Lower bound for sparse Euclidean spanners

Authors:
Pankaj K. Agarwal;Yusu Wang;Peng Yin
Affiliations:
Duke University, Durham, NC;Duke University, Durham, NC;Duke University, Durham, NC
Venue:
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2005

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Abstract

Given a one-dimensional graph G such that any two consecutive nodes are unit distance away, and such that the minimum number of links between any two nodes (the diameter of G) is O(log n), we prove an Ω(n log n/log log n) lower bound on the sum of lengths of all the edges (i.e., the weight of G). The problem is a variant of the widely studied partial sum problem. This in turn provides a lower bound on Euclidean spanner graphs with small diameter and low weight, showing that the upper bound from [1] is almost tight.