Conflict-free coloring for intervals: From offline to online

This paper studies deterministic algorithms for a frequency assignment problem in cellular networks. A cellular network consists of fixed-position base stations and moving agents. Each base station operates at a fixed frequency, and this allows an agent tuned at this frequency to communicate with the base station. Each agent has a specific range of communication (described as a geometric shape, e.g., a disc) that may contain one or several base stations. To avoid interference, the goal is to assign frequencies to base stations such that for any range, there exists a base station in the range with a frequency that is not reused by some other base station in the range. The base station with this unique (in the range) frequency serves the aforementioned range. Since using many frequencies is expensive, the optimization goal is to use as few frequencies as possible. The problem can be modeled as a special coloring problem for hypergraphs. Base stations are the vertices, ranges are the hyperedges, and colors (frequencies) must be assigned to vertices following the conflict-free property: In every hyperedge there is a color that occurs exactly once. We concentrate on the special case where the n base stations lie on the real line and ranges are the n(n + 1)/2 non-empty subsets of consecutive points. This problem is called conflict-free coloring for intervals. We introduce a hierarchy of four models for the above problem: (i) the static model, where the complete hypergraph is given and all vertices are colored simultaneously, (ii) the dynamic offline model, where the vertices appear in some order and the conflict-free property has to be maintained at all times, (iii) the online absolute positions model where the order is revealed in an online fashion and the final hypergraph and positions are known, and (iv) the online relative positions model where there is no knowledge about the final hypergraph and the final positions of vertices. In the case of intervals, the hierarchy is strict. In the dynamic offline model, we give a deterministic algorithm that uses at most log _3/2 n+1 colors and exhibit inputs that force any algorithm to use at least 2 log ₃ n + 1 colors. For the online absolute positions model, we give two deterministic algorithms that use at most 2[log ₂(n + 1)J and 3[log ₃n] colors, respectively. To the best of our knowledge, these are the first O(log ) deterministic online algorithms, in a non-trivial model. In the online relative positions model, we resolve an open problem by showing a tight analysis on the number of colors used by the natural greedy online algorithm, that at each step uses the smallest color possible. In the case of conflict-free coloring only with respect to intervals that contain either of the two extreme points, we show a strong separation between static and dynamic models and we provide tight bounds for all four models up to an additive term of two.