TY - JOUR

T1 - Deterministic conflict-free coloring for intervals

T2 - From offline to online

AU - Bar-Noy, Amotz

AU - Cheilaris, Panagiotis

AU - Smorodinsky, Shakhar

PY - 2008/8/1

Y1 - 2008/8/1

N2 - 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 log3/2 n + 1 ≈ 1.71 log2 n colors and show inputs that force any algorithm to use at least 3 log5 n + 1 1.29 log2 n colors. For the online absolute-positions model, we give a deterministic algorithm that uses at most 3⌈log3 n⌉ 1.89 log2 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.

AB - 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 log3/2 n + 1 ≈ 1.71 log2 n colors and show inputs that force any algorithm to use at least 3 log5 n + 1 1.29 log2 n colors. For the online absolute-positions model, we give a deterministic algorithm that uses at most 3⌈log3 n⌉ 1.89 log2 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.

KW - Cellular networks

KW - Coloring

KW - Conflict free

KW - Frequency assignment

KW - Online algorithms

UR - http://www.scopus.com/inward/record.url?scp=50849116105&partnerID=8YFLogxK

U2 - 10.1145/1383369.1383375

DO - 10.1145/1383369.1383375

M3 - Article

AN - SCOPUS:50849116105

VL - 4

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 4

M1 - 44

ER -