We address two open problems in sorting with priced information, introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to find a sorting algorithm with small competitive ratio, defined as the (worst-case) ratio of the algorithm's cost to the cost of the cheapest proof of the sorted order. 1) When all costs are in $\{0,1,n,\infty\}$, we give an algorithm that has $\widetilde{O}(n^{3/4})$ competitive ratio. Our result refutes the hypothesis that a widely cited $\Omega(n)$ lower bound on the competitive ratio for finding the maximum extends to sorting. This lower bound by [Gupta, Kumar, FOCS 2000] uses costs in $\{0,1,n, \infty\}$ and was claimed as the reason why sorting with arbitrary costs seemed bleak and hopeless. Our algorithm also generalizes the algorithms for generalized sorting (all costs in $\{1,\infty\}$), a version initiated by [Huang, Kannan, Khanna, FOCS 2011] and addressed recently by [Kuszmaul, Narayanan, FOCS 2021]. 2) We answer the problem of bichromatic sorting posed by [CFGKRS]: We are given two sets $A$ and $B$ of total size $n$, and the cost of an $A-A$ comparison or a $B-B$ comparison is higher than an $A-B$ comparison. The goal is to sort $A \cup B$. An $\Omega(\log n)$ lower bound on competitive ratio follows from unit-cost sorting. We give a randomized algorithm with an almost-optimal w.h.p. competitive ratio of $O(\log^{3} n)$. We also study generalizations of the problem \emph{universal sorting} and \emph{bipartite sorting} (a generalization of nuts-and-bolts). Here, we define a notion of \textit{instance optimality}, and develop an algorithm for bipartite sorting which is $O(\log^{3} n)$ instance-optimal. Our framework of instance optimality applies to other static problems and may be of independent interest.

翻译：我们解决了[Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]提出的定价信息排序中的两个开放问题。在该设定中，不同比较操作具有不同（可能为无穷大）的成本。目标是设计一种具有较小竞争比的排序算法，该竞争比定义为算法在最坏情况下的成本与排序序的最廉价证明成本之比。1）当所有成本属于集合$\{0,1,n,\infty\}$时，我们给出了一种竞争比为$\widetilde{O}(n^{3/4})$的算法。该结果驳斥了被广泛引用的$\Omega(n)$下界（该下界针对寻找最大值问题）可推广至排序问题的假设。[Gupta, Kumar, FOCS 2000]提出的该下界使用$\{0,1,n,\infty\}$中的成本值，并曾被认为是任意成本下排序问题前景黯淡且无解的原因。我们的算法同时推广了广义排序算法（所有成本属于$\{1,\infty\}$）——该版本由[Huang, Kannan, Khanna, FOCS 2011]提出，近期由[Kuszmaul, Narayanan, FOCS 2021]解决。2）我们解决了[CFGKRS]提出的双色排序问题：给定总规模为$n$的两个集合$A$和$B$，其中$A-A$比较或$B-B$比较的成本高于$A-B$比较。目标是对$A \cup B$进行排序。单位成本排序给出竞争比的$\Omega(\log n)$下界。我们提出了一种随机算法，其高概率竞争比为$O(\log^{3} n)$，接近最优。我们还研究了该问题的推广形式——\emph{通用排序}和\emph{二分排序}（螺栓螺母问题的推广）。在此框架下，我们定义了\textit{实例最优性}概念，并开发了一种对于二分排序问题具有$O(\log^{3} n)$实例最优性的算法。我们的实例最优性框架可应用于其他静态问题，可能具有独立研究价值。