Efficient Multidimensional Search: Generalizing Binary Search to Higher Dimensions

Core Concepts

This work generalizes the classical binary search problem to a d-dimensional domain, where the target element must be found using a minimum number of queries.

Abstract

The authors study a generalization of the classical binary search problem to a d-dimensional domain S1 × · · · × Sd, where Si = {0, 1, ..., ni - 1} and d ≥ 1. Given a target element (t1, ..., td), the result of a comparison of a selected element (x1, ..., xd) is the sequence of inequalities stating whether ti < xi or ti > xi for each i ∈ {1, ..., d}, where at least one inequality is correct but the algorithm does not know the specific dimension i on which the correct direction to the target is given. The authors provide a series of upper and lower bounds on the query complexity Q(n1, ..., nd) required to find the target: For the 2-dimensional case, they show that Q(m, n) ≤ 2n log2(m/n + 1) + 4, which is asymptotically tight. For the 3-dimensional case, they prove a lower bound of Ω(n2n3 log2(n1/n2 + 1)) and an almost matching upper bound of O(n2n3 log2(n1/n2 + 1)). For the general d-dimensional case, they show a lower bound of Ω(nd-1/d) and an upper bound of O(nd), leaving an open question about the exact query complexity Q(n×d) for an arbitrary dimension d. The problem is shown to be equivalent to the classical binary search in the 1-dimensional case and exhibits interesting differences in higher dimensions, such as the fact that if each of the d inequalities must be correct, the search can be completed in log2 max{n1, ..., nd} queries.

Stats

The number of queries needed to search one line segment in the 2-dimensional diagonal is at least 1 + ⌊log2 ⌊m/n⌋⌋. The number of queries needed to search the 3-dimensional plane H is at least 1/2 (n2 - 1) n3 log2((n1 - 1)/(n2 - 1) + 1). The number of queries needed to search the d-dimensional hyperplane H is at least 2⌊nd-1/(d-1)⌋.

Quotes

"One way to define the problem is to see it as an adaptive query-reply search game between two players called Algorithm and Adversary." "Besides a formal statement of our problem, we also give some remarks regarding alternative formulations." "We note that the majority of works focus solely on the query complexity. However, from the point of view of potential applications, it may be of interest to have an algorithm whose computational complexity per query is low."

Key Insights Distilled From

On multidimensional generalization of binary search

by Dari... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13193.pdf

Deeper Inquiries

How can the proposed algorithms be extended or adapted to handle noisy or uncertain comparisons in the multidimensional search setting

In the context of handling noisy or uncertain comparisons in multidimensional search settings, the proposed algorithms can be extended by incorporating probabilistic models or Bayesian inference techniques. By introducing probabilities to the comparison outcomes, the algorithms can adapt to uncertain information and make decisions based on the likelihood of each comparison being correct. This approach allows for more robust search strategies that can handle noisy data or imperfect comparisons. One way to implement this extension is to assign probabilities to the comparison outcomes, indicating the confidence level in each comparison result. The algorithm can then use these probabilities to update its beliefs about the target's location and make informed decisions on the next query. Bayesian updating can be employed to adjust the probabilities based on new information obtained from each comparison. Furthermore, techniques from machine learning, such as reinforcement learning, can be utilized to train the algorithm to optimize its query strategy in the presence of noisy comparisons. By learning from past interactions and feedback, the algorithm can improve its search efficiency and accuracy over time, even in the presence of uncertainty.

What are the implications of the query complexity bounds on the practical feasibility of the multidimensional search approach for real-world applications with varying dimensionality

The implications of the query complexity bounds on the practical feasibility of multidimensional search approaches for real-world applications are significant. The bounds provide insights into the minimum number of queries required to locate a target in a multidimensional space, which is crucial for optimizing search algorithms in various domains. For applications with varying dimensionality, the query complexity bounds offer a guideline for estimating the computational resources needed to perform efficient searches. Understanding the trade-off between the dimensionality of the search space and the query complexity helps in designing algorithms that are scalable and effective across different scenarios. In practical terms, the query complexity bounds can inform the development of search algorithms for tasks like image recognition, data retrieval, or optimization problems in high-dimensional spaces. By knowing the minimum number of queries needed to find a target, developers can tailor their algorithms to meet specific performance requirements and constraints. Overall, the query complexity bounds provide a theoretical foundation for multidimensional search algorithms, guiding their implementation and optimization for real-world applications with varying dimensionality.

Are there any connections between the multidimensional search problem and other areas of computer science, such as machine learning or data mining, that could lead to further insights or applications

The multidimensional search problem has connections to various areas of computer science, including machine learning and data mining, offering opportunities for cross-disciplinary insights and applications. In machine learning, the multidimensional search problem can be related to optimization tasks, where the goal is to find the optimal solution in a high-dimensional space. Search algorithms developed for multidimensional search can be adapted for tasks like hyperparameter tuning in machine learning models or feature selection in data analysis. In data mining, the multidimensional search problem aligns with the exploration of large datasets to discover patterns or relationships among data points. Efficient search algorithms can enhance data mining processes by enabling quick retrieval of relevant information from complex multidimensional datasets. Moreover, the concepts and techniques from the multidimensional search problem can be applied to spatial databases, geographic information systems, and recommendation systems, where efficient search strategies are essential for retrieving relevant information in large-scale datasets. By exploring these connections and leveraging insights from multidimensional search algorithms, researchers and practitioners can enhance the efficiency and effectiveness of various applications in machine learning, data mining, and related fields.