Kinetic sorted list


A kinetic sorted list is a kinetic data structure for maintaining a list of points under motion in sorted order. It is used as a kinetic predecessor data structure, and as a component in more complex kinetic data structures such as kinetic closest pair.

Implementation

This data structure maintains a list of the elements in sorted order, with the certificates enforcing the order between adjacent elements. When a certificate fails, the concerned elements are swapped. Then at most three certificates must be updated, the certificate of the swapped pair, and the two certificates involving the swapped elements and the elements of the sorted list which directly precede and follow the swapped pair.
For example, given a sorted list, the certificates will be , , , , . If the certificate fails, the list will be updated to, and the certificates , , and , will be replaced with , , and , respectively. The new set of certificates will be , , , ,

Analysis

This kinetic data structure is:
This data structure can be generalized to a kinetic data structure which can return a sorted list of points in time and processes events total, assuming pseudo algebraic trajectories, where is a parameter of the data structure. Thus, a maintenance-time versus query-time tradeoff can be made to tune to specific applications.
In the generalized data structure, the points are partitioned arbitrarily into m subsets of size, and kinetic sorted lists are maintained on the subsets. Each sorted sublist needs to process events maximum, since there are swaps of each of the pairs of elements. Thus the total time required to maintain the data structure is. Requests for the sorted list can then be answered in by merging the sorted sublists with mergesort.