Dynamization of Decomposable Searching Problems

In the complexity theory of geometric configurations (and other areas of algorithmic endeavor) one encounters a fair number of problems for which a very efficient static solution is known, but no alternative to complete reconstruction seems to come to mind when we wish to insert or delete even a single point. Bentley [1 3 recognized a large class of problems (which he called decomposable searching problems) for which there is hope that a reasonably efficient dynamic solution can be attained. Briefly, a searching problem is said to be decomposable if its solution can be synthesized at only nominal extra cost from the solutions of the very same problem for all distinct parts of some arbitrary partition of the original point-set. The question to determine which point of a given set is closest to some (varying) point x is a typical example of a decomposable searching problem. Bentley’s primary technique of dynamization for these problems consists of finding a partition of the set into pieces, each statically organised, such that insertions at the tow end (see Fig. 1) most often require a repartition of Fig. 1.