149,271 research outputs found
On the average running time of odd-even merge sort
This paper is concerned with the average running time of Batcher's odd-even merge sort when implemented on a collection of processors. We consider the case where , the size of the input, is an arbitrary multiple of the number of processors used. We show that Batcher's odd-even merge (for two sorted lists of length each) can be implemented to run in time on the average, and that odd-even merge sort can be implemented to run in time on the average. In the case of merging (sorting), the average is taken over all possible outcomes of the merging (all possible permutations of elements). That means that odd-even merge and odd-even merge sort have an optimal average running time if . The constants involved are also quite small
Zig-zag Sort: A Simple Deterministic Data-Oblivious Sorting Algorithm Running in O(n log n) Time
We describe and analyze Zig-zag Sort--a deterministic data-oblivious sorting
algorithm running in O(n log n) time that is arguably simpler than previously
known algorithms with similar properties, which are based on the AKS sorting
network. Because it is data-oblivious and deterministic, Zig-zag Sort can be
implemented as a simple O(n log n)-size sorting network, thereby providing a
solution to an open problem posed by Incerpi and Sedgewick in 1985. In
addition, Zig-zag Sort is a variant of Shellsort, and is, in fact, the first
deterministic Shellsort variant running in O(n log n) time. The existence of
such an algorithm was posed as an open problem by Plaxton et al. in 1992 and
also by Sedgewick in 1996. More relevant for today, however, is the fact that
the existence of a simple data-oblivious deterministic sorting algorithm
running in O(n log n) time simplifies the inner-loop computation in several
proposed oblivious-RAM simulation methods (which utilize AKS sorting networks),
and this, in turn, implies simplified mechanisms for privacy-preserving data
outsourcing in several cloud computing applications. We provide both
constructive and non-constructive implementations of Zig-zag Sort, based on the
existence of a circuit known as an epsilon-halver, such that the constant
factors in our constructive implementations are orders of magnitude smaller
than those for constructive variants of the AKS sorting network, which are also
based on the use of epsilon-halvers.Comment: Appearing in ACM Symp. on Theory of Computing (STOC) 201
Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker
Since the proof of the four color theorem in 1976, computer-generated proofs
have become a reality in mathematics and computer science. During the last
decade, we have seen formal proofs using verified proof assistants being used
to verify the validity of such proofs.
In this paper, we describe a formalized theory of size-optimal sorting
networks. From this formalization we extract a certified checker that
successfully verifies computer-generated proofs of optimality on up to 8
inputs. The checker relies on an untrusted oracle to shortcut the search for
witnesses on more than 1.6 million NP-complete subproblems.Comment: IMADA-preprint-c
Feedback Controlled Software Systems
Software systems generally suffer from a certain fragility in the face of disturbances such as bugs, unforeseen user input, unmodeled interactions with other software components, and so on. A single such disturbance can make the machine on which the software is executing hang or crash. We postulate that what is required to address this fragility is a general means of using feedback to stabilize these systems. In this paper we develop a preliminary dynamical systems model of an arbitrary iterative software process along with the conceptual framework for stabilizing it in the presence of disturbances. To keep the computational requirements of the controllers low, randomization and approximation are used. We describe our initial attempts to apply the model to a faulty list sorter, using feedback to improve its performance. Methods by which software robustness can be enhanced by distributing a task between nodes each of which are capable of selecting the best input to process are also examined, and the particular case of a sorting system consisting of a network of partial sorters, some of which may be buggy or even malicious, is examined
- …