Maintaining nets and net trees under incremental motion

Abstract

The problem of maintaining geometric structures for points in motion has been well studied over the years. Much theoretical work to date has been based on the assumption that point motion is continuous and predictable, but in practice, motion is typically presented incrementally in discrete time steps and may not be predictable. We consider the problem of maintaining a data structure for a set of points undergoing such incremental motion. We present a simple online model in which two agents cooperate to maintain the structure. One defines the data structure and provides a collection of certificates, which guarantee the structure's correctness. The other checks that the motion over time satisfies these certificates and notifies the first agent of any violations. We present efficient online algorithms for maintaining both nets and net trees for a point set undergoing incremental motion in a space of constant dimension. We analyze our algorithms' efficiencies by bounding their competitive ratios relative to an optimal algorithm. We prove a constant factor competitive ratio for maintaining a slack form of nets, and our competitive ratio for net trees is proportional to the square of the tree's height. © 2009 Springer-Verlag Berlin Heidelberg.