments showing that a slightly different insertion order leads to a better average case and we compare the algorithm to Manacher’s combination of merging and MergeInsertion as well as to the recent combined algorithm with (1,2)-Insertionsort by Iwama and Teruyama.

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