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Enhancing Binary Search via Overlapping Partitions

Kaan Buyukkalayci, Merve Karakas, Xinlin Li, Christina Fragouli

TL;DR

This work addresses the challenge of locating a target area under noisy binary decisions by introducing a binary search framework with tunable overlapping partitions. The method adds controlled redundancy through overlap parameter $\alpha$, enabling a direct trade-off between robustness (lower $P_e$) and search depth $\beta$ in both discrete and continuous domains. The authors derive a Voronoi-region-based analysis of the error event, provide a closed-form step-error expression under symmetric sensing, and specialize to a 1D setting with a continuous-limit interpretation. Numerical results in 1D illustrate the benefits of overlap, and sensor placement optimization highlights practical guidelines for reducing error. The framework offers a principled extension to noisy search paradigms with broad applicability to area localization in sensor networks and related decision problems.

Abstract

This paper considers the task of performing binary search under noisy decisions, focusing on the application of target area localization. In the presence of noise, the classical partitioning approach of binary search is prone to error propagation due to the use of strictly disjoint splits. While existing works on noisy binary search propose techniques such as query repetition or probabilistic updates to mitigate errors, they often lack explicit mechanisms to manage the trade-off between error probability and search complexity, with some providing only asymptotic guarantees. To address this gap, we propose a binary search framework with tunable overlapping partitions, which introduces controlled redundancy into the search process to enhance robustness against noise. We analyze the performance of the proposed algorithm in both discrete and continuous domains for the problem of area localization, quantifying how the overlap parameter impacts the trade-off between search tree depth and error probability. Unlike previous methods, this approach allows for direct control over the balance between reliability and efficiency. Our results emphasize the versatility and effectiveness of the proposed method, providing a principled extension to existing noisy search paradigms and enabling new insights into the interplay between partitioning strategies and measurement reliability.

Enhancing Binary Search via Overlapping Partitions

TL;DR

This work addresses the challenge of locating a target area under noisy binary decisions by introducing a binary search framework with tunable overlapping partitions. The method adds controlled redundancy through overlap parameter , enabling a direct trade-off between robustness (lower ) and search depth in both discrete and continuous domains. The authors derive a Voronoi-region-based analysis of the error event, provide a closed-form step-error expression under symmetric sensing, and specialize to a 1D setting with a continuous-limit interpretation. Numerical results in 1D illustrate the benefits of overlap, and sensor placement optimization highlights practical guidelines for reducing error. The framework offers a principled extension to noisy search paradigms with broad applicability to area localization in sensor networks and related decision problems.

Abstract

This paper considers the task of performing binary search under noisy decisions, focusing on the application of target area localization. In the presence of noise, the classical partitioning approach of binary search is prone to error propagation due to the use of strictly disjoint splits. While existing works on noisy binary search propose techniques such as query repetition or probabilistic updates to mitigate errors, they often lack explicit mechanisms to manage the trade-off between error probability and search complexity, with some providing only asymptotic guarantees. To address this gap, we propose a binary search framework with tunable overlapping partitions, which introduces controlled redundancy into the search process to enhance robustness against noise. We analyze the performance of the proposed algorithm in both discrete and continuous domains for the problem of area localization, quantifying how the overlap parameter impacts the trade-off between search tree depth and error probability. Unlike previous methods, this approach allows for direct control over the balance between reliability and efficiency. Our results emphasize the versatility and effectiveness of the proposed method, providing a principled extension to existing noisy search paradigms and enabling new insights into the interplay between partitioning strategies and measurement reliability.
Paper Structure (17 sections, 4 theorems, 35 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 4 theorems, 35 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Binary Search with Overlapping Partitions
  4. Main Results (Error Probability)
  5. Voronoi Description of the Error Event
  6. A Simplified Formula under Symmetric Measurements
  7. Analysis in One-Dimensional Search Spaces
  8. Evaluation
  9. Conclusion
  10. Proof of Lemma \ref{['thm:main']}
  11. Proof of Theorem \ref{['thm:symm']}
  12. Proof of Corollary \ref{['thm:cont']}
  13. Parameters Specification and Extended Results
  14. Amount of Overlap vs Error Probability
  15. Amount of Overlap vs. Tree Depth (Delay)
  16. ...and 2 more sections

Key Result

Lemma 1

For the case where $\boldsymbol{l_T} \notin \mathcal{H}^t_1 \cap \mathcal{H}^t_2$, assume $\mathcal{H}_m^t \subseteq \mathcal{H}^{t-1}$ contains the true target location $\boldsymbol{l}_T$, that is, $m \in \{1, 2\}$ is the index of the correct hypothesis set. Then the error probability at step $t$ i where $V_{\boldsymbol{k}}^t$ is the Voronoi cell corresponding to $\boldsymbol{h}_{\boldsymbol{k}}^

Figures (7)

  • Figure 1: Examples of decision trees for $n=8$ elements.
  • Figure 2: Error probability $P_e^t$ vs. overlap parameter $\alpha$ in a 1D setting, for increasing $n$ and fixed sensor location $s^t=\tfrac{L^{t-1}}{4}$. See Appendix \ref{['appx:plots']}.Appendix
  • Figure 3: Comparison of sensor positions optimized with respect to $P_e^t$ with an evolutionary algorithm and with uniform distribution described in \ref{['sec:num']}
  • Figure 4: Error probability in the final output of the algorithm vs. tree depth $\beta$.
  • Figure 5: Error probability $P_e^t$ vs. overlap parameter $\alpha$ with sensor locations specified by $s^t = \frac{L^{t-1}}{2}\left(\frac{1}{2} - \alpha\right)$.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1: Geometry of the Error Event
  • Theorem 1
  • Corollary 1
  • Corollary 2: 1D Error Probability $P_e^t$