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We study scheduling mechanisms that explore the trade-off between containing the spread of COVID-19 and performing in-person activity in organizations. Our mechanisms, referred to as group scheduling , are based on partitioning the population randomly into groups and scheduling each group on appropriate days with possible gaps (when no one is working and all are quarantined). Each group interacts with no other group and, importantly, any person who is symptomatic in a group is quarantined. We show that our mechanisms effectively trade-off in-person activity for more effective control of the COVID-19 virus spread. In particular, we show that a mechanism which partitions the population into two groups that alternatively work in-person for five days each, flatlines the number of COVID-19 cases quite effectively, while still maintaining in-person activity at 70% of pre-COVID-19 level. Other mechanisms that partitions into two groups with less continuous work days or more spacing or three groups achieve even more aggressive control of the virus at the cost of a somewhat lower in-person activity (about 50%). We demonstrate the efficacy of our mechanisms by theoretical analysis and extensive experimental simulations on various epidemiological models based on real-world data.

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower boundsmore »for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds.« less

We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT-1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparison-based (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT-1 Congest model if non-comparison-based algorithms are permitted. Anmore »important implication of this result is that one can use the synchronous nature of the KT-1 Congest model, using silence to convey information,and solve any graph problem using non-comparison-based algorithms with Õ(n) messages, but this takes an exponential number of rounds. In the asynchronous model, even this is not possible. In contrast, much less is known about the message complexity of local symmetry breaking problems such as coloring and MIS. Our paper fills this gap by presenting the following results. Lower bounds: In the KT-1 CONGEST model, we show that any comparison-based algorithm, even a randomized Monte Carlo algorithm with constant success probability, requires Ω(n 2) messages in the worst case to solve either (△ + 1)-coloring or MIS, regardless of the number of rounds. We also show that Ω(n) is a lower bound on the number ofmessages for any (△ + 1)-coloring or MIS algorithm, even non-comparison-based, and even with nodes having initial knowledge of up to a constant radius. Upper bounds: In the KT-1 CONGEST model, we present the following randomized non-comparison-based algorithms for coloring that, with high probability, use o(m) messages and run in polynomially many rounds.(a) A (△ + 1)-coloring algorithm that uses Õ(n1.5) messages, while running in Õ(D + √ n) rounds, where D is the graph diameter. Our result also implies an asynchronous algorithm for (△ + 1)-coloring with the same message bound but running in Õ(n) rounds. (b) For any constantε > 0, a (1+ε)△-coloring algorithm that uses Õ(n/ε 2 ) messages, while running in Õ(n) rounds. If we increase our input knowledge slightly to radius 2, i.e.,in the KT-2 CONGEST model, we obtain:(c) A randomized comparison-based MIS algorithm that uses Õ(n 1.5) messages. while running in Õ( √n) rounds. While our lower bound results can be viewed as counterparts to the classical result of Awerbuch, Goldreich, Peleg, and Vainish [JACM 90], but for local problems, our algorithms are the first-known algorithms for coloring and MIS that take o(m) messages and run in polynomially many rounds.« less

We study several fundamental problems in the k-machine model, a message-passing model for large-scale distributed computations where k ≥ 2 machines jointly perform computations on a large input of size N, (typically, N ≫ k). The input is initially partitioned (randomly or in a balanced fashion) among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main result is a general technique for designing efficient deterministic distributed algorithms in the k-machine model using PRAM algorithms. Our technique works by efficiently simulating PRAM algorithms in the k-machine model in a deterministic way. This simulation allows us to arrive at new algorithms in the k-machine model for some problems for which no efficient k-machine algorithms are known before and also improve on existing results in the k-machine model for some problems. While our simulation allows us to obtain k-machine algorithms for any problem with a known PRAM algorithm, we mainly focus on graph problems. For an input graph on n vertices and m edges, we obtain Õ(m/k 2 ) round 4 algorithms for various graph problems such as r-connectivity for r = 1,more »2, 3, 4, minimum spanning tree (MST), maximal independent set (MIS), (Δ + 1)-coloring, maximal matching, ear decomposition, and spanners under the assumption that the edges of the input graph are partitioned (randomly, or in an arbitrary, but balanced, fashion) among the k machines. For problems such as connectivity and MST, the above bound is (essentially) the best possible (up to logarithmic factors). Our simulation technique allows us to obtain the first known efficient deterministic algorithms in the k-machine model for other problems with known deterministic PRAM algorithms.« less

