Introduction and Survey; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems --- Sorting and Searching; E.5 [Data]: Files --- Sorting/searching; G.3 [Mathematics of Computing]: Probability and Statistics --- Probabilistic algorithms; E.2 [Data Storage Representation]: Composite structures, linked representations. General Terms: Algorithms, Theory. Additional Key Words and Phrases: Adaptive sorting algorithms, Comparison trees, Measures of disorder, Nearly sorted sequences, Randomized algorithms. A Survey of Adaptive Sorting Algorithms 2 CONTENTS INTRODUCTION I.1 Optimal adaptivity I.2 Measures of disorder I.3 Organization of the paper 1.WORST-CASE ADAPTIVE (INTERNAL) SORTING ALGORITHMS 1.1 Generic Sort 1.2 Cook--Kim division 1.3 Partition Sort 1.4 Exponential Search 1.5 Adaptive Merging 2.EXPECTED-CASE ADAPTIV

This paper summarizes its evolution

Black-box optimization algorithms cannot use the specific parameters of the problem instance, i.e., of the fitness function f. Their run time is measured as the number of f-evaluations. This implies that the usual algorithmic complexity of a problem cannot be used in the black-box scenario. Therefore, a new framework for the valuation of algorithms for black-box optimization is presented allowing the notion of the black-box complexity of a problem. For several problems upper and lower bounds on their black-box complexity are presented. Moreover, it can can be concluded that randomized search heuristics whose (worst-case) expected optimization time for some problem is close to the black-box complexity of the problem are provably efficient (in the black-box scenario). The new approach is applied to several problems based on typical example functions and further interesting problems. Run times of general EAs for these problems are compared with the black-box complexity of the problem.

Writing portable programs that perform well on multiple platforms or for varying input sizes and types can be very difficult because performance is often sensitive to the system architecture, the runtime environment, and input data characteristics. This is even more challenging on parallel and distributed systems due to the wide variety of system architectures. One way to address this problem is to adaptively select the best parallel algorithm for the current input data and system from a set of functionally equivalent algorithmic options. Toward this goal, we have developed a general framework for adaptive algorithm selection for use in the Standard Template Adaptive Parallel Library (STAPL). Our framework uses machine learning techniques to analyze data collected by STAPL installation benchmarks and to determine tests that will select among algorithmic options at run-time. We apply a prototype implementation of our framework to two important parallel operations, sorting and matrix multiplication, on multiple platforms and show that the framework determines run-time tests that correctly select the best performing algorithm from among several competing algorithmic options in 86-100 % of the cases studied, depending on the operation and the system.

: We describe a sorting algorithm that is optimally adaptive with respect to several important measures of presortedness. In particular, the algorithm requires O(n+k log k) time on n-sequences X that have a longest ascending subsequence of length n \Gamma k and for which Rem(X) = k; O(n log(k=n)) time on sequences with k inversions; and O(n log k) time on sequences that can be decomposed into k monotone shuffles. The algorithm makes use of an adaptive merging operation that can be implemented using finger search trees. 1 Introduction An adaptive algorithm is one which requires fewer resources to solve `easy' problem instances than it does to solve `hard'. For sorting an adaptive algorithm should run in O(n) time if presented with a sorted n-sequence, and in O(n log n) time for all n- sequences, with the time for any particular sequence depending upon the `nearness' of the sequence to being sorted. Mannila [7] established the notion of a measure of presortedness to quantify the disord...

Quicksort was first introduced in 1961 by Hoare. Many variants have been developed, the best of which are among the fastest generic sorting algorithms available, as testified by the choice of Quicksort as the default sorting algorithm in most programming libraries. Some sorting algorithms are adaptive, i.e. they have a complexity analysis which is better for inputs which are nearly sorted, according to some specified measure of presortedness. Quicksort is not among these, as it uses Ω(n log n) comparisons even when the input is already sorted. However, in this paper we demonstrate empirically that the actual running time of Quicksort is adaptive with respect to the presortedness measure Inv. Differences close to a factor of two are observed between instances with low and high Inv value. We then show that for the randomized version of Quicksort, the number of element swaps performed is provably adaptive with respect to the measure Inv. More precisely, we prove that randomized Quicksort performs expected O(n(1+log(1+ Inv/n))) element swaps, where Inv denotes the number of inversions in the input sequence. This result provides a theoretical explanation for the observed behavior, and gives new insights on the behavior of the Quicksort algorithm. We also give some empirical results on the adaptive behavior of Heapsort and Mergesort.

We introduce the concept of presorting algorithms, quantifying and evaluating the performance of such algorithms with the average reduction in number of inversions. Stages of well-known algorithms such as Shellsort and quicksort are evaluated in such a framework and shown to cause a meaning drop in the inversion statistic. The expected value, variance and generating function for the decrease in number of inversions are computed. The possibility of "presorting" a sorting algorithm is also investigated under a similar framework.

We give the first sorting algorithm with bounds in terms of higher-order entropies: let S be a sequence of length m containing n distinct elements and let H # (S) be the #th-order empirical entropy of S, log n # O(m); our algorithm sorts S using (H # (S) + O(1))m comparisons.