Markov Incremental Constructions

Abstract

A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be "locally" random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology.We generalize Mulmuley's theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson-Shor type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to average-case analysis in computational geometry.

Data Structures on Event Graphs

“…If the walk is adversarial, we can achieve a similar result with near-linear storage. The former result is in the same spirit as previous work by the authors on randomized incremental construction (RIC) for Markov sources [3]. RIC is a fundamental algorithmic paradigm in computational geometry that uses randomness for the construction of certain geometric objects, and we showed that there is no significant loss of efficiency if the randomness comes from a Markov chain with sufficiently high conductance.…”