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].

翻译：本文将二分搜索问题推广到$d$维域$S_1\times\cdots\times S_d$，其中$S_i=\{0, 1, \ldots, n_i-1\}$且$d\geq 1$，具体方式如下：给定待查找目标元素$(t_1,\ldots,t_d)$，对选定元素$(x_1,\ldots,x_d)$进行比较的结果是一系列不等式，每个不等式表明对于$i\in\{1,\ldots,d\}$，要么$t_i < x_i$，要么$t_i > x_i$，其中至少有一个不等式正确，且算法不知道给出正确目标方向的坐标$i$。在其他情况中，我们展示了当$n_i=n$时，查询复杂度的下界和上界渐近几乎匹配，分别为$\Omega(n^{d-1}/d)$和$O(n^d)$。特别地，对于固定的$d$，这些界渐近完全匹配。该问题在一维情况下等价于经典二分搜索，并在更高维度中展现出有趣的差异。例如，若强制要求所有$d$个不等式均正确，则可在$\log_2\max\{n_1,\ldots,n_d\}$次查询内完成搜索。在另一种中间模型中，当算法知道哪个不等式正确时，所需查询次数为$\log_2(n_1\cdot\ldots\cdot n_d)$。后者源于Emamjomeh-Zadeh等人[STOC 2016]提出的图搜索模型。