TY - GEN

T1 - A Linear Time Algorithm for L(2,1)-Labeling of Trees

AU - Hasunuma, Toru

AU - Ishii, Toshimasa

AU - Ono, Hirotaka

AU - Uno, Yushi

PY - 2009

Y1 - 2009

N2 - 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 λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ4.5 n) for more than a decade, and an O( min {n 1.75,Δ1.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n=|V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.

AB - 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 λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ4.5 n) for more than a decade, and an O( min {n 1.75,Δ1.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n=|V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.

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U2 - 10.1007/978-3-642-04128-0_4

DO - 10.1007/978-3-642-04128-0_4

M3 - Conference contribution

AN - SCOPUS:70350378256

SN - 3642041272

SN - 9783642041273

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 35

EP - 46

BT - Algorithms - ESA 2009 - 17th Annual European Symposium, Proceedings

T2 - 17th Annual European Symposium on Algorithms, ESA 2009

Y2 - 7 September 2009 through 9 September 2009

ER -