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

Tight Mobile Byzantine Tolerant Atomic Storage

This paper proposes the first implementation of an atomic storage tolerant to mobile Byzantine agents. Our implementation is designed for the round-based synchronous model where the set of Byzantine nodes changes from round to round. In this model we explore the feasibility of multi-writer multi-reader atomic register prone to various mobile Byzantine behaviors. We prove upper and lower bounds for solving the atomic storage in all the explored models. Our results, significantly different from the static case, advocate for a deeper study of the main building blocks of distributed computing while the system is prone to mobile Byzantine failures

Distributed Exact Shortest Paths in Sublinear Time

The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in $O(n)$ time, where $n$ is the number of vertices in the input graph $G$. Peleg and Rubinovich (FOCS'99) showed a lower bound of $\tilde{\Omega}(D + \sqrt{n})$ for this problem, where $D$ is the hop-diameter of $G$. Whether or not this problem can be solved in $o(n)$ time when $D$ is relatively small is a major notorious open question. Despite intensive research \cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires $O((n \log n)^{5/6})$ time, for $D = O(\sqrt{n \log n})$, and $O(D^{1/3} \cdot (n \log n)^{2/3})$ time, for larger $D$. This running time is sublinear in $n$ in almost the entire range of parameters, specifically, for $D = o(n/\log^2 n)$. For the all-pairs shortest paths problem, our algorithm requires $O(n^{5/3} \log^{2/3} n)$ time, regardless of the value of $D$. We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset $G"$ of a skeleton graph $G'$ of $G$ without first computing $G'$ itself. We then conduct a Bellman-Ford exploration in $G' \cup G"$, while computing the required edges of $G'$ on the fly. As a result, our algorithm computes exactly those edges of $G'$ that it really needs, rather than computing approximately the entire $G'$

Positional Encoding by Robots with Non-Rigid Movements

Consider a set of autonomous computational entities, called \emph{robots}, operating inside a polygonal enclosure (possibly with holes), that have to perform some collaborative tasks. The boundary of the polygon obstructs both visibility and mobility of a robot. Since the polygon is initially unknown to the robots, the natural approach is to first explore and construct a map of the polygon. For this, the robots need an unlimited amount of persistent memory to store the snapshots taken from different points inside the polygon. However, it has been shown by Di Luna et al. [DISC 2017] that map construction can be done even by oblivious robots by employing a positional encoding strategy where a robot carefully positions itself inside the polygon to encode information in the binary representation of its distance from the closest polygon vertex. Of course, to execute this strategy, it is crucial for the robots to make accurate movements. In this paper, we address the question whether this technique can be implemented even when the movements of the robots are unpredictable in the sense that the robot can be stopped by the adversary during its movement before reaching its destination. However, there exists a constant $\delta > 0$, unknown to the robot, such that the robot can always reach its destination if it has to move by no more than $\delta$ amount. This model is known in literature as \emph{non-rigid} movement. We give a partial answer to the question in the affirmative by presenting a map construction algorithm for robots with non-rigid movement, but having $O(1)$ bits of persistent memory and ability to make circular moves

Stable Leader Election in Population Protocols Requires Linear Time

A population protocol *stably elects a leader* if, for all $n$, starting from an initial configuration with $n$ agents each in an identical state, with probability 1 it reaches a configuration $\mathbf{y}$ that is correct (exactly one agent is in a special leader state $\ell$) and stable (every configuration reachable from $\mathbf{y}$ also has a single agent in state $\ell$). We show that any population protocol that stably elects a leader requires $\Omega(n)$ expected "parallel time" --- $\Omega(n^2)$ expected total pairwise interactions --- to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.Comment: accepted to Distributed Computing special issue of invited papers from DISC 2015; significantly revised proof structure and intuitive explanation

Deterministic Digital Clustering of Wireless Ad Hoc Networks

We consider deterministic distributed communication in wireless ad hoc networks of identical weak devices under the SINR model without predefined infrastructure. Most algorithmic results in this model rely on various additional features or capabilities, e.g., randomization, access to geographic coordinates, power control, carrier sensing with various precision of measurements, and/or interference cancellation. We study a pure scenario, when no such properties are available. As a general tool, we develop a deterministic distributed clustering algorithm. Our solution relies on a new type of combinatorial structures (selectors), which might be of independent interest. Using the clustering, we develop a deterministic distributed local broadcast algorithm accomplishing this task in $O(\Delta \log^*N \log N)$ rounds, where $\Delta$ is the density of the network. To the best of our knowledge, this is the first solution in pure scenario which is only polylog$(n)$ away from the universal lower bound $\Omega(\Delta)$, valid also for scenarios with randomization and other features. Therefore, none of these features substantially helps in performing the local broadcast task. Using clustering, we also build a deterministic global broadcast algorithm that terminates within $O(D(\Delta + \log^* N) \log N)$ rounds, where $D$ is the diameter of the network. This result is complemented by a lower bound $\Omega(D \Delta^{1-1/\alpha})$, where $\alpha > 2$ is the path-loss parameter of the environment. This lower bound shows that randomization or knowledge of own location substantially help (by a factor polynomial in $\Delta$) in the global broadcast. Therefore, unlike in the case of local broadcast, some additional model features may help in global broadcast