Incremental non-dominated sorting with O(N) insertion for the two-dimensional case

We propose a new algorithm for incremental non-dominated sorting of two-dimensional points. The data structure which stores non-dominating layers is based on a tree of Cartesian trees. If there are N points in M layers, the running time for of an insertion is O(M(1 + log(N/M)) + log M log(N/ log M)), which is O(N) in the worst case. This algorithm can be a basic building block for efficient implementations of steady-state multiobjective algorithms such as NSGA-II.