Distributed Strong Diameter Network Decomposition

For a pair of positive parameters $D,\chi$, a partition ${\cal P}$ of the vertex set $V$ of an $n$-vertex graph $G = (V,E)$ into disjoint clusters of diameter at most $D$ each is called a $(D,\chi)$ network decomposition, if the supergraph ${\cal G}({\cal P})$, obtained by contracting each of the clusters of ${\cal P}$, can be properly $\chi$-colored. The decomposition ${\cal P}$ is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diameter at most $D$, i.e., if for every cluster $C \in {\cal P}$ and every two vertices $u,v \in C$, the distance between them in the induced graph $G(C)$ of $C$ (resp., in $G$) is at most $D$. Network decomposition is a powerful construct, very useful in distributed computing and beyond. It was shown by Awerbuch \etal \cite{AGLP89} and Panconesi and Srinivasan \cite{PS92}, that strong $(2^{O(\sqrt{\log n})},2^{O(\sqrt{\log n})})$ network decompositions can be computed in $2^{O(\sqrt{\log n})}$ distributed time. Linial and Saks \cite{LS93} devised an ingenious randomized algorithm that constructs {\em weak} $(O(\log n),O(\log n))$ network decompositions in $O(\log^2 n)$ time. It was however open till now if {\em strong} network decompositions with both parameters $2^{o(\sqrt{\log n})}$ can be constructed in distributed $2^{o(\sqrt{\log n})}$ time. In this paper we answer this long-standing open question in the affirmative, and show that strong $(O(\log n),O(\log n))$ network decompositions can be computed in $O(\log^2 n)$ time. We also present a tradeoff between parameters of our network decomposition. Our work is inspired by and relies on the "shifted shortest path approach", due to Blelloch \etal \cite{BGKMPT11}, and Miller \etal \cite{MPX13}. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation

Uniting General-Graph and Geometric-Based Radio Networks via Independence Number Parametrization

In the study of radio networks, the tasks of broadcasting (propagating a message throughout the network) and leader election (having the network agree on a node to designate ‘leader’) are two of the most fundamental global problems, and have a long history of work devoted to them. This work has two divergent strands: some works focus on exploiting the geometric properties of wireless networks based in physical space, while others consider general graphs. Algorithmic results in each of these avenues have often used quite different techniques, and produced bounds using incomparable parametrizations. In this work, we unite the study of general-graph and geometric-based radio networks, by adapting the broadcast and leader election algorithm of Czumaj and Davies (JACM ’21) to achieve a running-time parametrized by the independence number of the network (i.e., the size of the maximum independent set). This parametrization preserves the running time on general graphs, matching the best known, but also improves running times to near-optimality across a wide range of geometric-based graph classes. As part of this algorithm, we also provide the first algorithm for computing a maximal independent set in general-graph radio networks. This algorithm runs in O(log3 n) time-steps, only a log n factor away from the Ω(log2 n) lower bound

The Weakest Failure Detector for Genuine Atomic Multicast

Atomic broadcast is a group communication primitive to order messages across a set of distributed processes. Atomic multicast is its natural generalization where each message m is addressed to dst(m), a subset of the processes called its destination group. A solution to atomic multicast is genuine when a process takes steps only if a message is addressed to it. Genuine solutions are the ones used in practice because they have better performance. Let ? be all the destination groups and ? be the cyclic families in it, that is the subsets of ? whose intersection graph is hamiltonian. This paper establishes that the weakest failure detector to solve genuine atomic multicast is ? = (?_{g,h ? ?} ?_{g ? h}) ? (?_{g ? ?} ?_g) ? ?, where ?_P and ?_P are the quorum and leader failure detectors restricted to the processes in P, and ? is a new failure detector that informs the processes in a cyclic family f ? ? when f is faulty. We also study two classical variations of atomic multicast. The first variation requires that message delivery follows the real-time order. In this case, ? must be strengthened with 1^{g ? h}, the indicator failure detector that informs each process in g ? h when g ? h is faulty. The second variation requires a message to be delivered when the destination group runs in isolation. We prove that its weakest failure detector is at least ? ? (?_{g, h ? ?} ?_{g ? h}). This value is attained when ? = ?

Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

Every Bit Counts in Consensus

Consensus enables n processes to agree on a common valid L-bit value, despite t < n/3 processes being faulty and acting arbitrarily. A long line of work has been dedicated to improving the worst-case communication complexity of consensus in partial synchrony. This has recently culminated in the worst-case word complexity of O(n^2). However, the worst-case bit complexity of the best solution is still O(n^2 L + n^2 kappa) (where kappa is the security parameter), far from the \Omega(n L + n^2) lower bound. The gap is significant given the practical use of consensus primitives, where values typically consist of batches of large size (L > n). This paper shows how to narrow the aforementioned gap while achieving optimal linear latency. Namely, we present a new algorithm, DARE (Disperse, Agree, REtrieve), that improves upon the O(n^2 L) term via a novel dispersal primitive. DARE achieves O(n^{1.5} L + n^{2.5} kappa) bit complexity, an effective sqrt{n}-factor improvement over the state-of-the-art (when L > n kappa). Moreover, we show that employing heavier cryptographic primitives, namely STARK proofs, allows us to devise DARE-Stark, a version of DARE which achieves the near-optimal bit complexity of O(n L + n^2 poly(kappa)). Both DARE and DARE-Stark achieve optimal O(n) latency

Simple and Efficient Single Round Almost Perfectly Secure Message Transmission Tolerating Generalized Adversary

Patra et al. gave a necessary and sufficient condition for the possibility of almost perfectly secure message transmission protocols tolerating general, non-threshold Q^2 adversary structure. However, their protocol requires at least three rounds and performs exponential (exponential in the size of the adversary structure) computation and communication. Moreover, they have left it as an open problem to design efficient protocol for almost perfectly secure message transmission, tolerating Q^2 adversary structure. In this paper, we show the first single round almost perfectly secure message transmission protocol tolerating Q^2 adversary structure. The computation and communication complexities of the protocol are both polynomial} in the size of underlying linear secret sharing scheme (LSSS) and adversary structure. This solves the open problem raised by Patra et al.. When we restrict our general protocol to threshold adversary with n=2t+1, we obtain a single round, communication optimal almost secure message transmission protocol tolerating threshold adversary, which is much more computationally efficient and relatively simpler than the previous communication optimal protocol of Srinathan et al

On the Complexity of Local Graph Transformations

We consider the problem of transforming a given graph G_s into a desired graph G_t by applying a minimum number of primitives from a particular set of local graph transformation primitives. These primitives are local in the sense that each node can apply them based on local knowledge and by affecting only its 1-neighborhood. Although the specific set of primitives we consider makes it possible to transform any (weakly) connected graph into any other (weakly) connected graph consisting of the same nodes, they cannot disconnect the graph or introduce new nodes into the graph, making them ideal in the context of supervised overlay network transformations. We prove that computing a minimum sequence of primitive applications (even centralized) for arbitrary G_s and G_t is NP-hard, which we conjecture to hold for any set of local graph transformation primitives satisfying the aforementioned properties. On the other hand, we show that this problem admits a polynomial time algorithm with a constant approximation ratio