Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs
International Symposium on Computational Geometry
We present the first near-linear-time (1 + ε)-approximation algorithm for the diameter of a weighted unit-disk graph of n vertices, running in O n log 2 n time, for any constant ε > 0, improving the near-O n 3/2 -time algorithm of Gao and Zhang [STOC 2003]. Using similar ideas, we can construct a (1 + ε)-approximate distance oracle for weighted unit-disk graphs with O(1) query time, with a similar improvement in the preprocessing time, from near O n 3/2 to O n log 3 n . We also obtain new
... s for a number of other related problems in the weighted unit-disk graph metric, such as the radius and bichromatic closest pair. As a further application, we use our new distance oracle, along with additional ideas, to solve the (1 + ε)-approximate all-pairs bounded-leg shortest paths problem for a set of n planar points, with near O n 2.579 preprocessing time, O n 2 log n space, and O (log log n) query time, improving thus the near-cubic preprocessing bound by Roditty and Segal [SODA 2007]. Besides practical motivation from wireless networks, this collection of problems is interesting from the theoretical perspective as well since it connects computational geometry with graph data structures -indeed, our new algorithms will draw on ideas from both areas. Planar graph techniques. There has already been an extensive body of work devoted to distance oracles and shortest-path-related problems both for general and for planar graphs, the latter of which are of particular relevance to us. For example, Thorup  gave (1 + ε)approximate distance oracles for weighted, undirected planar graphs with O (n polylog n) preprocessing time and space, and O (1) query time, for any constant ε > 0, while subsequent work [19, 18, 16, 10] gave improvements and examined the dependency of the hidden factors on ε. Weimann and Yuster  presented a (1 + ε)-approximation algorithm for the diameter for weighted, undirected planar graphs, running in O n log 4 n time, for any constant ε > 0, improved later to O n log 2 n by Chan and Skrepetos , who also reduced the ε-dependency from exponential to polynomial (there has also been exciting recent breakthrough on exact algorithms for diameter and distance oracles in planar graphs, by Cabello  and subsequent researchers [15, 11, 14] ). All the above approximation results for planar graphs rely heavily on the concept of shortest-path separator: a set of shortest paths with common root, such that the removal of their vertices decomposes the graph into at least two disjoint subgraphs. Unfortunately, such separators do not seem directly applicable to unit-disk graphs, and not only because the latter may be dense. Indeed, by grid rounding, we can construct a sparse weighted graph G, such that it (i) approximately preserves distances in the original unit-disk graph G (e.g., see the proof of Lemma 2), and (ii) is "nearly planar", in the sense that each edge intersects at most a constant number of other edges. However, even for such a graph, it is not clear how to define a shortest-path separator that divides it cleanly into an inside and an outside because edges may "cross" over the separator. At least one prior paper  worked on extending shortest-path separators to unit-disk graphs, but the construction was complicated and achieved only constant approximation factors. Gao and Zhang's WSPD technique. In a seminal paper, Gao and Zhang  obtained the first nontrivial set of results on shortest-path problems in weighted unit-disk graphs, by adapting a familiar technique in computational geometry -namely, the well-separated pair decomposition (WSPD), introduced by Callahan and Kosaraju  for addressing proximity problems in the Euclidean (or L p ) metric and has since found countless applications. Gao and Zhang proposed a new variant of WSPDs for the weighted unit-disk graph metric and showed that any n-point set in two dimensions has a WSPD of near-linear (O (n log n)) size under the new definition. Consequently, they obtained a (1 + ε)-approximate distance oracle with O (n log n) size and O(1) query time, for any constant ε > 0. Unfortunately, its preprocessing time, O n 3/2 √ log n , is quite high and becomes the bottleneck when the technique is applied to offline problems such as computing the diameter. However, the issue is not constructing the WSPD itself, which can be done in near-linear time, but computing the shortest-path distances of a near-linear number of vertex pairs in the "nearly planar" graph G mentioned above, which takes almost n 3/2 time, by adapting a known exact distance oracle for planar graphs  (noting that G has balanced separators [22, 12] ). Cabello  has given an improved algorithm for computing multiple distances in planar graphs, and if it could be adapted here, the running time would be reduced to around n 4/3 . However, near-linear time still seems out of reach with current techniques.