## Abstract

For a pair of parameters α, β ≥ 1, a spanning tree T of a weighted undirected n-vertex graph G = (V,E,w) is called an (α,β)- shallow-light tree} (shortly, (α,β)-SLT)of G with respect to a designated vertex rt ∈ V if (1) it approximates all distances from rt to the other vertices up to a factor of α, and(2) its weight is at most β times the weight of the minimum spanning tree MST(G) of G. The parameter α (respectively, β) is called the root-distortion (resp., lightness) of the tree T. Shallow-light trees (SLTs) constitute a fundamental graph structure, with numerous theoretical and practical applications. In particular, they were used for constructing spanners, in network design, for VLSI-circuit design, for various data gathering and dissemination tasks in wireless and sensor networks, in overlay networks, and in the message-passing model of distributed computing. Tight tradeoffs between the parameters of SLTs were established by Awer buch et al. [5], [6] and Khuller et al. [33]. They showed that for any ε gt; 0 there always exist (1+ ε, O(1/ε))-SLTs, and that the upper bound β = O(1/ε) on the lightness of SLTs cannot be improved. In this paper we show that using Steiner points one can build SLTs with logarithmic lightness, i.e., β = O(log 1/ε). This establishes an exponential separation between spanning SLTs and Steiner ones. One particularly remarkable point on our tradeoff curve is ε =0. In this regime our construction provides a shortest-path tree with weight at most O(log n)·w(MST(G)). Moreover, we prove matching lower bounds that show that all our results are tight up to constant factors. Finally, on our way to these results we settle (up to constant factors) a number of open questions that were raised by Khuller et al. [33] in SODA'93.

Original language | English |
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Title of host publication | Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |

Pages | 373-382 |

Number of pages | 10 |

DOIs | |

State | Published - 2011 |

Externally published | Yes |

Event | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States Duration: 22 Oct 2011 → 25 Oct 2011 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Conference

Conference | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |
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Country/Territory | United States |

City | Palm Springs, CA |

Period | 22/10/11 → 25/10/11 |

## Keywords

- Steiner points
- Steiner trees
- minimum spanning tree
- shortest-path tree