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

Mayank Goswami, Riko Jacob

TL;DR

This work generalizes nuts-and-bolts to bipartite sorting, where $n$ nuts and $m$ bolts must be ordered through cross-type comparisons alone. It introduces a neighborhood-based notion of instance-optimality to capture the wide variability across instances, and the main algorithm InversionSort achieves an $O((\log (n+m))^3)$-approximate instance-optimal performance with high probability. The authors extend bipartite sorting to DAG sorting, connecting to generalized sorting and the sorting with priced information framework, and show that a naive lower bound does not extend to price-based models. On the pricing side, they develop a $0$-$1$-$F$-$\infty$ algorithm with a tilde $O(n^{3/4})$ competitive ratio, using Hamiltonian by predecessors and a staged DAG-based approach, illustrating how instance-aware strategies can outperform worst-case guarantees. Overall, the paper unifies several strands of instance-optimality and pricing-based querying, offering a fine-grained, neighborhood-centric lens for static problems and potentially informing broader algorithm design beyond sorting problems.

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, López-Ortiz and Munro, SODA 2000), and the recent concept of universal optimality for graphs (Haeupler, Hladík, Rozhoň, Tarjan and Tětek, 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).

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

TL;DR

This work generalizes nuts-and-bolts to bipartite sorting, where nuts and bolts must be ordered through cross-type comparisons alone. It introduces a neighborhood-based notion of instance-optimality to capture the wide variability across instances, and the main algorithm InversionSort achieves an -approximate instance-optimal performance with high probability. The authors extend bipartite sorting to DAG sorting, connecting to generalized sorting and the sorting with priced information framework, and show that a naive lower bound does not extend to price-based models. On the pricing side, they develop a --- algorithm with a tilde competitive ratio, using Hamiltonian by predecessors and a staged DAG-based approach, illustrating how instance-aware strategies can outperform worst-case guarantees. Overall, the paper unifies several strands of instance-optimality and pricing-based querying, offering a fine-grained, neighborhood-centric lens for static problems and potentially informing broader algorithm design beyond sorting problems.

Abstract

We generalize the classical nuts and bolts problem to a setting where the input is a collection of nuts and 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, López-Ortiz and Munro, SODA 2000), and the recent concept of universal optimality for graphs (Haeupler, Hladík, Rozhoň, Tarjan and Tětek, 2023). As our main result on bipartite sorting, we give a randomized algorithm that is within a factor of 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).
Paper Structure (25 sections, 20 theorems, 5 equations, 2 figures, 3 algorithms)

This paper contains 25 sections, 20 theorems, 5 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Results and Technical Overview
  3. Bipartite Sorting
  4. Defining Instance-Optimality
  5. InversionSort: An Almost Instance Optimal Algorithm
  6. Unexpected Result: Generalized Sorting and Sorting with Priced Information
  7. Problem Definitions
  8. Results on Bipartite Sorting
  9. Defining Instance-Optimality
  10. Lower Bounds on OPT
  11. InversionSort and its $O(\log^3 n)$ instance-optimal guarantee
  12. Description of InversionSort
  13. Putting everything together: Proof of Theorem \ref{['thm:bipartiteinversionsort']}
  14. Result on Sorting with Priced Information: Lower bound does not extend
  15. Algorithm details
  16. ...and 10 more sections

Key Result

Theorem 1

[Instance Optimality of InversionSort] There exists a constant $c>0$, such that for every instance $\mathcal{I}$, the cost of InversionSort on $\mathcal{I}$ is, with probability at least $1-1/N$, at most $c(\log N)^3 \mathop{\mathrm{OPT}}\nolimits(\mathcal{I})$, where $\mathop{\mathrm{OPT}}\nolimits

Figures (2)

  • Figure 1: An example output to an instance of the bipartite sorting problem. Continuous runs of incomparable nuts and bolts are called "stripes".
  • Figure 2: The landscape of sorting with priced information. Solid arrows go from a problem to its special case. HA✓ indicates that the Hamiltonian path is assumed to exist and HA$\times$ indicates that a Hamiltonian path may not exist. Problems shaded in gray are introduced or studied in this paper for the first time. Dotted arrows highlight our results, arrows with $O$ show algorithms carrying over from one problem to another, and $\Omega$ show lower bounds not carrying over.

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Definition 3: Bipartite Sorting
  • Definition 4: Implied and essential edge, transitive reduction aho1972transitive
  • Definition 5: DAG Sorting, arbitrary costs
  • Definition 6: Sorting with Priced Information charikar2002query
  • Definition 7: One-inversion Instance
  • Definition 8: $\alpha$-Instance Optimality
  • Definition 9: Instance Optimality Distribution
  • Lemma 9: Transitive Reduction or Verification lower bound
  • ...and 21 more