On Combinatorial Depth Measures

Given a set P = {p1, . . . , pn} of points and a point q in the plane, we define a function ψ(q) that provides a combinatorial characterization of the multiset of values {|P ∩Hi|}, where for each i ∈ {1, . . . , n}, Hi is the open half-plane determined by q and pi. We introduce two new natural measures of depth, perihedral depth and eutomic depth, and we show how to express these and the well-known simplicial and Tukey depths concisely in terms of ψ(q). The perihedral and eutomic depths of q with respect to P correspond respectively to the number of subsets of P whose convex hull contains q, and the number of combinatorially distinct bisections of P determined by a line through q. We present algorithms to compute the depth of an arbitrary query point in O(n log n) time and medians (deepest points) with respect to these depth measures in O(n) and O(n) time respectively. For comparison, these results match or slightly improve on the corresponding best-known running times for simplicial depth, whose definition involves similar combinatorial complexity.