A Linear Time Euclidean Spanner on Imprecise Points

Ans-spanner on a setS ofn points in R d is a graph onS where for every two points p;q2 S, there exists a path between them in G whose length is less than or equal to sj pqj wherejpqj is the Euclidean distance between p and q. In this paper, we consider the construction of a Euclidean spanner for imprecise points where we take advantage of prior, inexact knowledge of our input. In particular, in the rst phase, we preprocess n ddimensional balls with radius r that are approximations of the input points with the guarantee that each input point lies within its respective ball. In the second phase, the specic points are revealed and we quickly compute a spanner using data from the preprocessing phase. We can compute (or update) the (1 + ")-spanner in time O(n(r + 1 ) d log(r + 1 )) after O(n(r + 1 ) d log ) preprocessing time where is the ratio between the furthest and the closest pair of points. Our algorithm does not have any restrictions on the distribution of the points. It is the rst such algorithm with linear (update) running time.