On building fast kd-Trees for Ray Tracing, and on doing that in O(N log N)

Though a large variety of efficiency structures for ray tracing exist, kd-trees today seem to slowly become the method of choice. In particular, kd-trees built with cost estimation functions such as a surface area heuristic (SAH) seem to be important for reaching high performance. Unfortunately, most algorithms for building such trees have a time complexity of O(N log
²
N), or even O(N
²
). In this paper, we analyze the state of the art in building good kd-trees for ray tracing, and eventually propose an algorithm that builds SAH kd-trees in O(N log N), the theoretical lower bound