3 research outputs found
On Online Labeling with Polynomially Many Labels
In the online labeling problem with parameters n and m we are presented with
a sequence of n keys from a totally ordered universe U and must assign each
arriving key a label from the label set {1,2,...,m} so that the order of labels
(strictly) respects the ordering on U. As new keys arrive it may be necessary
to change the labels of some items; such changes may be done at any time at
unit cost for each change. The goal is to minimize the total cost. An
alternative formulation of this problem is the file maintenance problem, in
which the items, instead of being labeled, are maintained in sorted order in an
array of length m, and we pay unit cost for moving an item.
For the case m=cn for constant c>1, there are known algorithms that use at
most O(n log(n)^2) relabelings in total [Itai, Konheim, Rodeh, 1981], and it
was shown recently that this is asymptotically optimal [Bul\'anek, Kouck\'y,
Saks, 2012]. For the case of m={\Theta}(n^C) for C>1, algorithms are known that
use O(n log n) relabelings. A matching lower bound was claimed in [Dietz,
Seiferas, Zhang, 2004]. That proof involved two distinct steps: a lower bound
for a problem they call prefix bucketing and a reduction from prefix bucketing
to online labeling. The reduction seems to be incorrect, leaving a (seemingly
significant) gap in the proof. In this paper we close the gap by presenting a
correct reduction to prefix bucketing. Furthermore we give a simplified and
improved analysis of the prefix bucketing lower bound. This improvement allows
us to extend the lower bounds for online labeling to the case where the number
m of labels is superpolynomial in n. In particular, for superpolynomial m we
get an asymptotically optimal lower bound {\Omega}((n log n) / (log log m - log
log n)).Comment: 15 pages, Presented at European Symposium on Algorithms 201
Online List Labeling with Predictions
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