## Tree spanners (1995)

Venue: | SIAM J. Discrete Math |

Citations: | 60 - 1 self |

### BibTeX

@ARTICLE{Cai95treespanners,

author = {Leizhen Cai and Derek Corneil},

title = {Tree spanners},

journal = {SIAM J. Discrete Math},

year = {1995},

volume = {8},

pages = {359--387}

}

### Years of Citing Articles

### OpenURL

### 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

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Citation Context ...ational number t > 1. In this subsection we assume that all edge weights are positive rational numbers, and that t > 1 is a fixed rational number. Recall that an instance (U, C) of 3SAT (cf. [LO2] in =-=[19]-=-) consists of a set U of n distinct Boolean variables and a collection C of m 3-element clauses over U. For any variable u ∈ U, both u and u are literals; and for a truth assignment ξ, a 1 x Z1 1 4 2 ... |

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Citation Context ...le set; end for; 13. if T is a tree spanner then compute the stretch index t of T else output “No”; end TREE-SPANNER. We now consider the complexity of the above algorithm. Line (1) takes O(m+n) (cf. =-=[1, 30]-=-). Vertex vi,H at line (10) can be found in O(|N + (i)|) time by keeping track of the set Hk for each out-neighbour k of i at line (7). By Theorem 2.1-(b), line (13) can be carried out in linear time.... |

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Citation Context ...her values of t. Furthermore, we present linear time algorithms for verifying tree and quasitree t-spanners for all values of t. 1.2 Notation and definitions We use the terminology of Bondy and Murty =-=[9]-=-. Graphs in this paper can be either weighted or unweighted, directed or undirected; they are connected graphs without loops, multiedges and multiarcs. For any graph G, V (G) denotes the vertex set of... |

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Data Structures and Network Algorithms
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Citation Context ...le set; end for; 13. if T is a tree spanner then compute the stretch index t of T else output “No”; end TREE-SPANNER. We now consider the complexity of the above algorithm. Line (1) takes O(m+n) (cf. =-=[1, 30]-=-). Vertex vi,H at line (10) can be found in O(|N + (i)|) time by keeping track of the set Hk for each out-neighbour k of i at line (7). By Theorem 2.1-(b), line (13) can be carried out in linear time.... |

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Citation Context ...er, it seems that the algorithm using statement (2) is more efficient for constructing H, especially when pairwise distances are given; using the fastest known algorithm to compute pairwise distances =-=[17]-=-, we can implement the algorithm in O(mn + n 2 log n) time. Theorem 3.6 The minimum (or optimal) 1-spanner of a weighted graph can be found in O(mn + n 2 log n) time. Clearly we can find the tree 1-sp... |

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Citation Context ...as well. Regarding the complexity of the algorithm, we see that line (1) is trivial. To carry out the computation of line (3), we use a linear time least common ancestor algorithm of Harel and Tarjan =-=[21]-=-. Clearly, line (4.2) takes O(1) time. In the next two subsections, we will discuss efficient implementations of line (2) and line (4.1) for undirected graphs and digraphs so as to obtain the results ... |

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Citation Context ...panner problem on unweighted digraphs. One can also consider the tree t-spanner problem for restricted families of graphs. For partial k-trees, it is easily deduced from the results of Arnborg et al. =-=[4]-=- that the problem is polynomial time solvable for any fixed t, since it is a monadic second order problem. However, the problem is open for planar graphs, bounded degree graphs, and many other interes... |

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Citation Context ...tree, then it is a tree 2-spanner; otherwise G contains no tree 2-spanner. The algorithm can be implemented in linear time by using the triconnected component finding algorithm of Hopcroft and Tarjan =-=[22]-=- and standard techniques. Theorem 4.5 A tree 2-spanner (if it exists) of a graph can be found in O(m + n) time. 4.2 The skeleton tree A tree spanner may be required to have some additional properties,... |

