  A growing line of work shows how learned predictions can be used to break
through worst-case barriers to improve the running time of an algorithm.
However, incorporating predictions into data structures with strong theoretical
guarantees remains underdeveloped. This paper takes a step in this direction by
showing that predictions can be leveraged in the fundamental online list
labeling problem. In the problem, n items arrive over time and must be stored
in sorted order in an array of size Theta(n). The array slot of an element is
its label and the goal is to maintain sorted order while minimizing the total
number of elements moved (i.e., relabeled). We design a new list labeling data
structure and bound its performance in two models. In the worst-case
learning-augmented model, we give guarantees in terms of the error in the
predictions. Our data structure provides strong guarantees: it is optimal for
any prediction error and guarantees the best-known worst-case bound even when
the predictions are entirely erroneous. We also consider a stochastic error
model and bound the performance in terms of the expectation and variance of the
error. Finally, the theoretical results are demonstrated empirically. In
particular, we show that our data structure has strong performance on real
temporal data sets where predictions are constructed from elements that arrived
in the past, as is typically done in a practical use case.
