Instant approximate 1-center on road networks via embeddings

Abstract

We study the 1-center problem on road networks, an important problem in GIS. Using Euclidean embeddings, and reduction to fast nearest neighbor search, we devise an approximation algorithm for this problem. Our initial experiments on real world data sets indicate fast computation of constant factor approximate solutions for query sets much larger than previously computable using exact techniques.

Geometric Clustering and its Applications

“…Using Euclidean embeddings, and reduction to fast nearest neighbor search, we devise an approximation algorithm for this problem. Our initial experiments on real world data sets indicate fast computation of constant factor approximate solutions for query sets much larger than previously computable using exact techniques [3,16].…”

Fast k-clustering queries on embeddings of road networks

“…We wish to emphasize the difference between the problem that we examine in this paper and those two important areas: In the parlance of this paper, both of those areas require that P ⊂ Q, whereas for us P and Q can be and are distinct. We are instead following up on problems suggested by [15] as well as [13,7,15]. In effect, we show that these various problems can be reduced to the well-studied clustering approaches discussed above.…”

“…This algorithm is not new here it was described in [13,7] in the context of the 1-center problem and in context of the 1-median and 1-center problem in [15]. In [15], the authors give a proof that the approximation factor for the algorithm in the 1-centers case is √ 2 and that this bound is tight.…”

“…The authors would like to thank Dr. David Mount for initial discussions on this problem and for pointing out that in case of 1-means, our technique could actually give the exact answer instead of an approximation. We would like to thank the anonymous reviewers for their helpful suggestions, as well as Bradley Neff and Samidh Chatterjee for discussions, help with the implementation, and code that they made available to us [7].…”

“…[7] have recently shown that the shortest path metric on road networks, d(·), can be embedded in Euclidean spaces with low distortion. That is done by making use of Multidimensional Scaling ("MDS") [2] wherein the data (in this case nodes on the road network) are given pairwise similarities (in this case pair-wise distances) and the representatives of those data in Euclidean space are chosen in such a way as to minimize some functional.…”