On multidimensional generalization of binary search

TL;DR

This paper extends binary search to a $d$-dimensional grid $\mathcal{S}=S_1\times\cdots\times S_d$, where each query $q$ yields a vector of inequalities for the target $t$, with the index of the correct coordinate unknown. It establishes asymptotically tight bounds on the query complexity $Q(n_1,\dots,n_d)$, showing $Q(n^{\times d})=\Omega\bigl(\dfrac{n^{d-1}}{d}\bigr)$ and $Q(n^{\times d})=O(n^d)$ for equal sizes, with matching behavior for fixed $d$; special cases include $\log_2\max\{n_i\}$ when all inequalities are correct and $\log_2(n_1\cdots n_d)$ when the correct inequality is known. The authors derive a tight 2D bound, and provide lower bounds via adversarial plane/hyperplane strategies in higher dimensions, complemented by an inductive upper-bound construction using byproduct grids. Overall, the work clarifies how multidimensionality affects search complexity, connects to graph-search models like STOC 2016 Emamjomeh-Zadeh, and delineates regimes where the multidimensional problem approaches or diverges from classical 1D binary search. These insights have implications for multi-criteria search and decision-making scenarios where multiple directional comparisons provide partial but incomplete information.

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 $Ω(n^{d-1}/d)$ 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].

Figures (4)

• Figure 1: A diagonal on which Adversary hides the target.
• Figure 2: Possible configuration after the binary search (with $m_{10} = m$).
• Figure 3: Points that have to be searched to find the target on the grid with dimensions $6 \times 5 \times 4$.
• Figure 4: A view from the top of the grid with dimensions $6 \times 5 \times 4$ (each color is a different level of the grid).

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2