On multidimensional generalization of binary search
Dariusz Dereniowski, Przemys{\l}aw Gordinowicz, Karolina Wr\'obel

TL;DR
This paper extends binary search to multiple dimensions, analyzing the query complexity for locating a target in a $d$-dimensional grid with partial information, providing bounds that match asymptotically for fixed dimensions.
Contribution
It introduces a multidimensional generalization of binary search, establishing asymptotically tight bounds on query complexity for various informational models.
Findings
Asymptotically matching lower and upper bounds of $oldsymbol{rac{oldsymbol{n^{d-1}}}{d}}$ and $oldsymbol{n^d}$ for query complexity.
Special cases where all inequalities are correct or only one is known, leading to logarithmic query bounds.
Demonstrates differences between classical binary search and the multidimensional generalization.
Abstract
This work generalizes the binary search problem to a -dimensional domain , where and , in the following way. Given , the target element to be found, the result of a comparison of a selected element is the sequence of inequalities each stating that either or , for , for which at least one is correct, and the algorithm does not know the coordinate on which the correct direction to the target is given. Among other cases, we show asymptotically almost matching lower and upper bounds of the query complexity to be in and for the case of . In particular, for fixed these bounds asymptotically do match. This problem is equivalent to the classical binary search in case of one dimension and shows interesting…
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Taxonomy
TopicsData Management and Algorithms · Metaheuristic Optimization Algorithms Research · Fuzzy Systems and Optimization
