Deterministic conflict-free coloring for intervals

Abstract

We investigate deterministic algorithms for a frequency assignment problem in cellular networks. The problem can be modeled as a special vertex coloring problem for hypergraphs: In every hyperedge there must exist a vertex with a color that occurs exactly once in the hyperedge (the conflict-free property). We concentrate on a special case of the problem, called conflict-free coloring for intervals. We introduce a hierarchy of four models for the aforesaid problem: (i) static, (ii) dynamic offline, (iii) dynamic online with absolute positions, and (iv) dynamic online with relative positions. In the dynamic offline model, we give a deterministic algorithm that uses at most log 3/2 n + 1 ≈ 1.71 log 2 n colors and show inputs that force any algorithm to use at least 3 log 5 n + 1 ≈ 1.29 log 2 n colors. For the online absolute-positions model, we give a deterministic algorithm that uses at most 3⌈log 3 n ⌉ ≈ 1.89 log 2 n colors. To the best of our knowledge, this is the first deterministic online algorithm using O (log n ) colors in a nontrivial online 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 first-fit greedy online algorithm. We also consider conflict-free coloring only with respect to intervals that contain at least one of the two extreme points.