Analyzing the average-case complexity of algorithms is a very practical but very difficult problem in computer science. In the past few years, we have demonstrated that Kolmogorov complexity is an important tool for analyzing the average-case complexity of algorithms. We have developed the incompressibility method [7]. In this paper, we use several simple examples to further demonstrate the power and simplicity of such method. We prove bounds on the average-case number of stacks (queues) required for sorting sequential or parallel Queueusort or Stacksort.

When someone designs a new data structure, they want to know how well it performs. Previously, the only way to do this involves finding, coding and testing some applications to act as benchmarks. This can be tedious and time-consuming. Worse, how a benchmark uses a data structure may considerably affect the efficiency of the data structure. Thus, the choice of benchmarks may bias the results. For these reasons, new data structures developed for functional languages often pay little attention to empirical performance. We solve these problems by developing a benchmarking tool, Auburn, that can generate benchmarks across a fair distribution of uses. We precisely define "the use of a data structure", upon which we build the core algorithms of Auburn: how to generate a benchmark from a description of use, and how to extract a description of use from an application. We consider how best to use these algorithms to benchmark competing data structures. Finally, we test Auburn by benchmarking ...

Abstract not found

We reinvestigate the oblivious RAM concept introduced by Goldreich and Ostrovsky, which enables a client, that can store locally only a constant amount of data, to store remotely n data items, and access them while hiding the identities of the items which are being accessed. Oblivious RAM is often cited as a powerful tool, which can be used, for example, for search on encrypted data or for preventing cache attacks. However, oblivious RAM it is also commonly considered to be impractical due to its overhead, which is asymptotically efficient but is quite high: each data request is replaced by O(log 4 n) requests, or by O(log 3 n) requests where the constant in the “O ” notation is a few thousands. In addition, O(n log n) external memory is required in order to store the n data items. We redesign the oblivious RAM protocol using modern tools, namely Cuckoo hashing and a new oblivious sorting algorithm. The resulting protocol uses only O(n) external memory, and replaces each data request by only O(log 2 n) requests (with a small constant). This analysis is validated by experiments that we ran. Keywords: Secure two-party computation, oblivious RAM.

Implementations of map-reduce are being used to perform many operations on very large data. We explore alternative ways that a system could use the environment and capabilities of map-reduce implementations such as Hadoop, yet perform operations that are not identical to map-reduce. In particular, we look at strategies for taking the join of several relations and sorting large sets. The centerpiece of this exploration is a computational model that captures the essentials of the environment in which systems like Hadoop operate. Files are unordered sets of tuples that can be read and/or written in parallel; processes are limited in the amount of input/output they can perform, and processors are available in essentially unlimited supply. In our study, we focus on communication among processes and processing time costs, both total and elapsed. We show tradeoffs among them depending on the computational limits we invoke on the processes. 1.

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.

Sorting is one of the most important and well studied problems in Computer Science. Many good algorithms are known which offer various trade-offs in efficiency, simplicity, memory use, and other factors. However, these algorithms do not take into account features of modern computer architectures that significantly influence performance. Caches and branch predictors are two such features, and while there has been a significant amount of research into the cache performance of general purpose sorting algorithms, there has been little research on their branch prediction properties. In this paper we empirically examine the behaviour of the branches in all the most common sorting algorithms. We also consider the interaction of cache optimization on the predictability of the branches in these algorithms. We find insertion sort to have the fewest branch mispredictions of any comparison-based sorting algorithm, that bubble and shaker sort operate in a fashion which makes their branches highly unpredictable, that the unpredictability of shellsort’s branches improves its caching behaviour and that several cache optimizations have little effect on mergesort’s branch mispredictions. We find also that optimizations to quicksort – for example the choice of pivot – have a strong influence on the predictability of its branches. We point out a simple way of removing branch instructions from a classic heapsort implementation, and show also that unrolling a loop in a cache optimized heapsort implementation improves the predicitability of its branches. Finally, we note that when sorting random data two-level adaptive branch predictors are usually no better than simpler bimodal predictors. This is despite the fact that two-level adaptive predictors are almost always superior to bimodal predictors in general.

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 survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewicz-sort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks.

We analyze the (3, 2, 1)-Shell Sort algorithm under the usual random permutation model.

This work describes our approach to creating a fast and low-cost sorting system. The goal of this work is to win the 2002 PennySort and Performance/Price sort. We have designed a sorting program called DMSort that is capable of more than double the performance of previously published results when run on our system configuration. This paper discusses the DMSort system alongside a discussion of topics relevant to PennySort and Performance/Price sort. In particular, the DMSort sorting algorithm, hardware system, system implementation, and performance characteristics are discussed in detail.