## Abstract

We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph G and a root node r, we seek a single spanning tree T of minimum stretch, where the stretch of T is defined to be the maximum ratio, over all terminal sets X, of the cost of the minimal sub-tree T_{X} of T that connects X to r to the cost of an optimal Steiner tree connecting X to r in G. Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. We provide a polynomial time UST construction for general graphs with 2 ^{O(√log n)}-stretch. We also give a polynomial time polylogarithmic-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first sub polynomial-stretch (o(n^{ε}) for any ε > 0) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs.

Original language | English (US) |
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Article number | 6375285 |

Pages (from-to) | 81-90 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: Oct 20 2012 → Oct 23 2012 |

## Keywords

- graph clustering
- hierarchical graph partition
- minor-free graphs
- universal Steiner tree

## ASJC Scopus subject areas

- Computer Science(all)