TY - JOUR

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 - 2009/9/6

Y1 - 2009/9/6

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 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 (Δ4.5 n) 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 Δ = Ω (sqrt(n)), which leads to a linear time algorithm for computing λ (T) under this condition. We then show that λ (T) can be computed in O (Δ1.5 n) time for any tree T. Combining these, we finally obtain an O (n1.75) 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 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 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 (Δ4.5 n) 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 Δ = Ω (sqrt(n)), which leads to a linear time algorithm for computing λ (T) under this condition. We then show that λ (T) can be computed in O (Δ1.5 n) time for any tree T. Combining these, we finally obtain an O (n1.75) time algorithm, which substantially improves upon previously known results.

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U2 - 10.1016/j.tcs.2009.04.025

DO - 10.1016/j.tcs.2009.04.025

M3 - Article

AN - SCOPUS:68249127938

VL - 410

SP - 3702

EP - 3710

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 38-40

ER -