Abstract
Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let sin P be a given source node. Each node p can transmit information to all other nodes within unit distance, provided p is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source s can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, s must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width w. We describe several algorithms for both the regular and the hop-bounded versions, and show that both problems are solvable in polynomial time in strips of small constant width. These results complement the hardness results in a companion paper (de Berg et al. in Algorithmica, 2017).
Highlights
Wireless networks give rise to a host of interesting algorithmic problems
In the traditional model of a wireless network each node is modeled as a point p ∈ R2, which is the center of a disk δ( p) whose radius equals the transmission range of p
We are interested in broadcast problems, where the desired property is that a given source node can reach any other node in the communication graph
Summary
Wireless networks give rise to a host of interesting algorithmic problems. In the traditional model of a wireless network each node is modeled as a point p ∈ R2, which is the center of a disk δ( p) whose radius equals the transmission range of p. √ Theorem 1 The broadcast problem inside a strip of width at most 3/2 can be solved in O(n log n) time As remarked earlier, this result implies an O(dminn log n) algorithm for Connected Dominating Set in narrow strip unit disk graphs, where dmin is the minimum degree in the graph. The key step toward getting this algorithm is a structural lemma stating that, except in some small-diameter instances that can be handled separately, there is always an optimal broadcast tree that induces a path in the underlying unit disk graph In this case, the problem boils down to computing a shortest path from the source to some specific sets of points that are “far enough” to the right or to the left of the source.
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