Minimum energy broadcast on rectangular grid wireless networks

Authors:
Atsushi Murata;Akira Matsubayashi
Affiliations:
Division of Electrical and Computer Engineering, Kanazawa University, Kanazawa, Japan;Division of Electrical Engineering and Computer Science, Kanazawa University, Kanazawa, Japan
Venue:
ALGOSENSORS'10 Proceedings of the 6th international conference on Algorithms for sensor systems, wireless adhoc networks, and autonomous mobile entities
Year:
2010

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Abstract

The minimum energy broadcast problem is to assign a transmission range to each node in an ad hoc wireless network to construct a spanning tree rooted at a given source node such that any nonroot node resides within the transmission range of its parent. The objective is to minimize the total energy consumption, i.e., the sum of the δth powers of a transmission range (δ ≥ 1). In this paper, we consider the case that δ = 2, and that nodes are located on a 2-dimensional rectangular grid. We prove that the minimum energy consumption for an n-node k × l-grid with n = kl and k ≤ l is at most n/π + O(n/k0.68) and at least n/π + Ω(n/k) - O(k). Our bounds close the previously known gap of upper and lower bounds for square grids. Moreover, our lower bound is n/3 - O(1) for 3 ≤ k ≤ 18, which matches a naive upper bound within a constant term for k ≡ 0 (mod 3).