Dominance Products and Faster Algorithms for High-Dimensional Closest Pair under L∞

We give improved algorithmic time bounds for two fundamental problems, and establish a new complexity connection between them. The first is computing dominance product: given a set of n points p1, . . . , pn in R , compute a matrix D, such that Dri, js “ ˇ ˇtk | pirks ď pjrksu ˇ ; this is the number of coordinates at which pj dominates pi. Dominance product computation has often been applied in algorithm design over the last decade. The second problem is the L8 Closest Pair in high dimensions: given a set S of n points in R , find a pair of distinct points in S at minimum distance under the L8 metric. When d is constant, there are efficient algorithms that solve this problem, and fast approximate solutions are known for general d. However, obtaining an exact solution in very high dimensions seems to be much less understood. We significantly simplify and improve previous results, showing that the problem can be solved by a deterministic strongly-polynomial algorithm that runs in OpDP pn, dq log nq time, whereDP pn, dq is the time bound for computing the dominance product for n points in R. For integer coordinates from some interval r ́M,M s, and for d “ n for some r ą 0, we obtain an algorithm that runs in Õ ` mintMnωp1,r,1q, DP pn, dqu ̆ time, where ωp1, r, 1q is the exponent of multiplying an nˆ n matrix by an n ˆ n matrix.