Cache oblivious search trees via binary trees of small height
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A Sparse Table Implementation of Priority Queues
Proceedings of the 8th Colloquium on Automata, Languages and Programming
Two Simplified Algorithms for Maintaining Order in a List
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
A locality-preserving cache-oblivious dynamic dictionary
Journal of Algorithms
A Tight Lower Bound for Online Monotonic List Labeling
SIAM Journal on Discrete Mathematics
SIAM Journal on Computing
Information Processing Letters
Tight lower bounds for the online labeling problem
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
SIGACT news online algorithms column 21: APPROX and ALGO
ACM SIGACT News
On randomized online labeling with polynomially many labels
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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In the online labeling problem with parameters n and m we are presented with a sequence of nkeys 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 c1, there are known algorithms that use at most O(n log(n)2) relabelings in total [9], and it was shown recently that this is asymptotically optimal [1]. For the case of m=θ(nC) for C1, algorithms are known that use O(n logn) relabelings. A matching lower bound was claimed in [7]. 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 Ω((n logn) / (loglogm−loglogn)).