Fast and compact self-stabilizing verification, computation, and fault detection of an MST

This paper demonstrates the usefulness of distributed local verification of proofs, as a tool for the design of self-stabilizing algorithms.In particular, it introduces a somewhat generalized notion of distributed local proofs, and utilizes it for improving the time complexity significantly, while maintaining space optimality. As a result, we show that optimizing the memory size carries at most a small cost in terms of time, in the context of Minimum Spanning Tree (MST). That is, we present algorithms that are both time and space efficient for both constructing an MST and for verifying it.This involves several parts that may be considered contributions in themselves.First, we generalize the notion of local proofs, trading off the time complexity for memory efficiency. This adds a dimension to the study of distributed local proofs, which has been gaining attention recently. Specifically, we design a (self-stabilizing) proof labeling scheme which is memory optimal (i.e., $O(\log n)$ bits per node), and whose time complexity is $O(\log ^2 n)$ in synchronous networks, or $O(\Delta \log ^3 n)$ time in asynchronous ones, where $\Delta$ is the maximum degree of nodes. This answers an open problem posed by Awerbuch and Varghese (FOCS 1991). We also show that $\Omega(\log n)$ time is necessary, even in synchronous networks. Another property is that if $f$ faults occurred, then, within the requireddetection time above, they are detected by some node in the $O(f\log n)$ locality of each of the faults.Second, we show how to enhance a known transformer that makes input/output algorithms self-stabilizing. It now takes as input an efficient construction algorithm and an efficient self-stabilizing proof labeling scheme, and produces an efficient self-stabilizing algorithm. When used for MST, the transformer produces a memory optimal self-stabilizing algorithm, whose time complexity, namely, $O(n)$, is significantly better even than that of previous algorithms. (The time complexity of previous MST algorithms that used $\Omega(\log^2 n)$ memory bits per node was $O(n^2)$, and the time for optimal space algorithms was $O(n|E|)$.) Inherited from our proof labelling scheme, our self-stabilising MST construction algorithm also has the following two properties: (1) if faults occur after the construction ended, then they are detected by some nodes within $O(\log ^2 n)$ time in synchronous networks, or within $O(\Delta \log ^3 n)$ time in asynchronous ones, and (2) if $f$ faults occurred, then, within the required detection time above, they are detected within the $O(f\log n)$ locality of each of the faults. We also show how to improve the above two properties, at the expense of some increase in the memory

Experiments towards model-based testing using Plan 9: Labelled transition file systems, stacking file systems, on-the-fly coverage measuring

We report on experiments that we did on Plan 9/Inferno to gain more experience with the file-system-as-tool-interface approach. We reimplemented functionality that we earlier worked on in Unix, trying to use Plan 9 file system interfaces. The application domain for those experiments was model-based testing.\ud \ud The idea we wanted to experiment with consists of building small, reusable pieces of functionality which are then composed to achieve the intended functionality. In particular we want to experiment with the idea of 'stacking' file servers (fs) on top of each other, where the upper fs acts as a 'filter' on the data and structure provided by the lower fs.\ud \ud For this experiment we designed a file system interface (ltsfs) that gives fine-grained access to a labelled transition system, and made two implementations of it.\ud We developed a small fs that, when 'stacked' on top of the ltsfs, extends it with additional files, and an application that uses the resulting file system.\ud \ud The hope was that an interface like the one offered by ltsfs could be used as a general interface between (specification language specific) programs that give access to state spaces and (specification language independent) programs that use (walk) those state spaces like simulators, model checkers, or test derivation programs.\ud \ud Initial results (obtained on a less-than-modern machine) suggest that, although the approach by itself is definitely feasible in principle, in practice the fine-grained access offered by ltsfs may involve many file (9p) transactions which may seriously affect performance. In Unix we used a more conservative approach where the access was less fine-grained which likely explains why there we did not suffer from this problem.\ud \ud In addition we report on experiments to use acid to obtain coverage information that is updated on-the-fly while the program is running. This worked quite well. The main observation from those experiments is that the basic block notion of this approach, which has a more 'semantical' nature, differs from the more 'syntactical' nature of the basic block notion in Unix coverage measurement tools\ud like tcov or gcov