Approximate Nearest Neighbor Search in Metrics of Planar Graphs

We investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G = (V, E) on m vertices and a small “dataset” D ⊂ V of size n m, so that given a query point q ∈ V , one can quickly approximate dG(q, D) (the distance from q to its closest vertex in D) and find a vertex a ∈ D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices. For planar graphs G with maximum degree ∆, we show how to efficiently construct a compact data structure – of size ˜O(n(∆ + 1/ )) – that answers (1 + )-NNS queries in time ˜O(∆ + 1/ ). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into 2. We ...

• Type: Journal
• Year: 2015
• Where: APPROX
• Authors: Ittai Abraham, Shiri Chechik, Robert Krauthgamer, Udi Wieder
• Keywords: Algorithms, APPROX 2015