2006
DOI: 10.1007/11780823_18
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Abstract: Abstract. The Minimum Energy Broadcast problem consists in finding the minimum-energy range assignment for a given set S of n stations of an ad hoc wireless network that allows a source station to perform broadcast operations over S. We prove a nearly tight asymptotical bound on the optimal cost for the Minimum Energy Broadcast problem on square grids. We emphasize that finding tight bounds for this problem restriction is far to be easy: it involves the Gauss's Circle problem and the Apollonian Circle Packing.… Show more

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Cited by 4 publications
(12 citation statements)
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“…Specifically, we prove that a broadcast on an n-node square grid based on Apollonian gaskets achieves a cost of n π + O(n S 2 +ǫ ), where S is the Hausdorff dimension of an Apollonian gasket. Because it is well known that S < 1.314534 [9], our upper bound matches the lower bound of [8] within an o(n) term. We also generalize these results to rectangular grids.…”
Section: Introductionsupporting
confidence: 49%
See 3 more Smart Citations
“…Specifically, we prove that a broadcast on an n-node square grid based on Apollonian gaskets achieves a cost of n π + O(n S 2 +ǫ ), where S is the Hausdorff dimension of an Apollonian gasket. Because it is well known that S < 1.314534 [9], our upper bound matches the lower bound of [8] within an o(n) term. We also generalize these results to rectangular grids.…”
Section: Introductionsupporting
confidence: 49%
“…Our algorithm to construct a broadcast on an m × m-grid is based on an idea mentioned in [8] of naturally generalizing an Apollonian gasket to a circle packing of the square Q with side length m − 1 bounding the grid. Specifically, the algorithm, called AGBS, is defined as follows: …”
Section: Broadcast On Square Gridsmentioning
confidence: 99%
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“…While [56] assumed that the cost of broadcasting a message to all other nodes is 1, we assume that nodes communicate via multi-hop paths, leading to a cost of broadcast of O( n π − √ n) [21]. Similarly, we assume that the cost of sending a message to a single node is proportional to the network diameter, i.e., O( √ n).…”
Section: Performance Evaluation and Comparisonmentioning
confidence: 99%