Distributed MIS in O(log log n) Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best known (randomized) MIS algorithms have O(log n) round complexity on general graphs [Luby, STOC 1986] (where n is the number of nodes), while the best known lower bound is [EQUATION] [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the O(log n) round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with o(log n) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of O(log n) and also bypassing its fundamental [EQUATION] round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in O(log log n) awake complexity with high probability.1 However, this algorithm has a round complexity that is O(poly(n)). We then show how to drastically improve the round complexity at the cost of a slight increase in awake complexity by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability computes an MIS in O((log log n) log* n) awake complexity and O((log3 n)(log log n) log* n) round complexity. Our algorithms work in the CONGEST model where messages of size O(log n) bits can be sent per edge per round

Distributed MIS in O(log log n) Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best known (randomized) MIS algorithms have O(log n) round complexity on general graphs [Luby, STOC 1986] (where n is the number of nodes), while the best known lower bound is [EQUATION] [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the O(log n) round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with o(log n) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of O(log n) and also bypassing its fundamental [EQUATION] round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in O(log log n) awake complexity with high probability.1 However, this algorithm has a round complexity that is O(poly(n)). We then show how to drastically improve the round complexity at the cost of a slight increase in awake complexity by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability computes an MIS in O((log log n) log* n) awake complexity and O((log3 n)(log log n) log* n) round complexity. Our algorithms work in the CONGEST model where messages of size O(log n) bits can be sent per edge per round

Gossip in a Smartphone Peer-to-Peer Network

In this paper, we study the fundamental problem of gossip in the mobile telephone model: a recently introduced variation of the classical telephone model modified to better describe the local peer-to-peer communication services implemented in many popular smartphone operating systems. In more detail, the mobile telephone model differs from the classical telephone model in three ways: (1) each device can participate in at most one connection per round; (2) the network topology can undergo a parameterized rate of change; and (3) devices can advertise a parameterized number of bits about their state to their neighbors in each round before connection attempts are initiated. We begin by describing and analyzing new randomized gossip algorithms in this model under the harsh assumption of a network topology that can change completely in every round. We prove a significant time complexity gap between the case where nodes can advertise $0$ bits to their neighbors in each round, and the case where nodes can advertise $1$ bit. For the latter assumption, we present two solutions: the first depends on a shared randomness source, while the second eliminates this assumption using a pseudorandomness generator we prove to exist with a novel generalization of a classical result from the study of two-party communication complexity. We then turn our attention to the easier case where the topology graph is stable, and describe and analyze a new gossip algorithm that provides a substantial performance improvement for many parameters. We conclude by studying a relaxed version of gossip in which it is only necessary for nodes to each learn a specified fraction of the messages in the system.Comment: Extended Abstract to Appear in the Proceedings of the ACM Conference on the Principles of Distributed Computing (PODC 2017

Beeping a Deterministic Time-Optimal Leader Election

The beeping model is an extremely restrictive broadcast communication model that relies only on carrier sensing. In this model, we solve the leader election problem with an asymptotically optimal round complexity of O(D + log n), for a network of unknown size n and unknown diameter D (but with unique identifiers). Contrary to the best previously known algorithms in the same setting, the proposed one is deterministic. The techniques we introduce give a new insight as to how local constraints on the exchangeable messages can result in efficient algorithms, when dealing with the beeping model. Using this deterministic leader election algorithm, we obtain a randomized leader election algorithm for anonymous networks with an asymptotically optimal round complexity of O(D + log n) w.h.p. In previous works this complexity was obtained in expectation only. Moreover, using deterministic leader election, we obtain efficient algorithms for symmetry-breaking and communication procedures: O(log n) time MIS and 5-coloring for tree networks (which is time-optimal), as well as k-source multi-broadcast for general graphs in O(min(k,log n) * D + k log{(n M)/k}) rounds (for messages in {1,..., M}). This latter result improves on previous solutions when the number of sources k is sublogarithmic (k = o(log n))

On Query-To-Communication Lifting for Adversary Bounds

We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows: 1) We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 2) Turning to quantum models, we show a connection between lifting theorems for quantum adversary bounds and secure 2-party quantum computation in a certain "honest-but-curious" model. Under the assumption that such secure 2-party computation is impossible, we show that a simplified version of the positive-weight adversary bound lifts to a quantum communication lower bound using a constant-sized gadget. We also give an unconditional lifting theorem which lower bounds bounded-round quantum communication protocols. 3) Finally, we give some new results in query complexity. We show that the classical adversary and the positive-weight quantum adversary are quadratically related. We also show that the positive-weight quantum adversary is never larger than the square of the approximate degree. Both relations hold even for partial functions

Simple and Optimal Randomized Fault-Tolerant Rumor Spreading

We revisit the classic problem of spreading a piece of information in a group of $n$ fully connected processors. By suitably adding a small dose of randomness to the protocol of Gasienic and Pelc (1996), we derive for the first time protocols that (i) use a linear number of messages, (ii) are correct even when an arbitrary number of adversarially chosen processors does not participate in the process, and (iii) with high probability have the asymptotically optimal runtime of $O(\log n)$ when at least an arbitrarily small constant fraction of the processors are working. In addition, our protocols do not require that the system is synchronized nor that all processors are simultaneously woken up at time zero, they are fully based on push-operations, and they do not need an a priori estimate on the number of failed nodes. Our protocols thus overcome the typical disadvantages of the two known approaches, algorithms based on random gossip (typically needing a large number of messages due to their unorganized nature) and algorithms based on fair workload splitting (which are either not {time-efficient} or require intricate preprocessing steps plus synchronization).Comment: This is the author-generated version of a paper which is to appear in Distributed Computing, Springer, DOI: 10.1007/s00446-014-0238-z It is available online from http://link.springer.com/article/10.1007/s00446-014-0238-z This version contains some new results (Section 6