182 |
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Citation Context ...ner. This distance approximation property makes spanners quite useful in areas such as communication networks, distributed systems, motion planning, network design, and parallel machine architectures =-=[5, 3, 6, 16, 27, 28, 29, 25]-=-. For example, a sparse spanner (a spanner with few edges) of small stretch factor can be used to plan efficient routing schemes in a communication network while maintaining succinct routing tables [2... |

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Citation Context ...ee 3-spanner in a graph is still unknown. Remark Stronger NP-completeness results hold for the minimum t-spanner problem on unweighted graphs. In fact, the problem is NP-complete for any fixed t ≥ 2 (=-=[12, 26]-=-), even when restricted to graphs of bounded degree [15]. Theorem 4.10 For any fixed t ≥ 4, the tree t-spanner problem is NP-complete. Proof. It is clear that the problem is in NP. To establish the NP... |

139 | A tradeoff between space and efficiency for routing tables
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Citation Context ...ner. This distance approximation property makes spanners quite useful in areas such as communication networks, distributed systems, motion planning, network design, and parallel machine architectures =-=[5, 3, 6, 16, 27, 28, 29, 25]-=-. For example, a sparse spanner (a spanner with few edges) of small stretch factor can be used to plan efficient routing schemes in a communication network while maintaining succinct routing tables [2... |

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Citation Context ...n A t-spanner of a graph G is a spanning subgraph H in which the distance between every pair of vertices is at most t times their distance in G. This notion was introduced in 1987 by Peleg and Ullman =-=[27]-=-, who showed that spanners can be used to construct synchronizers for transforming synchronous algorithms into asynchronous ones. A similar notion appeared in 1986 when Chew [16] studied approximation... |

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Citation Context ...7 by Peleg and Ullman [27], who showed that spanners can be used to construct synchronizers for transforming synchronous algorithms into asynchronous ones. A similar notion appeared in 1986 when Chew =-=[16]-=- studied approximations of complete Euclidean graphs by their planar subgraphs. The key idea behind the notion of spanners is the approximation of pairwise vertex-to-vertex distances in the original g... |

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Citation Context ...gorithm for constructing the tree 1-spanner of G: We first find a minimum spanning tree T of G; and then verify if T is a 1-spanner of G. A minimum spanning tree can be found in O(m log β(m, n)) time =-=[18]-=-, where β(m, n) = min{i| log (i) n ≤ m/n} and log (i) n is defined by log (0) n = n, log (i) n = log log (i−1) n for i ≥ 1. Since verification takes linear time by Theorem 2.1-(a), we have the followi... |

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Citation Context ...ner. This distance approximation property makes spanners quite useful in areas such as communication networks, distributed systems, motion planning, network design, and parallel machine architectures =-=[5, 3, 6, 16, 27, 28, 29, 25]-=-. For example, a sparse spanner (a spanner with few edges) of small stretch factor can be used to plan efficient routing schemes in a communication network while maintaining succinct routing tables [2... |

33 |
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Citation Context ...ee 3-spanner in a graph is still unknown. Remark Stronger NP-completeness results hold for the minimum t-spanner problem on unweighted graphs. In fact, the problem is NP-complete for any fixed t ≥ 2 (=-=[12, 26]-=-), even when restricted to graphs of bounded degree [15]. Theorem 4.10 For any fixed t ≥ 4, the tree t-spanner problem is NP-complete. Proof. It is clear that the problem is in NP. To establish the NP... |

31 |
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Citation Context |

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Citation Context ...taining succinct routing tables [28]. Such a spanner can also be used as a substitute for its original network to reduce the construction cost of the network while keeping similar communication costs =-=[29, 24, 25, 23]-=-. In motion planning, when the input of a simple polygon is inaccurate, a special spanner of the visibility graph of the input polygon, called the visibility skeleton, can be used to plan collision-fr... |

16 |
Tree spanners: Spanning trees that approximate the distances
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Citation Context ... in [19]). Skeleton trees were also used in the design of a polynomial time algorithm for determining whether a 2-connected graph contains a tree 2-spanner isomorphic to a given treesTree Spanners 26 =-=[11, 13]-=-. We note again that the corresponding isomorphic spanning tree problem is NP-complete ([ND8] in [19]). Further applications of skeleton trees can be found in the construction of quasitree 2-spanners ... |

