# The Cardinal Complexity of Comparison-based Online Algorithms

We consider ordinal online problems, i.e., those tasks that only depend on the pairwise comparisons between elements in the input. E.g., the secretary problem and the game of googol. The natural approach to these tasks is to use ordinal online algorithms that at each step only consider relative ranking among the arrived elements, without looking at the numerical values of the input. We formally study the question of how cardinal algorithms (that can use numerical values of the input) can improve upon ordinal algorithms. We give a universal construction of the input distribution for any ordinal online problem, such that the advantage of the cardinal algorithms over the ordinal algorithms is at most 1+ε for arbitrary small ε> 0. However, the value range of the input elements in this construction is huge: O(n^3· n!/ε)↑↑ (n-1) for an input sequence of length n. Surprisingly, we also identify a natural family of hardcore problems that achieve a matching advantage of 1+ Ω(1/log^(c)N), where log^(c)N=loglog…log N with c iterative logs and c is an arbitrary constant c≤ n-2. We also consider a simpler variant of the hardcore problem, which we call maximum guessing and is closely related to the game of googol. We provide a much more efficient construction with cardinal complexity O(1/ε)^n-1 for this easier task. Finally, we study the dependency on n of the hardcore problem. We provide an efficient construction of size O(n), if we allow cardinal algorithms to have constant factor advantage against ordinal algorithms.

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