The Energy Complexity of Broadcast

Energy is often the most constrained resource in networks of battery-powered devices, and as devices become smaller, they spend a larger fraction of their energy on communication (transceiver usage) not computation. As an imperfect proxy for true energy usage, we define energy complexity to be the number of time slots a device transmits/listens; idle time and computation are free. In this paper we investigate the energy complexity of fundamental communication primitives such as broadcast in multi-hop radio networks. We consider models with collision detection (CD) and without (No-CD), as well as both randomized and deterministic algorithms. Some take-away messages from this work include: 1. The energy complexity of broadcast in a multi-hop network is intimately connected to the time complexity of leader election in a single-hop (clique) network. Many existing lower bounds on time complexity immediately transfer to energy complexity. For example, in the CD and No-CD models, we need $\Omega(\log n)$ and $\Omega(\log^2 n)$ energy, respectively. 2. The energy lower bounds above can almost be achieved, given sufficient ($\Omega(n)$) time. In the CD and No-CD models we can solve broadcast using $O(\frac{\log n\log\log n}{\log\log\log n})$ energy and $O(\log^3 n)$ energy, respectively. 3. The complexity measures of Energy and Time are in conflict, and it is an open problem whether both can be minimized simultaneously. We give a tradeoff showing it is possible to be nearly optimal in both measures simultaneously. For any constant $\epsilon>0$, broadcast can be solved in $O(D^{1+\epsilon}\log^{O(1/\epsilon)} n)$ time with $O(\log^{O(1/\epsilon)} n)$ energy, where $D$ is the diameter of the network

How to Wake up Your Neighbors: Safe and Nearly Optimal Generic Energy Conservation in Radio Networks

Recent work [Chang et al., 2018; Chang et al., 2020; Varsha Dani et al., 2021] has shown that it is sometimes feasible to significantly reduce the energy usage of some radio-network algorithms by adaptively powering down the radio receiver when it is not needed. Although past work has focused on modifying specific network algorithms in this way, we now ask the question of whether this problem can be solved in a generic way, treating the algorithm as a kind of black box. We are able to answer this question in the affirmative, presenting a new general way to modify arbitrary radio-network algorithms in an attempt to save energy. At the expense of a small increase in the time complexity, we can provably reduce the energy usage to an extent that is provably nearly optimal within a certain class of general-purpose algorithms. As an application, we show that our algorithm reduces the energy cost of breadth-first search in radio networks from the previous best bound of 2^O(?{log n}) to polylog(n), where n is the number of nodes in the network A key ingredient in our algorithm is hierarchical clustering based on additive Voronoi decomposition done at multiple scales. Similar clustering algorithms have been used in other recent work on energy-aware computation in radio networks, but we believe the specific approach presented here may be of independent interest

Energy Complexity of Distance Computation in Multi-hop Networks

Energy efficiency is a critical issue for wireless devices operated under stringent power constraint (e.g., battery). Following prior works, we measure the energy cost of a device by its transceiver usage, and define the energy complexity of an algorithm as the maximum number of time slots a device transmits or listens, over all devices. In a recent paper of Chang et al. (PODC 2018), it was shown that broadcasting in a multi-hop network of unknown topology can be done in $\text{poly} \log n$ energy. In this paper, we continue this line of research, and investigate the energy complexity of other fundamental graph problems in multi-hop networks. Our results are summarized as follows. 1. To avoid spending $\Omega(D)$ energy, the broadcasting protocols of Chang et al. (PODC 2018) do not send the message along a BFS tree, and it is open whether BFS could be computed in $o(D)$ energy, for sufficiently large $D$. In this paper we devise an algorithm that attains $\tilde{O}(\sqrt{n})$ energy cost. 2. We show that the framework of the ${\Omega}(n)$ round lower bound proof for computing diameter in CONGEST of Abboud et al. (DISC 2017) can be adapted to give an $\tilde{\Omega}(n)$ energy lower bound in the wireless network model (with no message size constraint), and this lower bound applies to $O(\log n)$-arboricity graphs. From the upper bound side, we show that the energy complexity of $\tilde{O}(\sqrt{n})$ can be attained for bounded-genus graphs (which includes planar graphs). 3. Our upper bounds for computing diameter can be extended to other graph problems. We show that exact global minimum cut or approximate $s$--$t$ minimum cut can be computed in $\tilde{O}(\sqrt{n})$ energy for bounded-genus graphs

Wake Up and Join Me! An Energy-Efficient Algorithm for Maximal Matching in Radio Networks

We consider networks of small, autonomous devices that communicate with each other wirelessly. Minimizing energy usage is an important consideration in designing algorithms for such networks, as battery life is a crucial and limited resource. Working in a model where both sending and listening for messages deplete energy, we consider the problem of finding a maximal matching of the nodes in a radio network of arbitrary and unknown topology. We present a distributed randomized algorithm that produces, with high probability, a maximal matching. The maximum energy cost per node is $O(\log^2 n)$, where $n$ is the size of the network. The total latency of our algorithm is $O(n \log n)$ time steps. We observe that there exist families of network topologies for which both of these bounds are simultaneously optimal up to polylog factors, so any significant improvement will require additional assumptions about the network topology. We also consider the related problem of assigning, for each node in the network, a neighbor to back up its data in case of node failure. Here, a key goal is to minimize the maximum load, defined as the number of nodes assigned to a single node. We present a decentralized low-energy algorithm that finds a neighbor assignment whose maximum load is at most a polylog($n$) factor bigger that the optimum.Comment: 14 pages, 2 figures, 3 algorithm

Robust and Listening-Efficient Contention Resolution

This paper shows how to achieve contention resolution on a shared communication channel using only a small number of channel accesses -- both for listening and sending -- and the resulting algorithm is resistant to adversarial noise. The shared channel operates over a sequence of synchronized time slots, and in any slot agents may attempt to broadcast a packet. An agent's broadcast succeeds if no other agent broadcasts during that slot. If two or more agents broadcast in the same slot, then the broadcasts collide and both broadcasts fail. An agent listening on the channel during a slot receives ternary feedback, learning whether that slot had silence, a successful broadcast, or a collision. Agents are (adversarially) injected into the system over time. The goal is to coordinate the agents so that each is able to successfully broadcast its packet. A contention-resolution protocol is measured both in terms of its throughput and the number of slots during which an agent broadcasts or listens. Most prior work assumes that listening is free and only tries to minimize the number of broadcasts. This paper answers two foundational questions. First, is constant throughput achievable when using polylogarithmic channel accesses per agent, both for listening and broadcasting? Second, is constant throughput still achievable when an adversary jams some slots by broadcasting noise in them? Specifically, for $N$ packets arriving over time and $J$ jammed slots, we give an algorithm that with high probability in $N+J$ guarantees $\Theta(1)$ throughput and achieves on average $O(\texttt{polylog}(N+J))$ channel accesses against an adaptive adversary. We also have per-agent high-probability guarantees on the number of channel accesses -- either $O(\texttt{polylog}(N+J))$ or $O((J+1) \texttt{polylog}(N))$, depending on how quickly the adversary can react to what is being broadcast