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From Amortized to Worst Case Delay in Enumeration Algorithms

Florent Capelli, Yann Strozecki

TL;DR

This work investigates turning algorithms with polynomial amortized delay into algorithms with polynomial worst-case delay for enumeration problems, introducing geometric regularization to achieve this with polynomial-space complexity. It proves the surprising result that $ ext{DelayP}^{ ext{poly}}= ext{AmDelayP}^{ ext{poly}}$, and extends to incremental delay under adaptive schemes, while establishing robust lower bounds that highlight intrinsic limits when preserving output order or operating under blackbox access. The authors also provide detailed RAM-based implementations to realize these regularization schemes and discuss the practical and theoretical implications of their results. Overall, the paper offers a comprehensive framework for designing space-efficient, delay-guaranteed enumerators and delineates the boundaries of what is achievable under blackbox access.

Abstract

The quality of enumeration algorithms is often measured by their delay, that is, the maximal time spent between the output of two distinct solutions. If the goal is to enumerate $t$ distinct solutions for any given $t$, then another relevant measure is the maximal time needed to output $t$ solutions divided by $t$, a notion we call the amortized delay of the algorithm, since it can be seen as the amortized complexity of the problem of enumerating $t$ elements in the set. In this paper, we study the relation between these two notions of delay, showing different schemes allowing one to transform an algorithm with polynomial amortized delay for which one has a blackbox access into an algorithm with polynomial delay. We complement our results by providing several lower bounds and impossibility theorems in the blackbox model.

From Amortized to Worst Case Delay in Enumeration Algorithms

TL;DR

This work investigates turning algorithms with polynomial amortized delay into algorithms with polynomial worst-case delay for enumeration problems, introducing geometric regularization to achieve this with polynomial-space complexity. It proves the surprising result that , and extends to incremental delay under adaptive schemes, while establishing robust lower bounds that highlight intrinsic limits when preserving output order or operating under blackbox access. The authors also provide detailed RAM-based implementations to realize these regularization schemes and discuss the practical and theoretical implications of their results. Overall, the paper offers a comprehensive framework for designing space-efficient, delay-guaranteed enumerators and delineates the boundaries of what is achievable under blackbox access.

Abstract

The quality of enumeration algorithms is often measured by their delay, that is, the maximal time spent between the output of two distinct solutions. If the goal is to enumerate distinct solutions for any given , then another relevant measure is the maximal time needed to output solutions divided by , a notion we call the amortized delay of the algorithm, since it can be seen as the amortized complexity of the problem of enumerating elements in the set. In this paper, we study the relation between these two notions of delay, showing different schemes allowing one to transform an algorithm with polynomial amortized delay for which one has a blackbox access into an algorithm with polynomial delay. We complement our results by providing several lower bounds and impossibility theorems in the blackbox model.

Paper Structure

This paper contains 39 sections, 27 theorems, 11 equations, 1 figure, 7 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related work.
  4. Organization of the paper.
  5. Preliminaries
  6. Enumeration problems.
  7. Computational model.
  8. Enumeration process and delay.
  9. Complexity classes.
  10. Blackbox oracle access.
  11. Regularization scheme.
  12. Buffer Based Regularization
  13. Classical Method
  14. Adaptative Buffering
  15. Amortizing Self-Reducible Problems
  16. ...and 24 more sections

Key Result

Theorem 1

For every enumeration process $\mathcal{I}$, we have $\mathsf{avg} \mathsf{delay}(\mathcal{I}) \leq \mathsf{am} \mathsf{delay}(\mathcal{I}) \leq \mathsf{delay}(\mathcal{I})$.

Figures (1)

  • Figure 1: A traversal of the tree of partial solutions by the flashlight search. We represent the subtree visited between time $t$ when the algorithm visits $u$ and time $t'$ when the algorithm visits $v$. The subproblems completely solved recursively are in blue, the path between $u$ and $v$ in red.

Theorems & Definitions (49)

  • Theorem 1
  • proof
  • Definition 1: Polynomial Amortized Delay
  • Definition 2: Polynomial Delay
  • Lemma 1: Folklore
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • ...and 39 more