Design patterns in beeping algorithms: Examples, emulation, and analysis

Abstract

We consider networks of processes which interact using beeps. In the basic model defined by Cornejo and Kuhn (2010), processes can choose in each round either to beep or to listen. Those who beep are unable to detect simultaneous beeps. Those who listen can only distinguish between silence and the presence of at least one beep. We refer to this model as BL (beep or listen). Stronger models exist where the nodes can detect collision while they are beeping (B cd L), listening (BL cd ), or both (B cd L cd ). Beeping models are weak in essence and even simple tasks are difficult or unfeasible within such models. We present a set of generic building blocks (design patterns) which seem to occur frequently in the design of beeping algorithms. They include multi-slot phases: the fact of dividing the main loop into a number of specialised slots; exclusive beeps: having a single node beep at a time in a neighbourhood (within one or two hops); adaptive probability: increasing or decreasing the probability of beeping to produce more exclusive beeps; internal (resp. peripheral) collision detection: for detecting collision while beeping (resp. listening); and emulation of collision detection when it is not available as a primitive. Based on these patterns, we provide algorithms for a number of basic problems, including colouring, 2-hop colouring, degree computation, 2-hop MIS, and collision detection (in BL). The patterns make it possible to formulate these algorithms in a rather concise and elegant way. Their analyses are more technical; one of them significantly reduces the constant factor in the analysis of the best known MIS algorithm by Jeavons et al. (2016). Finally, inspired by a technique from Afek et al. (2013), our last contribution is to show that any Las Vegas algorithm relying on collision detection can be transposed into a Monte Carlo algorithm without collision detection, through emulation of this primitive at the cost of a logarithmic slowdown. We prove that this is optimal by showing a matching lower bound.

Noisy Beeping Networks

“…In particular, we show how to perform coloring in O(∆ log n + log 2 n) rounds. This follows by emulating a B cd L coloring protocol by [CMRZ19]. Surprisingly, a lower bound of Ω(n log n) for coloring a clique of size n in the BL model is given by [CDT17].…”

“…Among these are algorithms for finding an MIS [AAB + 11, AAB + 13, JSX16, HL16, BBDK18, CMRZ19], (node) coloring of the network [CK10, JSX16, CDT17, CMRZ19], leader-election [GH13, FSW14, DBB18, CD19b], selfstabilization and synchronization [GM15], and broadcasting a message [CD19a]. Simulating beeping algorithms with collision-detection (B cd L cd , B cd L, and BL cd ) over beeping networks without collision-detection BL was shown by [CMRZ19]; the simulation incurs an O(log n) factor.…”

Design Patterns and Electric Vehicle Charging Software

“…A. Casteigts et al [9] formulate multiple algorithms for degree computation, MIS and coloring, using the design patterns such as: peripheral collision detection, adaptive probability, exclusive beeps and multi-slot phases. They work to improve the complexities of the algorithms and prove their convertibility.…”

A handshake algorithm for scheduling communications in wireless sensor networks

“…This algorithm upgrades the algorithm in Reference 7 by minimizing the number of sent messages and transforming it into a fault-tolerant algorithm. The asynchronous fault-tolerant handshake algorithm (ATH) is more effective compared with other algorithms 13 thanks to its asynchronous aspect. When a network contains some nodes with faults, they will affect just their direct neighbors.…”