14 |
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Citation Context ...-spanner of a communication network can be used as a substitute for the original network without introducing extra delay in communication. It is also closely related to the metric realization problem =-=[2, 31, 20]-=- (to construct a graph with a minimum total weight that realizes an n-by-n symmetric distance matrix M = (mi,j)). To see this, we construct a complete graph G(M) on n vertices such that w(ij) = mi,j f... |

12 |
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Citation Context ...-spanner of a communication network can be used as a substitute for the original network without introducing extra delay in communication. It is also closely related to the metric realization problem =-=[2, 31, 20]-=- (to construct a graph with a minimum total weight that realizes an n-by-n symmetric distance matrix M = (mi,j)). To see this, we construct a complete graph G(M) on n vertices such that w(ij) = mi,j f... |

8 |
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Citation Context |

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Citation Context ...-spanner of a communication network can be used as a substitute for the original network without introducing extra delay in communication. It is also closely related to the metric realization problem =-=[2, 31, 20]-=- (to construct a graph with a minimum total weight that realizes an n-by-n symmetric distance matrix M = (mi,j)). To see this, we construct a complete graph G(M) on n vertices such that w(ij) = mi,j f... |

5 |
G.: Cycles in weighted graphs
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Citation Context ...d lines [10]. There are also some surprising connections between tree 2-spanners and cycles in graphs: certain cycle-extremal weighted graphs can be represented as a weighted union of tree 2-spanners =-=[8]-=- (where they were called tritrees), and graphs that contain tree 2-spanners [7] (where they were called trigraphs) appear to be the only graphs that require a large number of cycles to cover the edges... |

5 |
Spanners in Graphs of Bounded Degree
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Citation Context ...-completeness results hold for the minimum t-spanner problem on unweighted graphs. In fact, the problem is NP-complete for any fixed t ≥ 2 ([12, 26]), even when restricted to graphs of bounded degree =-=[15]-=-. Theorem 4.10 For any fixed t ≥ 4, the tree t-spanner problem is NP-complete. Proof. It is clear that the problem is in NP. To establish the NP-completeness, we present a polynomial transformation fr... |

2 |
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Citation Context ...put of a simple polygon is inaccurate, a special spanner of the visibility graph of the input polygon, called the visibility skeleton, can be used to plan collision-free paths inside the real polygon =-=[14]-=-. In most applications, the sparseness of a spanner is the main concern; and the sparsest t-spanner in a connected graph is a tree t-spanner, that is, a t-spanner which is a tree. Therefore, as far as... |

2 |
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Citation Context ...taining succinct routing tables [28]. Such a spanner can also be used as a substitute for its original network to reduce the construction cost of the network while keeping similar communication costs =-=[29, 24, 25, 23]-=-. In motion planning, when the input of a simple polygon is inaccurate, a special spanner of the visibility graph of the input polygon, called the visibility skeleton, can be used to plan collision-fr... |

1 |
Spanning 2-trees
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Citation Context ...ery. The existence of a tree 2-spanner in a 2-connected network guarantees that the communication amongst operative sites will not be affected by any isolated failure of communication sites and lines =-=[10]-=-. There are also some surprising connections between tree 2-spanners and cycles in graphs: certain cycle-extremal weighted graphs can be represented as a weighted union of tree 2-spanners [8] (where t... |

1 |
Isomorphic tree spanner problems. Algorithmica
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(Show Context)
Citation Context ... in [19]). Skeleton trees were also used in the design of a polynomial time algorithm for determining whether a 2-connected graph contains a tree 2-spanner isomorphic to a given treesTree Spanners 26 =-=[11, 13]-=-. We note again that the corresponding isomorphic spanning tree problem is NP-complete ([ND8] in [19]). Further applications of skeleton trees can be found in the construction of quasitree 2-spanners ... |