Kinetic heap-ordered trees: Tight analysis and improved algorithms

The most natural kinetic data structure for maintaining the maximum of a collection of continuously changing numbers is the kinetic heap. Basch, Guibas, and Ramkumar proved that the maximum number of events processed by a kinetic heap with n numbers changing as linear functions of time is O(n log n) and Ω(n log n). We prove that this number is actually Θ(n log n). In the kinetic heap, a linear number of events are stored in a priority queue, consequently, it takes O(log n) time to determine the next event at each iteration. We also present a modified version of the kinetic heap that processes O(n log n/ log log n) events, with the same O(log n) time complexity to determine the next event.