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Hairpin Completion Distance Lower Bound

Itai Boneh, Dvir Fried, Shay Golan, Matan Kraus

TL;DR

This work proves a conditional lower bound for computing the hairpin completion distance under SETH, showing no $O(n^{2-\varepsilon})$-time algorithm exists unless SETH fails, thereby matching the known $O(n^{2})$ upper bound up to subpolynomial factors. The authors construct a linear-time reduction from the LCS problem on ternary strings to the hairpin distance problem using a rich gadget-based encoding that yields $\mathsf{HDD}(x,y) = \sum_{\alpha} D(\alpha)(\#_\alpha(S)+\#_\alpha(T)) - \mathsf{LCS}(S,T)\cdot B$, and then bridge to the original hairpin distance via a linear transformation to $x',y'$. This reduction establishes that efficiency gains below quadratic time are unlikely under SETH, and the argument applies even on a small alphabet, highlighting the practical relevance for DNA-inspired string problems. The paper thus tightens the complexity landscape for hairpin-based string operations and clarifies the computational limits of distance computations in this model.

Abstract

Hairpin completion, derived from the hairpin formation observed in DNA biochemistry, is an operation applied to strings, particularly useful in DNA computing. Conceptually, a right hairpin completion operation transforms a string $S$ into $S\cdot S'$ where $S'$ is the reverse complement of a prefix of $S$. Similarly, a left hairpin completion operation transforms a string $S$ into $S'\cdot S$ where $S'$ is the reverse complement of a suffix of $S$. The hairpin completion distance from $S$ to $T$ is the minimum number of hairpin completion operations needed to transform $S$ into $T$. Recently Boneh et al. showed an $O(n^2)$ time algorithm for finding the hairpin completion distance between two strings of length at most $n$. In this paper we show that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the hairpin completion distance problem unless the Strong Exponential Time Hypothesis (SETH) is false. Thus, under SETH, the time complexity of the hairpin completion distance problem is quadratic, up to sub-polynomial factors.

Hairpin Completion Distance Lower Bound

TL;DR

This work proves a conditional lower bound for computing the hairpin completion distance under SETH, showing no -time algorithm exists unless SETH fails, thereby matching the known upper bound up to subpolynomial factors. The authors construct a linear-time reduction from the LCS problem on ternary strings to the hairpin distance problem using a rich gadget-based encoding that yields , and then bridge to the original hairpin distance via a linear transformation to . This reduction establishes that efficiency gains below quadratic time are unlikely under SETH, and the argument applies even on a small alphabet, highlighting the practical relevance for DNA-inspired string problems. The paper thus tightens the complexity landscape for hairpin-based string operations and clarifies the computational limits of distance computations in this model.

Abstract

Hairpin completion, derived from the hairpin formation observed in DNA biochemistry, is an operation applied to strings, particularly useful in DNA computing. Conceptually, a right hairpin completion operation transforms a string into where is the reverse complement of a prefix of . Similarly, a left hairpin completion operation transforms a string into where is the reverse complement of a suffix of . The hairpin completion distance from to is the minimum number of hairpin completion operations needed to transform into . Recently Boneh et al. showed an time algorithm for finding the hairpin completion distance between two strings of length at most . In this paper we show that for any there is no -time algorithm for the hairpin completion distance problem unless the Strong Exponential Time Hypothesis (SETH) is false. Thus, under SETH, the time complexity of the hairpin completion distance problem is quadratic, up to sub-polynomial factors.
Paper Structure (14 sections, 31 theorems, 1 equation)

This paper contains 14 sections, 31 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Reduction
  4. Intuition for the Reduction Correctness
  5. Well-Behaved Paths
  6. Properties of Paths in Gₓ
  7. Transitions Between Gadegets
  8. Cost of Well-Behaved Steps
  9. Correctness
  10. Reduction to Hairpin Deletion
  11. Missing Proofs from Section 4
  12. Missing proofs of Section 4.1
  13. Missing proofs of Section 4.2
  14. Analysis of Deletion of Synchronized Gadgets

Key Result

Theorem 2

Let $\varepsilon>0$. If there is an algorithm that computes the hairpin completion distance from $y$ to $x$ in $O(|x|^{2-\varepsilon})$ time, then $\mathsf{SETH}$ is false. This holds even if the input strings are over an alphabet of size $4$.

Theorems & Definitions (37)

  • Theorem 2
  • Definition 3: Hairpin Operations
  • Definition 4: Hairpin Operations, Modified definition
  • Definition 5: Hairpin Deletion Graph
  • Lemma 6: Reduction Correctness
  • Definition 7: Well-Behaved Path
  • Lemma 7: Optimal Well-Behaved Path
  • Corollary 9
  • Lemma 9: Always $01 *$ or $*\inv{10}$
  • Corollary 10
  • ...and 27 more