On multidimensional generalization of binary search

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 $d$-dimensional domain $S_1\times\cdots\times S_d$, where $S_i=\{0, 1, \ldots,n_i-1\}$ and $d\geq 1$, in the following way. Given $(t_1,\ldots,t_d)$, the target element to be found, the result of a comparison of a selected element $(x_1,\ldots,x_d)$ is the sequence of inequalities each stating that either $t_i < x_i$ or $t_i>x_i$, for $i\in\{1,\ldots,d\}$, for which at least one is correct, and the algorithm does not know the coordinate $i$ 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 $\Omega(n^{d-1}/d)$ and $O(n^d)$ for the case of $n_i=n$. In particular, for fixed $d$ these bounds asymptotically do match. This problem is equivalent to the classical binary search in case of one dimension and shows interesting differences for higher dimensions. For example, if one would impose that each of the $d$ inequalities is correct, then the search can be completed in $\log_2\max\{n_1,\ldots,n_d\}$ queries. In an intermediate model when the algorithm knows which one of the inequalities is correct the sufficient number of queries is $\log_2(n_1\cdot\ldots\cdot n_d)$. The latter follows from a graph search model proposed by Emamjomeh-Zadeh et al. [STOC 2016].