A Refined Analysis of Online Path Coloring in Trees
Our results are on the online version of path coloring in trees where each request is a path to be colored online, and two paths that share an edge must get different colors. For each T, we come up with a hierarchical partitioning of its edges with a minimum number of parts, denoted by h(T), and design an O(h(T))-competitive online algorithm. We then use the lower bound technique of Bartal and Leonardi  along with a structural property of the hierarchical partitioning, to show a lower bound of \(\varOmega (h(T)/\log (4h(T)))\) for each tree T on the competitive ratio of any deterministic online algorithm for the problem. This gives us an insight into online coloring of paths on each tree T, whereas the current tight lower bound results are known only for special trees like paths and complete binary trees.
- 10.Kumar, V., Schwabe, E.J.: Improved access to optical bandwidth in trees. In Proceedings of SODA 1997, pp. 437–444 (1997)Google Scholar
- 11.Mihail, M., Kaklamanis, C., Rao, S.: Efficient access to optical bandwidth. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science, FOCS 1995, p. 548. IEEE Computer Society, Washington, DC (1995)Google Scholar
- 13.Pemmaraju, S.V., Raman, R., Varadarajan, K.R.: Buffer minimization using max-coloring. In: Ian Munro, J. (ed.) SODA, pp. 562–571. SIAM (2004)Google Scholar
- 14.Raghavan, P., Upfal, E.: Efficient routing in all-optical networks. In: Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, STOC 1994, pp. 134–143. ACM, New York (1994)Google Scholar
- 17.West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River, NJ (2000)Google Scholar