Groovin’ with the Big Band(width)

This paper is concerned with efficiently assigning bandwidth to radio transmitters, so as to avoid interference. We assume that the area (city, state, etc) is divided into a honeycomb hexagonal grid, and that transmitters are placed at the centers of the hexagons, which have side length s. The bandwidth assigned to a transmitter will be represented by a channel number. To avoid interference, the following constraints must be met: two transmitters within 2s must be assigned channels that differ by at least 2, and transmitters within 4s must be assigned channels that differ by at least 1. We seek to find the span of a network, which is the smallest integer, n, such that a proper assignment configuration exists on the grid that uses no channel higher than n. We find that with these constraints, the span of a grid (indeed, the entire plane), is 9. To prove this result, we must show that a proper assignment does not exist using number of at most 8, and we must also demonstrate how the assignments can be made using the integers 1 through 9. We show that, in fact, there is a unique such pattern which meets the constraints. We then generalize to the constraint that channels for transmitters within 2s must differ by at least k, while channels for transmitters within 4s must still differ by at least 1. We prove that for k=1, the span is 7; for k=2, the span is 9 (our original case); for k=3, the span is 12; and for k>3, the span is 2k+7. In each case, we show that the span cannot be smaller, and also give an example with that span. Each example is a simple pattern that can be efficiently extended to grids of arbitrarily large size, and these patterns guarantee that the bandwidth used is the smallest possible. We again generalize to the constraint that channels for transmitters within 2s must differ by at least k, while channels for transmitters within 4s must differ by at least m. We demonstrate a minimum bound for the span, 1+2k+4m, and also provide an example with span 1+2k+6m. This example is again a simple pattern that can be efficiently extended to larger grids. We also demonstrate that, though we have not proven that 1+2k+6m is the span, it is still an effective way to minimize the bandwidth used. Thirdly, we generalize to 3 levels of interference, so that we have the constraints that channels for transmitters within 2s must differ by at least k, channels for transmitters within 4s must differ by at least m, and channels for transmitters within 6s must differ by at least n. We demonstrate a method for building up this 3-level interference from an assignment configuration which is valid for a 2-layer interference. We show that the span, for all k, m, and n, is less than 1+2k+6m+18n, and we motivate that this is an efficient way to minimize bandwidth allocation. In summary, we completely solve parts A,B, and C of the problem statement, as well as providing several efficient, useful, and beautiful generalizations of the problem.