An Optimal Balanced Partitioning of a Set of 1D Intervals

Given a set of 1D intervals and a desired partition number, this paper studies on how to make an optimal partitioning of these intervals, such that the number of intervals between the largest partition and smallest partition is minimal among all possible partitioning schemes. Though seemingly easy at the first glance, this problem has its difficulty due to the fact that an interval ``striding'' multiple partitions should be counted multiple times. Previously we have given an approximated solution to this problem by employing a simulated annealing approach [Yang & Chiueh, 2006], which could give satisfactory results in most cases; however, there is no theoretical guarantee on its optimality. In this paper, we propose a method that could both optimally and deterministically partition a given set of 1D intervals into a given number of partitions. This problem originates from dealing with the partition of a large volume data, where a more balanced partitioning could facilitate efficient out-of-core volume visualization faced in the computer graphics world. We will show that some types of load balancing problem could also be formulated as a balanced interval partitioning problem. As a result, seeking better solutions to this problem has become an important issue.