Deterministic Digital Clustering of Wireless Ad Hoc Networks

Abstract

We consider deterministic distributed communication in wireless ad hoc networks of identical weak devices under the SINR model without predefined infrastructure. Most algorithmic results in this model rely on various additional features or capabilities, e.g., randomization, access to geographic coordinates, power control, carrier sensing with various precision of measurements, and/or interference cancellation. We study a pure scenario, when no such properties are available.The key difficulty in the considered pure scenario stems from the fact that it is not possible to distinguish successful delivery of message sent by close neighbors from those sent by nodes located on transmission boundaries. This problem has been avoided by appropriate model assumptions in many results, which simplified algorithmic solutions.As a general tool, we develop a deterministic distributed clustering algorithm, which splits nodes of a multi-hop network into clusters such that: (i) each cluster is included in a ball of constant diameter; (ii) each ball of diameter 1 contains nodes from O(1) clusters. Our solution relies on a new type of combinatorial structures (selectors), which might be of independent interest.Using the clustering, we develop a deterministic distributed local broadcast algorithm accomplishing this task in O(∆ log * N log N ) rounds, where ∆ is the density of the network. To the best of our knowledge, this is the first solution in pure scenario which is only polylog(n) away from the universal lower bound Ω(∆), valid also for scenarios with randomization and other features. Therefore, none of these features substantially helps in performing the local broadcast task.Using clustering, we also build a deterministic global broadcast algorithm that terminates within O(D(∆ + log * N ) log N ) rounds, where D is the diameter of the network. This result is complemented by a lower bound Ω(D∆ 1−1/α ), where α > 2 is the path-loss parameter of the environment. This lower bound, in view of previous work, shows that randomization or knowledge of own location substantially help (by a factor polynomial in ∆) in the global broadcast. Therefore, unlike in the case of local broadcast, some additional model features may help in global broadcast. with a set W ⊂ W such that O(1) (and at least one!) nodes from each cluster of W belongs to W . Using sparsification algorithm, we develop a tool for imperfect labeling of clusters, which results in assigning temporary IDs (tempID) in range O(∆) to nodes such that O(1) nodes in each cluster have the same tempID. Moreover, an efficient radius reduction algorithm is presented, which transforms r-clustering for r > 1 into 1-clustering.Two communication primitives are essential for efficient implementation of our tools, namely, Sparse Network Schedule (SNS) and Close Neighbors Schedule (CNS). Sparse Network Schedule is a communication protocol of length O(log N ) which guarantees that, given an arbitrary set of nodes X with constant density, each element v of X performs local broadcast (i.e., there i...