Sorting with Priced Comparisons: The General, the Bichromatic, and the Universal

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.