On Randomized Online Labeling with Polynomially Many Labels

Abstract

Abstract. We prove an optimal lower bound on the complexity of randomized algorithms for the online labeling problem with polynomially many labels. All previous work on this problem (both upper and lower bounds) only applied to deterministic algorithms, so this is the first paper addressing the (im)possibility of faster randomized algorithms. Our lower bound Ω(n log(n)) matches the complexity of a known deterministic algorithm for this setting of parameters so it is asymptotically optimal. 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, 2, . . . , m} so that the order of labels (strictly) respects the ordering 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.

Online List Labeling: Breaking the $\log^2n$ Barrier

“…A surprising aspect of our results is how they contrast with the polynomial regime m = n 1+Θ (1) where randomized and deterministic algorithms are asymptotically equivalent [6,7,23]. Our final upper-bound result considers a continuum between these regimes, where m = ω(n) ∩ n o (1) .…”

“…In the polynomial regime, when m n = n Θ (1) , the amortized cost becomes O(log n) [2,33,43]. These bounds are known to be tight for both deterministic and randomized algorithms [6,7,23].…”

“…Starting in 1990, there has been a long line of work towards establishing a matching Ω(log 2 n) lower bound [22,23,[26][27][28]. It is known that any deterministic algorithm requires Ω(log 2 n) amortized cost per insertion [22].…”

“…These lower bounds tell us that, if an algorithm is to beat the O(log 2 n) bound, then the algorithm must be both randomized and non-smooth. Whether or not any such algorithm is possible has remained the central open question [22,23,[26][27][28]40] in this research area (see also discussion of the problem in open-problem surveys and textbooks [32,48,64]). Several sources [26][27][28] have conjectured that Θ(log 2 n) cost is optimal in general.…”

On Online Labeling with Large Label Set

“…All of the work mentioned so far (both upper and lower bounds) is for the case of deterministic algorithms. In work subsequent to this paper, a subset of the authors [5] showed that the Ω(n log n) lower bound for the case m = n 1+C of polynomially many labels extends for randomized algorithms. The results in that paper do not subsume this one because (a) The lower bounds for randomized algorithms don't extend to the range that m is superpolynomial in n (as do the results here) and (b) The lower bound proofs here are conceptually simpler.…”