TY - GEN

T1 - An O(n1.75) algorithm for L(2,1)-labeling of trees

AU - Hasunuma, Toru

AU - Ishii, Toshimasa

AU - Ono, Hirotaka

AU - Uno, Yushi

N1 - Funding Information:
I An extended abstract of this article was presented in Proceedings of the 11th Scandinavian Workshop on Algorithm Theory, SWAT 2008, in: Lecture Notes in Computer Science, vol. 5124, Springer, 2008, pp. 185–197. II This research is partly supported by INAMORI FOUNDATION, Asahi Glass Foundation and Grant-in-Aid for Scientific Research (KAKENHI), Nos.

PY - 2008

Y1 - 2008

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 assignment 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. Tree is one of a few classes for which the problem is polynomially solvable, but still only an time algorithm for a tree T has been known so far, where Δ is the maximum degree of T and n∈=∈|V(T)|. In this paper, we first show that an existent necessary condition for λ(T)∈=∈Δ∈+∈1 is also sufficient for a tree T with , which leads a linear time algorithm for computing λ(T) under this condition. We then show that λ(T) can be computed in time for any tree T. Combining these, we finally obtain an time algorithm, which substantially improves upon previously known results.

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 assignment 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. Tree is one of a few classes for which the problem is polynomially solvable, but still only an time algorithm for a tree T has been known so far, where Δ is the maximum degree of T and n∈=∈|V(T)|. In this paper, we first show that an existent necessary condition for λ(T)∈=∈Δ∈+∈1 is also sufficient for a tree T with , which leads a linear time algorithm for computing λ(T) under this condition. We then show that λ(T) can be computed in time for any tree T. Combining these, we finally obtain an time algorithm, which substantially improves upon previously known results.

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U2 - 10.1007/978-3-540-69903-3_18

DO - 10.1007/978-3-540-69903-3_18

M3 - Conference contribution

AN - SCOPUS:54249122280

SN - 3540699007

SN - 9783540699002

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

SP - 185

EP - 197

BT - Algorithm Theory - SWAT 2008 - 11th Scandinavian Workshop on Algorithm Theory, Proceedings

T2 - 11th Scandinavian Workshop on Algorithm Theory, SWAT 2008

Y2 - 2 July 2008 through 4 July 2008

ER -