On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting

TL;DR

This paper introduces bipartite sorting, a generalized nuts and bolts problem, providing an instance-optimal randomized algorithm within a polylogarithmic factor and extending the concept to DAG sorting, impacting related sorting problems.

Contribution

It defines a neighborhood-based notion of instance optimality for bipartite sorting and presents a near instance-optimal randomized algorithm, also extending the framework to DAG sorting.

Findings

The algorithm is within O(log^3(n+m)) of instance-optimal w.h.p.

Bipartite sorting exhibits wide complexity variation among instances.

The DAG sorting approach challenges existing lower bounds on sorting with priced information.

Abstract

We generalize the classical nuts and bolts problem to a setting where the input is a collection of $n$ nuts and $m$ bolts, and there is no promise of any matching pairs. It is not allowed to compare a nut directly with a nut or a bolt directly with a bolt, and the goal is to perform the fewest nut-bolt comparisons to discover the partial order between the nuts and bolts. We term this problem \emph{bipartite sorting}. We show that instances of bipartite sorting of the same size exhibit a wide range of complexity, and propose to perform a fine-grained analysis for this problem. We rule out straightforward notions of instance-optimality as being too stringent, and adopt a \emph{neighborhood-based} definition. Our definition may be of independent interest as a unifying lens for instance-optimal algorithms for other static problems existing in literature. This includes problems like sorting (Estivill-Castro and Woods, ACM Comput. Surv. 1992), convex hull (Afshani, Barbay and Chan, JACM 2017), adaptive joins (Demaine, L\'{o}pez-Ortiz and Munro, SODA 2000), and the recent concept of universal optimality for graphs (Haeupler, Hlad\'ik, Rozho\v{n}, Tarjan and T\v{e}tek, 2023). As our main result on bipartite sorting, we give a randomized algorithm that is within a factor of $O(\log ^3 (n+m))$ of being instance-optimal w.h.p., with respect to the neighborhood-based definition. As our second contribution, we generalize bipartite sorting to DAG sorting, when the underlying DAG is not necessarily bipartite. As an unexpected consequence of a simple algorithm for DAG sorting, we rule out a potential lower bound on the widely-studied problem of \emph{sorting with priced information}, posed by (Charikar, Fagin, Guruswami, Kleinberg, Raghavan and Sahai, STOC 2000).