Rüdiger Reischuk

Renaming in an asynchronous environment

This paper is concerned with the solvability of the problem of processor renaming in unreliable, completely asynchronous distributed systems. Fischer et al. prove in [8] that “nontrivial consensus” cannot be attained in such systems, even when only a single, benign processor failure is possible. In contrast, this paper shows that problems of processor renaming can be solved even in the presence of up to <italic>t</italic> < <italic>n</italic>/2 faulty processors, contradicting the widely held belief that no nontrivial problem can be solved in such a system. The problems deal with <italic>renaming processors</italic> so as to reduce the size of the initial name space. When only uniqueness of the new names is required, we present a lower bound of <italic>n</italic> + 1 on the size of the new name space, and a renaming algorithm that establishes an upper bound on <italic>n</italic> + <italic>t</italic>. If the new names are required also to preserve the original order, a tight bound of 2′(<italic>n</italic> - <italic>t</italic> + 1) - 1 is obtained.

Probabilistic Parallel Algorithms for Sorting and Selection

Probabilistic parallel algorithms are described to sort n keys and to select the k-smallest element among them. For each problem we construct a probabilistic parallel decision tree. The tree for selection finishes with high probability in constant time and the sorting tree in time

On alternation

O(\log n)

A fast probabilistic parallel sorting algorithm

. The same time bound for sorting can also be achieved by a probabilistic parallel machine consisting of n RAMs, each with small private memory, and a common memory of size

A new solution for the byzantine generals problem

O(n)

Reliable computation with noisy circuits and decision trees-a general n log n lower bound

. These algorithms meet the information theoretic lower bounds.

Exact lower time bounds for computing Boolean functions on CREW PRAMs

SummaryEvery alternating t(n)-time bounded multitape Turing machine can be simulated by an alternating t(n)-time bounded 1-tape Turing machine. Every nondeterministic t(n)-time bounded 1-tape Turing machine can be simulated by an alternating (n + (t(n))1/2)-timebounded 1-tape Turing machine. For wellbehaved functions t(n) every nondeterministic t(n)-time bounded 1-tape Turing machine can be simulated by a deterministic ((nlogn)1/2 + (t(n))1/2)-tape bounded off-line Turing machine. These results improve or extend results by Chandra-Stockmeyer, Lipton-Tarjan and Paterson.

On alternation II

We describe a probabilistic parallel algorithm to sort n keys drawn from some arbitrary total ordered set. This algorithm can be implemented on a parallel computer consisting of n RAMs, each with small private memory, and a common memory of size O(n) such that the average runtime is bounded by O(log n). Hence for this algorithm the product of time and number of processors meets the information theoretic lower bound for sorting.

On different modes of communication

We define a new model for algorithms to reach Byzantine Agreement. It allows to measure the complexity more accurately, to differentiate between processor faults and to include communication link failures. A deterministic algorithm is presented that exhibits early stopping by phase 2f+4 in the worst case, where f is the actual number of faults, under less stringent conditions than the ones of previous algorithms. Also its average performance can easily be analysed making realistic assumptions on random distributions of faults. We show that it stops with high probability after a small number of phases.

The Sublogarithmic Alternating Space World

Boolean circuits in which gates independently make errors with probability (at most) epsilon are considered. It is shown that the critical number crit(f) of a function f yields lower bound Omega (crit(f) log crit (f)) for the noisy circuit size. The lower bound is proved for an even stronger computational model, static Boolean decision trees with erroneous answers. A decision tree is static if the questions it asks do not depend on previous answers. The depth of such a tree provides a lower bound on the number of gates that depend directly on some input and hence on the size of a noisy circuit. Furthermore, it is shown that an Omega (n log n) lower bound holds for almost all Boolean n-input functions with respect to the depth of noisy dynamic decision trees. This bound is the best possible and implies that almost all n-input Boolean functions have noisy decision tree complexity Theta (n log n) in the static as well as in the dynamic case.<<ETX>>