Byzantine Lattice Agreement in Synchronous Systems

In this paper, we study the Byzantine lattice agreement problem in synchronous systems. The lattice agreement problem in crash failure model has been studied both in synchronous and asynchronous systems, which leads to the current best upper bound of $O(\log f)$ rounds in both systems. However, very few algorithmic results are known for the lattice agreement problem in Byzantine failure model. The paper [Nowak et al., DISC, 2019] first gives an algorithm for a variant of the lattice agreement problem on cycle-free lattices that tolerates up to $f < n/(h(X) + 1)$ Byzantine faults, where $n$ is the number of processes and $h(X)$ is the height of the input lattice $X$. The recent preprint by Di et al. studies this problem with a slightly modified validity condition in asynchronous systems. They present a $O(f)$ rounds algorithm by using the reliable broadcast primitive as a first step and following the similar algorithmic framework as the algorithms in crash failure model. In this paper, we propose three algorithms for the Byzantine lattice agreement problem in synchronous systems. The first algorithm takes $\min \{3h(X) + 6,6\sqrt{f} + 6\})$ rounds and $O(n^2 \min\{h(X), \sqrt{f}\})$ messages, where $h(X)$ is the height of the input lattice $X$, $n$ is the total number of processes. The second algorithm runs in $3\log n + 3$ rounds and takes $O(n^2 \log n)$ messages. The third algorithm takes $4 \log f + 3$ rounds and takes $O(n^2 \log f)$ messages. All algorithms can tolerate up to $f < \frac{n}{3}$ Byzantine failures

Amortized Constant Round Atomic Snapshot in Message-Passing Systems

We study the lattice agreement (LA) and atomic snapshot problems in asynchronous message-passing systems where up to $f$ nodes may crash. Our main result is a crash-tolerant atomic snapshot algorithm with \textit{amortized constant round complexity}. To the best of our knowledge, the best prior result is given by Delporte et al. [TPDS, 18] with amortized $O(n)$ complexity if there are more scans than updates. Our algorithm achieves amortized constant round if there are $\Omega(\sqrt{k})$ operations, where $k$ is the number of actual failures in an execution and is bounded by $f$. Moreover, when there is no failure, our algorithm has $O(1)$ round complexity unconditionally. To achieve amortized constant round complexity, we devise a simple \textit{early-stopping} lattice agreement algorithm and use it to "order" the update and scan operations for our snapshot object. Our LA algorithm has $O(\sqrt{k})$ round complexity. It is the first early-stopping LA algorithm in asynchronous systems

Transactions on Red-black and AVL trees in NVRAM

Byte-addressable non-volatile memory (NVRAM) supports persistent storage with low latency and high bandwidth. Complex data structures in it ought to be updated transactionally, so that they remain recoverable at all times. Traditional database technologies such as keeping a separate log, a journal, or shadow data work on a coarse-grained level, where the whole transaction is made visible using a final atomic update operation. These methods typically need significant additional space overhead and induce non-trivial overhead for log pruning, state maintenance, and resource (de-)allocation. Thus, they are not necessarily the best choice for NVRAM, which supports fine-grained, byte-addressable access. We present a generic transaction mechanism to update dynamic complex data structures `in-place' with a constant memory overhead. It is independent of the size of the data structure. We demonstrate and evaluate our approach on Red-Black Trees and AVL Trees with a redo log of constant size (4 resp. 2 cache lines). The redo log guarantees that each accepted (started) transaction is executed eventually despite arbitrary many system crashes and recoveries in the meantime. We update complex data structures in local and remote NVRAM providing exactly once semantics and durable linearizability for multi-reader single-writer access. To persist data, we use the available processor instructions for NVRAM in the local case and remote direct memory access (RDMA) combined with a software agent in the remote case