On the Hardness and Approximation of Euclidean DBSCAN

DBSCAN is a method proposed in 1996 for clustering multi-dimensional points, and has received extensive applications. Its computational hardness is still unsolved to this date. The original KDD‚96 paper claimed an algorithm of O(n log n) ”average runtime complexity„ (where n is the number of data points) without a rigorous proof. In 2013, a genuine O(n log n)-time algorithm was found in 2D space under Euclidean distance. The hardness of dimensionality d ≥3 has remained open ever since. This article considers the problem of computing DBSCAN clusters from scratch (assuming no existing indexes) under Euclidean distance. We prove that, for d ≥3, the problem requires ω(n 4/3) time to solve, unless very significant breakthroughs—ones widely believed to be impossible—could be made in theoretical computer science. Motivated by this, we propose a relaxed version of the problem called ρ-approximate DBSCAN, which returns the same clusters as DBSCAN, unless the clusters are ”unstable„ (i.e., they change once the input parameters are slightly perturbed). The ρ-approximate problem can be settled in O(n) expected time regardless of the constant dimensionality d. The article also enhances the previous result on the exact DBSCAN problem in 2D space. We show that, if the n data points have been pre-sorted on each dimension (i.e., one sorted list per dimension), the problem can be settled in O(n) worst-case time. As a corollary, when all the coordinates are integers, the 2D DBSCAN problem can be solved in O(n log log n) time deterministically, improving the existing O(n log n) bound.