An $\mbox{O}(n^{1.75})$ Algorithm for L(2,1)-Labeling of Trees

  • Authors:

  • Toru Hasunuma;Toshimasa Ishii;Hirotaka Ono;Yushi Uno

  • Affiliations:

  • Department of Mathematical and Natural Sciences, The University of Tokushima, Tokushima, Japan 770---8502;Department of Information and Management Science, Otaru University of Commerce, Otaru, Japan 047-8501;Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka, Japan 812-8581;Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Sakai, Japan 599-8531

  • Venue:
  • SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
  • Year:
  • 2008

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Abstract

An L(2,1)-labeling of a graph Gis an assignment ffrom the vertex set V(G) to the set of nonnegative integers such that |f(x) 茂戮驴 f(y)| 茂戮驴 2 if xand yare adjacent and |f(x) 茂戮驴 f(y)| 茂戮驴 1 if xand yare at distance 2 for all xand yin V(G). A k-L(2,1)-labeling is an assignment f:V(G)茂戮驴{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by 茂戮驴(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an $\mbox{O}(\Delta^{4.5} n)$ time algorithm for a tree Thas been known so far, where Δis the maximum degree of Tand n= |V(T)|. In this paper, we first show that an existent necessary condition for 茂戮驴(T) = Δ+ 1 is also sufficient for a tree Twith $\Delta=\Omega(\sqrt{n})$, which leads a linear time algorithm for computing 茂戮驴(T) under this condition. We then show that 茂戮驴(T) can be computed in $\mbox{O}(\Delta^{1.5}n)$ time for any tree T. Combining these, we finally obtain an time algorithm, which substantially improves upon previously known results.