We present a compositional program logic for call-by-value imperative higherorder functions with general forms of aliasing, which can arise from the use of reference names as function parameters, return values, content of references and part of data structures. The program logic

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.

1.2 Algorithms Should Be Cache-Conscious......................... 2

We introduce the notion of discrimination as a generalization of both sorting and partitioning and show that worst-case linear-time discrimination functions (discriminators) can be defined generically, by (co-)induction on an expressive language of order denotations. The generic definition yields discriminators that generalize both distributive sorting and multiset discrimination. The generic discriminator can be coded compactly using list comprehensions, with order denotations specified using Generalized Algebraic Data Types (GADTs). A GADT-free combinator formulation of discriminators is also given. We give some examples of the uses of discriminators, including a new most-significant-digit lexicographic sorting algorithm. Discriminators generalize binary comparison functions: They operate on n arguments at a time, but do not expose more information than the underlying equivalence, respectively ordering relation on the arguments. We argue that primitive types with equality (such as references in ML) and ordered types (such as the machine integer type), should expose their equality, respectively standard ordering relation, as discriminators: Having only a binary equality test on a type requires Θ(n 2) time to find all the occurrences of an element in a list of length n, for each element in the list, even if the equality test takes only constant time. A discriminator accomplishes this in linear time. Likewise, having only a (constant-time) comparison function requires Θ(n log n) time to sort a list of n elements. A discriminator can do this in linear time.

[2] Rodney M. Burstall. Some techniques for proving correctness of programs

Sorting is one of the most important and studied problems in computer science. Many good algorithms exist which offer various trade-offs in efficiency, simplicity and memory use. However most of these algorithms were discovered decades ago at a time when computer architectures were much simpler than today. Branch prediction and cache memories are two developments in computer architecture that have a particularly large impact on the performance of sorting algorithms. This report describes a study of the behaviour of sorting algorithms on branch predictors and caches. Our work on branch prediction is almost entirely new, and finds a number of important results. In particular we show that insertion sort causes the fewest branch mispredictions of any comparison-based algorithm, that optimizations- such as the choice of the pivot in quicksort- can have a large impact on the predictability of branches, and that advanced two-level branch predictors are usually worse at predicting branches in sorting algorithms than simpler branch predictors. In many cases it is possible to draw links between classical theoretical analyses of algorithms and their branch prediction behaviour. The other main work described in this report is an analysis of the behaviour of sorting algorithms on modern caches. Over the last decade there has been considerable interest in optimizing sorting algorithms to reduce the number of cache misses. We experimentally study the cache performance of both classical