Longest Increasing Subsequence (LIS) is a fundamental problem in combinatorics and computer science. Previously, there have been numerous works on both upper bounds and lower bounds of the time complexity of computing and approximating LIS, yet only a few on the equally important space complexity. In this paper, we further study the space complexity of computing and approximating LIS in various models. Specifically, we prove non-trivial space lower bounds in the following two models: (1) the adaptive query-once model or read-once branching programs, and (2) the streaming model where the order of streaming is different from the natural order. As far as we know, there are no previous works on the space complexity of LIS in these models. Besides the bounds, our work also leaves many intriguing open problems.

翻译：最长递增子序列（LIS）是组合数学与计算机科学中的基础问题。此前，已有大量工作研究计算与近似LIS的时间复杂度上界与下界，但仅有少数工作关注同样重要的空间复杂度。本文将进一步研究多种模型下计算与近似LIS的空间复杂度。具体而言，我们在以下两个模型中证明了非平凡的空间下界：（1）自适应单次查询模型或只读分支程序，以及（2）流顺序与自然顺序不同的流模型。据我们所知，此前尚无关于LIS在这些模型中空间复杂度的工作。除界限外，我们的工作还遗留了许多引人入胜的开放问题。