We give near-optimal algorithms for computing an ellipsoidal rounding of a convex polytope whose vertices are given in a stream. The approximation factor is linear in the dimension (as in John's theorem) and only loses an excess logarithmic factor in the aspect ratio of the polytope. Our algorithms are nearly optimal in two senses: first, their runtimes nearly match those of the most efficient known algorithms for the offline version of the problem. Second, their approximation factors nearly match a lower bound we show against a natural class of geometric streaming algorithms. In contrast to existing works in the streaming setting that compute ellipsoidal roundings only for centrally symmetric convex polytopes, our algorithms apply to general convex polytopes. We also show how to use our algorithms to construct coresets from a stream of points that approximately preserve both the ellipsoidal rounding and the convex hull of the original set of points.

翻译：我们提出近优算法，用于计算顶点以流式给出的凸多面体的椭球圆整。其近似因子在维度上呈线性关系（符合约翰定理），且仅在多面体的长宽比上损失一个额外的对数因子。我们的算法在两个方面近乎最优：首先，其运行时间几乎与已知最高效的离线版本算法相匹配；其次，其近似因子几乎达到我们对一类几何流式算法所证明的下界。与现有仅针对中心对称凸多面体计算椭球圆整的流式算法不同，我们的算法适用于一般凸多面体。我们还展示了如何利用这些算法从点流中构建核心集，这些核心集能近似保留原始点集的椭球圆整和凸包。