相关论文: Average-Case Complexity of Shellsort
We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is $\Omega (pn^{1 + 1/p})$ for every $p$. The proof…
We prove a lower bound expressed in the increment sequence on the average-case complexity of the number of inversions of Shellsort. This lower bound is sharp in every case where it could be checked. A special case of this lower bound yields…
The original Leapfrogging Samplesort operates on a sorted sample of size $s$ and an unsorted part of size $s+1$. We generalize this to a sorted sample of size $s$ and an unsorted part of size $(2^k-1)(s+1)$, where $k = O(1)$. We present a…
A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments $(h,g,1)$. In particular, when $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, the average running time is…
This paper studies the average complexity on the number of comparisons for sorting algorithms. Its information-theoretic lower bound is $n \lg n - 1.4427n + O(\log n)$. For many efficient algorithms, the first $n\lg n$ term is easy to…
Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard average-case analysis is ammenable to this method. Here we…
The average-case complexity of a branch-and-bound algorithms for Minimum Dominating Set problem in random graphs in the G(n,p) model is studied. We identify phase transitions between subexponential and exponential average-case complexities,…
Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its efficiency. The heart of the algorithm is the gap sequence chosen a priori and used during sorting. The selection of this gap…
In this paper, we describe randomized Shellsort--a simple, randomized, data-oblivious version of the Shellsort algorithm that always runs in O(n log n) time and, as we show, succeeds in sorting any given input permutation with very high…
We consider unidirectional data streams with restricted access, such as read-only and write-only streams. For read-write streams, we also introduce a new complexity measure called expansion, the ratio between the space used on the stream…
Sorting has a natural generalization where the input consists of: (1) a ground set $X$ of size $n$, (2) a partial oracle $O_P$ specifying some fixed partial order $P$ on $X$ and (3) a linear oracle $O_L$ specifying a linear order $L$ that…
In this master thesis we analyze the complexity of sorting a set of strings. It was shown that the complexity of sorting strings can be naturally expressed in terms of the prefix trie induced by the set of strings. The model of computation…
For any real number $p > 0$, we nearly completely characterize the space complexity of estimating $\|A\|_p^p = \sum_{i=1}^n \sigma_i^p$ for $n \times n$ matrices $A$ in which each row and each column has $O(1)$ non-zero entries and whose…
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…
We present numerical results for the probability of bad cases for Quicksort, i.e. cases of input data for which the sorting cost considerably exceeds that of the average. Dynamic programming was used to compute solutions of the recurrence…
Estimating the p-th frequency moment of data stream is a very heavily studied problem. The problem is actually trivial when p = 1, assuming the strict Turnstile model. The sample complexity of our proposed algorithm is essentially O(1) near…
Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quicksort…
In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\Omega\left(\epsilon^{-2/(3p+1)}\right)$…
The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand,…
Set cover, over a universe of size $n$, may be modelled as a data-streaming problem, where the $m$ sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only $O(n\, \mathrm{poly}\{\log n, \log…