Tree Spanners

Abstract

A tree t-spanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with m edges and n vertices is a minimum spanning tree and can be found in O(m log β(m, n)) time, where β(m, n) = min{i| log (i) n ≤ m/n}. On the other hand, for any fixed t > 1, the problem of determining the existence of a tree t-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree t-spanner is NP-complete for any fixed t ≥ 4. A theorem which captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an O((m + n)α(m, n)) algorithm is provided for finding a tree t-spanner with t as small as possible, where α(m, n) is a functional inverse of Ackerman's function. The results for tree spanners on undirected graphs are extended to "quasitree spanners" on digraphs. Furthermore, linear time algorithms are derived for verifying tree spanners and quasitree spanners.

Swapping labeled tokens on graphs

“…A tree α-spanner T of an unweighted graph G = (V, E) is a spanning tree of G such that sp T (v, w) ≤ α · sp G (v, w) for every pair of vertices v, w ∈ V [2]. Then, we have the following theorem.…”

“…However, Cai and Corneil [2] proved that deciding whether an unweighted graph G has a tree α-spanner is NP-complete for any fixed α ≥ 4, while it can be solved in polynomial time for α ≤ 2. Therefore, several approximation and FPT algorithms have been studied extensively.…”

“…Thus, one interesting research topic is minimizing the number of bars in a ladder lottery for a given permutation π. This minimization problem on ladder lottery can be solved by counting the number of "inversions" in π [8,10], where a pair (p i , p j ) in π is called an inversion in π if p i > p j and i < j; for example, there are four inversions in the permutation (4,2,1,3), that is, (4, 2), (4, 1), (4,3) and (2,1), and hence the ladder lottery in Fig. 2(a) has the minimum number of bars.…”

Navigating in a Graph by Aid of Its Spanning Tree

“…For r ≥ 0 and t ≥ 1, a spanning tree T of a graph G is called an additive tree r-spanner (a multiplicative tree t-spanner y)) for each pair x, y ∈ V [9]. By Observation 1(a), we obtain.…”

Computing Geometric Minimum-Dilation Graphs Is NP-Hard

“…Cai [3] and Cai and Corneil [4] have studied the problem of finding tree spanners of dilation ≤ δ in weighted graphs. They proved that the decision problem is NP-complete for any δ ≥ 4, but polynomially solvable for δ = 2, while the case δ = 3 seems to be open.…”