We consider the problem of performing a random walk in a distributed network. Given bandwidth constraints, the goal of the problem is to minimize the number of rounds required to obtain a random walk sample. Das Sarma et al. [PODC'10] show that a random walk of length $\ell$ on a network of diameter $D$ can be performed in $\tilde O(\sqrt{\ell D}+D)$ time. A major question left open is whether there exists a faster algorithm, especially whether the multiplication of $\sqrt{\ell}$ and $\sqrt{D}$ is necessary.
In this paper, we show a tight unconditional lower bound on the time complexity of distributed random walk computation. Specifically, we show that for any $n$, $D$, and $D\leq \ell \leq (n/(D^3\log n))^{1/4}$, performing a random walk of length $Θ(\ell)$ on an $n$-node network of diameter $D$ requires $Ω(\sqrt{\ell D}+D)$ time. This bound is {\em unconditional}, i.e., it holds for any (possibly randomized) algorithm. To the best of our knowledge, this is the first lower bound that the diameter plays a role of multiplicative factor. Our bound shows that the algorithm of Das Sarma et al. is time optimal.
Our proof technique introduces a new connection between {\em bounded-round} communication complexity and distributed algorithm lower bounds with $D$ as a trade-off parameter, strengthening the previous study by Das Sarma et al. [STOC'11]. In particular, we make use of the bounded-round communication complexity of the pointer chasing problem. Our technique can be of independent interest and may be useful in showing non-trivial lower bounds on the complexity of other fundamental distributed computing problems.
We study the problem of testing conductance in the setting of distributed computing and give a two-sided tester that takes $\mathcal{O}(\log(n) / (εΦ^2))$ rounds to decide if a graph has conductance at least $Φ$ or is $ε$-far from having conductance at least $Φ^2 / 1000$ in the distributed CONGEST model. We also show that $Ω(\log n)$ rounds are necessary for testing conductance even in the LOCAL model. In the case of a connected graph, we show that we can perform the test even when the number of vertices in the graph is not known a priori. This is the first two-sided tester in the distributed …
In this paper, we study the distributed sketching complexity of connectivity. In distributed graph sketching, an $n$-node graph $G$ is distributed to $n$ players such that each player sees the neighborhood of one vertex. The players then simultaneously send one message to the referee, who must compute some function of $G$ with high probability. For connectivity, the referee must output whether $G$ is connected. The goal is to minimize the message lengths. Such sketching schemes are equivalent to one-round protocols in the broadcast congested clique model. We prove that the expected average mes…
Ubiquitous sensing devices frequently disseminate data among them. The use of a distributed event-based system that decouples publishers from subscribers arises as an ideal candidate to implement the dissemination process. In this paper, we present a network architecture that merges the network and overlay layers of typical structured event-based systems. Directional random walks are used for the construction of this merged layer. Our strategy avoids using a specific network protocol that provides point-to-point communication. This implies that the topology of the network is not maintained, so…
We study the maximum $k$-set coverage problem in the following distributed setting. A collection of sets $S_1,\ldots,S_m$ over a universe $[n]$ is partitioned across $p$ machines and the goal is to find $k$ sets whose union covers the most number of elements. The computation proceeds in synchronous rounds. In each round, all machines simultaneously send a message to a central coordinator who then communicates back to all machines a summary to guide the computation for the next round. At the end, the coordinator outputs the answer. The main measures of efficiency in this setting are the approxi…
The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion of mixing of a random walk on a (undirected) graph, called local mixing. Informally, the local mixing with respect to a given node $s$, is the mixing of a random walk probability distribution restricted to a large enough subset of nodes --- say, a subset of size at least $n/β$ for a given parameter $β$ --- containing $s$. The time to mix over such a subset by…
Honest cooperation among individuals in a network can be achieved in different ways. In online networks with some kind of central authority, such as Ebay, Airbnb, etc. honesty is achieved through a reputation system, which is maintained and secured by the central authority. These systems usually rely on review mechanisms, through which agents can evaluate the trustworthiness of their interaction partners. These reviews are stored centrally and are tamper-proof. In decentralized peer-to-peer networks, enforcing cooperation turns out to be more difficult. One way of approaching this problem is b…
This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously O(m) messages and O($\sqrt$ n+D) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G's diameter. On the other hand, we show that any MST verification algorithm must send Ω(m) messages and incur Ω($\sqrt$ n + D) tim…
We consider a standard distributed optimisation setting where $N$ machines, each holding a $d$-dimensional function $f_i$, aim to jointly minimise the sum of the functions $\sum_{i = 1}^N f_i (x)$. This problem arises naturally in large-scale distributed optimisation, where a standard solution is to apply variants of (stochastic) gradient descent. We focus on the communication complexity of this problem: our main result provides the first fully unconditional bounds on total number of bits which need to be sent and received by the $N$ machines to solve this problem under point-to-point communic…
This paper demonstrates and proves that the coordination of actions in a distributed swarm can be enhanced by using quantum entanglement. In particular, we focus on - Global and local simultaneous random walks, using entangled qubits that collapse into the same (or opposite) direction, either random direction or totally controlled simultaneous movements. - Identifying eavesdropping from malicious eavesdroppers aimed at disturbing the simultaneous random walks by using entangled qubits that were sent at random or with predefined bases. - Identifying Byzantine robots or malicious robots that are…
We prove a lower bound of Omega(log n/loglog n) for the remote memory reference (RMR) complexity of abortable test-and-set (leader election) in the cache-coherent (CC) and the distributed shared memory (DSM) model. This separates the complexities of abortable and non-abortable test-and-set, as the latter has constant RMR complexity (Golab, Hendler, Woelfel, SIAM Journal of Computing Vol. 39, 2010). Golab, Hendler, Hadzilacos and Woelfel (Distributed Computing Vol. 25, 2012) showed that compare-and-swap can be implemented from registers and TAS objects with constant RMR complexity. We observe t…
The growing number of applications for distributed ledger technologies is driving both industry and academia to solve the limitations of blockchain, particularly its scalability issues. Recent distributed ledger technologies have replaced the blockchain linear structure with a more flexible directed acyclic graph in an attempt to accommodate a higher throughput. Despite the fast-growing diffusion of directed acyclic graph based distributed ledger technologies, researchers lack a basic understanding of their behavior. In this paper we analyze the Tangle, a directed acyclic graph that is used (w…
Personalized PageRank (PPR) has enormous applications, such as link prediction and recommendation systems for social networks, which often require the fully PPR to be known. Besides, most of real-life graphs are edge-weighted, e.g., the interaction between users on the Facebook network. However, it is computationally difficult to compute the fully PPR, especially on large graphs, not to mention that most existing approaches do not consider the weights of edges. In particular, the existing approach cannot handle graphs with billion edges on a moderate-size cluster. To address this problem, this…
It remains a challenging problem to tightly estimate the worst case response time of an application in a distributed embedded system, especially when there are dependencies between tasks. We discovered that the state-of-the art techniques considering task dependencies either fail to obtain a conservative bound or produce a loose upper bound. We propose a novel conservative performance analysis, called hybrid performance analysis, combining the response time analysis technique and the scheduling time bound analysis technique to compute a tighter bound fast. Through extensive experiments with ra…