This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in asynchronous networks. Kutten et al. (JACM 2015) presented a singularly near optimal randomized leader election algorithm for general synchronous networks that ran in O(D) time and used O(m log n) messages (where D, m, and n are the network’s diameter, number of edges and number of nodes, respectively) with high probability. Both bounds are near optimal (up to a logarithmic factor), since Ω(D) and Ω(m) are the respective lower bounds for time and messages for leader election even for synchronous networks and even for (Monte-Carlo) randomized algorithms. On the other hand, for general asynchronous networks, leader election algorithms are only known that are either time or message optimal, but not both. Kutten et al. (DISC 2020) presented a randomized asynchronous leader election algorithm that is singularly near optimal for complete networks, but left open the problem for general networks. This paper shows that singularly near optimal (up to polylogarithmic factors) bounds can be achieved for general asynchronous networks. We present a randomized singularly near optimal leader election algorithm that runs in O(D + log²more »n) time and O(m log² n) messages with high probability. Our result is the first known distributed leader election algorithm for asynchronous networks that is near optimal with respect to both time and message complexity and improves over a long line of results including the classical results of Gallager et al. (ACM TOPLAS, 1983), Peleg (JPDC, 1989), and Awerbuch (STOC, 89).« less

Maximal Independent Set (MIS) is one of the fundamental problems in distributed computing. The round (time) complexity of distributed MIS has traditionally focused on the worst-case time for all nodes to finish. The best-known (randomized) MIS algorithms take O(log n) worst-case rounds on general graphs (where n is the number of nodes). Breaking the O(log n) worst-case bound has been a longstanding open problem, while currently the best-known lower bound is [EQUATION] rounds. Motivated by the goal to reduce total energy consumption in energy-constrained networks such as sensor and ad hoc wireless networks, we take an alternative approach to measuring performance. We focus on minimizing the total (or equivalently, the average) time for all nodes to finish. It is not clear whether the currently best-known algorithms yield constant-round (or even o(log n)) node-averaged round complexity for MIS in general graphs. We posit the sleeping model, a generalization of the traditional model, that allows nodes to enter either "sleep" or "waking" states at any round. While waking state corresponds to the default state in the traditional model, in sleeping state a node is "offline", i.e., it does not send or receive messages (and messages sent to it are dropped as well)more »and does not incur any time, communication, or local computation cost. Hence, in this model, only rounds in which a node is awake are counted and we are interested in minimizing the average as well as the worst-case number of rounds a node spends in the awake state, besides the traditional worst-case round complexity (i.e., the rounds for all nodes to finish including both the awake and sleeping rounds). Our main result is that we show that MIS can be solved in (expected) O(1) rounds under node-averaged awake complexity measure in the sleeping model. In particular, we present a randomized distributed algorithm for MIS that has expected O(1)-rounds node-averaged awake complexity and, with high probability1 has O(log n)-rounds worst-case awake complexity and O(log3.41 n)-rounds worst-case complexity. Our work is a step towards understanding the node-averaged complexity of MIS both in the traditional and sleeping models, as well as designing energy-efficient distributed algorithms for energy-constrained networks.« less

The K-nearest neighbors is a basic problem in machine learning with numerous applications. In this problem, given a (training) set of n data points with labels and a query point q, we want to assign a label to q based on the labels of the K-nearest points to the query. We study this problem in the k-machine model, a model for distributed large-scale data. In this model, we assume that the n points are distributed (in a balanced fashion) among the k machines and the goal is to compute an answer given a query point to a machine using a small number of communication rounds. Our main result is a randomized algorithm in the k-machine model that runs in O(log K) communication rounds with high success probability (regardless of the number of machines k and the number of points n). The message complexity of the algorithm is small taking only O(k log K) messages. Our bounds are essentially the best possible for comparison-based algorithms. We also implemented our algorithm and show that it performs well in practice.