## Abstract

Consider a network of \(n\) processes, where each process inputs a \(d\)-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The *multidimensional Byzantine consensus* problem, for synchronous systems, requires processes to decide on a single \(d\)-dimensional vector \(v \in {\mathbb {R}}^d\), inside the *convex hull* of \(d\)-dimensional vectors that were input by the non-faulty processes. Also, the *multidimensional Byzantine approximate agreement* (MBAA) problem, for asynchronous systems, requires processes to decide on multiple \(d\)-dimensional vectors in \({\mathbb {R}}^d\), all within a fixed Euclidean distance \(\epsilon \) of each other, and inside the convex hull of \(d\)-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to \(f\) Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, \(n > \max \{3f, (d+1)f\}\) is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, \(n > (d+2)f\) is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.

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## Notes

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(M. Herlihy) Supported by NSF 0830491. (N. Vaidya and V. K. Garg) This research is supported in part by National Science Foundation awards CNS-1059540 and CNS-1115808, and the Cullen Trust for Higher Education. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the funding agencies or the U.S. government.

## Appendix: Tverberg’s Theorem

### Appendix: Tverberg’s Theorem

###
**Theorem 8**

(Tverberg’s Theorem [22, 26]) For any integer \(f\) with \(f \ge 1\) and any multiset \(X\) containing at least \((d+1)f+1\) points in \({\mathbb {R}}^d\), there exists a partition \(X_1,\ldots , X_{f+1}\) of \(X\) into \(f+1\) non-empty multisets such that \(\cap _{1 \le x \le f+1} {\mathrm {Poly}}({X_x}) \ne \emptyset \).

The points in the above multiset \(X\) are not necessarily distinct [22, 26], so the same point may occur multiple times in \(X\). Any partition in Theorem 8 is called a *Tverberg partition*, and the points in \(\cap _{1 \le x \le f+1} {\mathrm {Poly}}({X_x})\) in Theorem 8 are called *Tverberg points*. It can be shown that any Tverberg point of a particular multiset \(X\) is necessarily within the safe area \({\mathrm {Safe}}_{f}({X})\) [27].

Algorithm 5 requires non-faulty processes to choose identically *any point* in \(S\) as the output vector. The deterministic procedure in the algorithm could therefore return a Tverberg point. For arbitrary \(d\), no algorithm to compute a Tverberg point of an arbitrary multiset is currently known with polynomial complexity [2, 18, 19]. However, in some restricted cases, efficient algorithms *are* known (e.g., [15]).

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Mendes, H., Herlihy, M., Vaidya, N. *et al.* Multidimensional agreement in Byzantine systems.
*Distrib. Comput.* **28**, 423–441 (2015). https://doi.org/10.1007/s00446-014-0240-5

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DOI: https://doi.org/10.1007/s00446-014-0240-5