Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and a reversal takes an arbitrary substring of elements and reverses their order. For this problem we develop two algorithms: a greedy approximation algorithm that finds a solution provably close to optimal in O(n 2 ) time and O(n) space for an n element permutation, and a branch and bound exact algorithm that finds an optimal solution in O(mL(n;n)) time and O(n 2 ) space, where m is the size of the branch and bound search tree and L(n; n) is the time to solve a linear program of n variables and n constraints. The greedy algorithm is the first to come within a constant factor of the optimum; it guarantees a solution that uses no more than twice the min...

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

There is an upsurge in interest in the Markov model and also more general stationary ergodic stochastic distributions in theoretical computer science community recently (e.g. see [Vitter,KrishnanSl], [Karlin,Philips,Raghavan92], [Raghavan9 for use of Markov models for on-line algorithms, e.g., cashing and prefetching). Their results used the fact that compressible sources are predictable (and vise versa), and showed that on-line algorithms can improve their performance by prediction. Actual page access sequences are in fact somewhat compressible, so their predictive methods can be of benefit. This paper investigates the interesting idea of decreasing computation by using learning in the opposite way, namely to determine the difficulty of prediction. That is, we will ap proximately learn the input distribution, and then improve the performance of the computation when the input is not too predictable, rather than the reverse. To our knowledge,

We explore various techniques to compress a permutation π over n integers, taking advantage of ordered subsequences in π, while supporting its application π(i) and the application of its inverse π −1 (i) in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications π k (i) of it, of integer functions, and of inverted lists and suffix arrays.

This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts

: 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...

Number systems serve admirably as templates for container types: a container object of size n is modelled after the representation of the number n and operations on container objects are modelled after their number-theoretic counterparts. Binomial queues are probably the first data structure that was designed with this analogy in mind. In this paper we show how to express these so-called numerical representations as higher-order nested datatypes. A nested datatype allows to capture the structural invariants of a numerical representation, so that the violation of an invariant can be detected at compile-time. We develop a programming method which allows to adapt algorithms to the new representation in a mostly straightforward manner. The framework is employed to implement three different container types: binary random-access lists, binomial queues, and 2-3 finger search trees. The latter data structure, which is treated in some depth, can be seen as the main innovation from a data-struct...

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.

is applied to the task of evolving general recursive sorting algorithms. We studied the effects of language primitives and fitness functions on the success of the evolutionary process. For language primitives, these were the methods of a simple list processing package. Five different fitness functions based on sequence disorder were evaluated. The time complexity of the successfully evolved algorithms was measured experimentally in terms of the number of method invocations made, and for the best evolved individuals this was best approximated as O(n × log(n)). This is the first time that sorting algorithms of this complexity have been evolved. I.

A sorting algorithm is adaptive if its run time for inputs of the same size n varies smoothly from O(n) to O(n log n) as the disorder of the input varies. It is well accepted that files that are already sorted are often sorted again and that many files occur naturally in nearly sorted state. Recently, researchers have focused their attention on sorting algorithms that are optimally adaptive with respect to several measures of disorder, (since the type of disorder in the input is unknown), and illustrating a need to develop tools for constructing adaptive algorithms for large classes of measures. We present a generic sorting algorithm that uses divide-and-conquer in which the number of subproblems depends on the disorder of the input and for which we can establish adaptivity with respect to an abstract measure. We present applications of this generic algorithm obtaining optimal adaptivity for several specific measures of disorder. Moreover, we define a randomized version of our generic ...