Abstract
We consider the broadcasting problem in multi-radio multi-channel ad hoc networks. The objective is to minimize the total cost of the network-wide broadcast, where the cost can be of any form that is summable over all the transmissions (e.g., the transmission and reception energy, the price for accessing a specific channel). Our technical approach is based on a simplicial complex model that allows us to capture the broadcast nature of the wireless medium and the heterogeneity across radios and channels. Specifically, we show that broadcasting in multi-radio multi-channel ad hoc networks can be formulated as a minimum spanning problem in simplicial complexes. We establish the NP-completeness of the minimum spanning problem and propose two approximation algorithms with order-optimal performance guarantee. The first approximation algorithm converts the minimum spanning problem in simplical complexes to a minimum connected set cover (MCSC) problem. The second algorithm converts it to a node-weighted Steiner tree problem under the classic graph model. These two algorithms offer tradeoffs between performance and time-complexity. In a broader context, this work appears to be the first that studies the minimum spanning problem in simplicial complexes and weighted MCSC problem.
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Notes
The ‘reception energy’ denotes the energy consumed by the radio in reception mode.
Strictly speaking, a subcomplex should also be closed under the subset operation, but without loss of generality, we do not include this condition in the definition of minimum connected spanning subcomplex, which is also more relevant to the broadcasting problem at hand.
Although there is no unified definition of tree in simplicial complexes, a couple of definitions can be obtained by generalizing those equivalent definitions of tree in a graph. For example, simplicial trees can be defined based on the universal existence of leaves in any subgraph, or the uniqueness of simplicial facet paths.
Notice that since A is its own nonempty subset, the simplex A is also a face of A.
\({({\mathcal{S}},w)}\) suffices to denote the WSC since \({V\subseteq {\mathcal{S}}}\), but we use the redundant \({(V,{\mathcal{S}},w)}\) for convenience.
We say that the weight function satisfies the monotone property if for any two faces \(S_1\subseteq S_2,\, w(S_1)\leq w(S_2)\), i.e., the weight is monotone non-decreasing with respect to the dimension of the face.
The approximation ratio of the greedy algorithm for general weighted MCSC problem is still an open problem.
A dominating set of a graph is a subset of vertices such that every vertex of the graph is either in the subset or a neighbor of some vertex in the subset, and a connected dominating set (CDS) is a dominating set where the subgraph induced by the vertices in the dominating set is connected. The CDS problem asks for a CDS with the minimum total weight, and it is shown to be a special case of the MCSC problem [21].
A random simplicial complex \(\Updelta (n,D,{\bf p})\) with n vertices, dimension at most D, and a D-dimensional probability vector \({\bf p}=\{p_1,p_2,\ldots,p_D\}\) is constructed in a bottom-up manner: first n vertices are fixed, which are the 0-simplices of \(\Updelta\), and then higher-dimensional simplices are generated inductively. Specifically, for each 1 ≤ i ≤ D, after all the simplices with dimension lower than i have been generated, consider every i-tuple of vertices: if they have formed all the lower dimensional simplices, then an i-simplex consisting of them is generated with probability p i . Notice that a random simplicial complex \(\Updelta (n,1,p)\) is the random graph introduced by Erdős and Rényi [5].
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This work was supported by the Army Research Laboratory NS-CTA under Grant W911NF-09-2-0053.
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Ren, W., Zhao, Q., Ramanathan, R. et al. Broadcasting in multi-radio multi-channel wireless networks using simplicial complexes. Wireless Netw 19, 1121–1133 (2013). https://doi.org/10.1007/s11276-012-0522-4
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DOI: https://doi.org/10.1007/s11276-012-0522-4