Average Awake Complexity of MIS and Matching

Abstract

Chatterjee, Gmyr, and Pandurangan [PODC 2020] recently introduced the notion of awake complexity for distributed algorithms, which measures the number of rounds in which a node is awake. In the other rounds, the node is sleeping and performs no computation or communication. Measuring the number of awake rounds can be of significance in many settings of distributed computing, e.g., in sensor networks where energy consumption is of concern.In that paper, Chatterjee et al. provide an elegant randomized algorithm for the Maximal Independent Set (MIS) problem that achieves an O(1) node-averaged awake complexity. That is, the average awake time among the nodes is O(1) rounds. However, to achieve that, the algorithm sacrifices the more standard round complexity measure from the well-known O(log n) bound of MIS, due to Luby [STOC'85], to O(log 3.41 n) rounds.Our first contribution is to present a simple randomized distributed MIS algorithm that, with high probability, has O(1) node-averaged awake complexity and O(log n) worst-case round complexity. Our second, and more technical contribution, is to show algorithms with the same O(1) node-averaged awake complexity and O(log n) worst-case round complexity for (1 + ε)-approximation of maximum matching and (2+ε)-approximation of minimum vertex cover, where ε denotes an arbitrary small positive constant.

Distributed MIS in O(log log n) Awake Complexity

“…Our work answers a key question left open in [13], namely whether one can design MIS algorithms with (even) 𝑜 (log 𝑛) (worstcase) awake complexity. We note that prior results [13,22] presented algorithms in the sleeping model with 𝑂 (log 𝑛) awake complexity (see Section 1.5). Our results show that we can compute an MIS in an awake complexity that is exponentially better compared to the best known round complexity of 𝑂 (log 𝑛).…”

“…Several recent works (see Section 1.5) have designed distributed algorithms in the sleeping model for fundamental problems such as MIS, approximate matching and vertex cover, spanning tree, minimum spanning tree, coloring, and other problems [2,7,13,22].…”

“…An important question left open in [13] is whether one can design an MIS algorithm with 𝑜 (log 𝑛) worst-case awake complexity (even in the LOCAL model). 3 Subsequently Ghaffari and Portmann [22] developed a randomized MIS algorithm that has worst-case awake complexity of 𝑂 (log 𝑛), round complexity of 𝑂 (log 𝑛), while having 𝑂 (1) node-averaged awake complexity (all bounds hold with high probability). They also present algorithms for (1 +𝜀) approximation of maximum matching and (2 + 𝜀) approximation of minimum vertex cover with the same bounds on round complexity and node-averaged awake complexity.…”

“…This work defines the measure of energy complexity which is the same as (worst-case) awake complexity (i.e., both measures count only the rounds that a node is awake). While the awake complexity used here and several other papers [2,7,13,22] assumes the usual CONGEST (or LOCAL) communication model (and hence the model can be called SLEEPING-CONGEST (or SLEEPING-LOCAL)), the energy complexity measure used in [12] (and also papers mentioned above) has some additional communication restrictions that pertain to radio networks (and can be called SLEEPING-RADIO model). One important restriction in the radio model is that nodes can only broadcast messages (hence the same message is sent to all neighbors).…”

Energy-Efficient Distributed Algorithms for Synchronous Networks

“…In the CONGEST model, It was first shown [4] that mis has constant average awake complexity, thanks to a randomized algorithm running in O(polylog(n)) rounds. The round complexity was improved in [7] with a randomized algorithm running in O(log n) rounds. The (worst-case) awake complexity of mis was proved to be O(log log n) using a randomized Monte-Carlo algorithm running in O(poly(n)) rounds [6].…”