Competitive search for longest empty intervals

Paper in proceeding, 2008

A problem arising in statistical data analysis and pattern recognition is to find a longest interval free of data points, given a set of data points in the unit interval. We use the inverse length of the empty interval as a parameter in the complexity bounds, since it is small in statistically relevant cases. For sorted point sets we get nearly optimal strategies. While the asymptotic complexities are trivial, achieving an optimal number of operations appears to be difficult. Constant factors can be of practical interest for huge data sets. We derive deterministic and randomized upper and lower bounds. Matching bounds and smooth trade-offs between the different operations (reads, comparisons, subtractions) are open questions. For unsorted point sets, the complexity is at least linear. Therefore we also use statistical inference to get approximate solutions in sublinear time.