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

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 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 O(∆ 4.5 n) time algorithm for a tree T has been known so far, where ∆ is the maximum degree of T and n = |V(T )|. In this paper, we first show that an existent necessary condition for λ(T ) = ∆ + 1 is also sufficient for a tree T with ∆ = Ω( √ n), which leads a linear time algorithm for computing λ(T ) under this condition. We then show that λ(T ) can be computed in O(∆ 1.5 n) time for any tree T . Combining these, we finally obtain an O(n 1.75 ) time algorithm, which greatly improves the currently best known result.

-Total labelling of trees with sparse vertices of maximum degree

“…By The following sufficient condition such that λ(T ) = Δ + 1 was recently established in [6], which implies our Theorem 9 when Δ 8. Hasunuma et al 2008, [6].)…”

“…Chang and Kuo [2] presented a polynomial time algorithm to solve the L(2, 1)-labelling number λ(T ) for a tree T . Hasunuma et al [6] further proved that the L(2, 1)-labelling problem of a tree T can be solved in O (min{n 1.75 , Δ 1.5 n}) time, where n = |T |. Note that the problem of determining the (2, 1)-total number of a tree T is equivalent to determine the L(2, 1)-labelling number of its incidence graph T , i.e., λ t 2 (T ) = λ(T ).…”

Algorithmic aspects of distance constrained labeling: a survey

“…The principle of optimality requires to solve at each vertex of the tree the assignments of labels to subtrees, and the assignments are formulated as the maximum matching in a certain bipartite graph. This running time is improved into O(min{n 1.75 , ∆ 1.5 n}) [34], and recently, a linear time algorithm has been established [38]. They are based on the similar DP framework to Chang and Kuo's algorithm, but achieve their efficiency by reducing heavy computation of bipartite matching in Chang and Kuo's and by using an amortized analysis.…”

A sufficient condition for a tree to be $$(\Delta +1)$$ ( Δ + 1 ) - $$(2,1)$$ ( 2 , 1 ) -totally labelable

“…However, it remains open to characterize trees T with ≥ 4 such that λ t 2 (T ) = + 1 or λ t 2 (T ) = + 2. In 2009, Hasunuma et al (2009) …”