On online labeling with large label set ∗

In the online labeling problem with parameters n and m we are presented with a sequence of n items from a totally ordered universe U and must assign each arriving item a label from the label set {1, . . . ,m} so that the order of labels respects the order on U . As new items 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)) relabelings in total [16], and it was shown recently that this is asymptotically optimal [3, 4]. For the case of m = Θ(n) for constant C > 0, algorithms are known that use O(n log n) relabelings. A matching lower bound was provided in [12]. The lower bound ∗A preliminary version of this paper appeared in the Proceedings of the 2012 European Symposium on Algorithms [1] †The first three authors gratefully acknowledge support from the Charles University Grant Agency (grant No. 265 111 and 344 711) and SVV project no. 265 314. ‡Part of the work done while visiting Aarhus University, partially supported by the Sino-Danish Center CTIC (funded under the grant 61061130540). Supported in part by GA ČR P202/10/0854, grant IAA100190902 of GA AV ČR, Center of Excellence CE-ITI (P202/12/G061 of GA ČR) and RVO: 67985840. §The work of this author was done while on sabbatical at Princeton University and was also supported in part by NSF under grants CCF-0832787 CCF-1218711.