A lower bound for the complexity of the union-split-find problem

We prove a Theta(loglog n) (i.e. matching upper and lower) bound on the complexity of the Union-Split-Find problem, a variant of the Union-Find problem. Our lower bound holds for all pointer machine algorithms and does not require the separation assumption used in the lower bound arguments of Tarjan [T79] and Blum [B86]. We complement this with a Theta(log n) bound for the Split-Find problem under the separation assumption. This shows that the separation assumption can imply an exponential loss in efficiency

Approximate Motion Planning and the Complexity of the Boundary . . .

We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have running time f~(n²) where n is the number of obstacle corners. We introduce the tightness of a motion planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with running time O((~-._.!_ _ + 1)n(log n)²), where a> b ecrlt are the lenghts of the sides of a rectangle and ~crit is the tightness of the problem. We show further that the complexity ( = number of vertices) of the boundary of n bow-ties (c.f. Figure 1.1) is O(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented