Instance and Universally Optimal Bounds for Imprecise Pareto Fronts

In the imprecise geometry model, the input is an imprecise point set, which is a family of regions $F = (R_1, \ldots,R_n)$, where for each $R_i$ one may retrieve the true point $p_i \in R_i$. By preprocessing $F$, we can construct the output, in our case the Pareto front, on $P$ faster. We efficiently construct the Pareto front of an imprecise point set in the plane. Efficiency is interpreted in two ways: minimizing (i) the number of retrievals, and (ii) the computation time used to determine the set of regions that must be retrieved and to construct the Pareto front. We present an algorithm to construct the Pareto front for possibly overlapping rectangles that is \emph{instance-optimal} with respect to the number of retrievals, meaning that for every fixed input $(F, P)$, there is no algorithm that retrieves asymptotically fewer regions to compute the output. This is a strong algorithmic quality, as it means that our algorithm is competitive even to clairvoyant algorithms which know a correct guess of the output and only have to verify its correctness. In terms of algorithmic running time, instance-optimality is provably unobtainable. We instead present an algorithm which is within a $\log n$-factor of instance-optimality. This generalizes earlier results to overlapping input regions, at only a minor cost in running time. For unit squares, we present an algorithm that is not only instance-optimal in the number of retrievals, but also \emph{universally} optimal in terms of running time, meaning that for any fixed set of regions $F$, no algorithm has a better worst-case running time for all possible point sets $P$. This is the first universally optimal algorithm for overlapping planar input. Compared to previous work, this result improves the degree of overlap, the preprocessing time, the number of retrievals, and the running time.