An O(n1.75) 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 770-8502, Japan;Department of Information and Management Science, Otaru University of Commerce, Otaru 047-8501, Japan;Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812-8581, Japan;Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Sakai 599-8531, Japan

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x)-f(y)|=2 if x and y are adjacent and |f(x)-f(y)|=1 if x and y are at distance 2 for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)-{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by @l(G), among all possible L(2,1)-labelings. 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 O(@D^4^.^5n) time algorithm for a tree T has been known so far, where @D is the maximum degree of T and n=|V(T)|. In this paper, we first show that an existent necessary condition for @l(T)=@D+1 is also sufficient for a tree T with @D=@W(n), which leads to a linear time algorithm for computing @l(T) under this condition. We then show that @l(T) can be computed in O(@D^1^.^5n) time for any tree T. Combining these, we finally obtain an O(n^1^.^7^5) time algorithm, which substantially improves upon previously known results.