Traditional orthogonal range problems allow queries over a static set of points, each with some value. Dynamic variants allow points to be added or removed, one at a time. To support more powerful updates, we introduce the Grid Range class of data structure problems over integer arrays in one or more dimensions. These problems allow range updates (such as filling all cells in a range with a constant) and queries (such as finding the sum or maximum of values in a range). In this work, we consider these operations along with updates that replace each cell in a range with the minimum, maximum, or sum of its existing value, and a constant. In one dimension, it is known that segment trees can be leveraged to facilitate any $n$ of these operations in $\tilde{O}(n)$ time overall. Other than a few specific cases, until now, higher dimensional variants have been largely unexplored. We show that no truly subquadratic time algorithm can support certain pairs of these updates simultaneously without falsifying several popular conjectures. On the positive side, we show that truly subquadratic algorithms can be obtained for variants induced by other subsets. We provide two approaches to designing such algorithms that can be generalised to online and higher dimensional settings. First, we give almost-tight $\tilde{O}(n^{3/2})$ time algorithms for single-update variants where the update operation distributes over the query operation. Second, for other variants, we provide a general framework for reducing to instances with a special geometry. Using this, we show that $O(m^{3/2-\epsilon})$ time algorithms for counting paths and walks of length 2 and 3 between vertex pairs in sparse graphs imply truly subquadratic data structures for certain variants; to this end, we give an $\tilde{O}(m^{(4\omega-1)/(2\omega+1)}) = O(m^{1.478})$ time algorithm for counting simple 3-paths between vertex pairs.

中文翻译：

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多维范围更新和查询的算法和难度

传统的正交范围问题允许查询一组静态的点，每个点都有一些值。动态变体允许一次添加或删除点。为了支持更强大的更新，我们在一个或多个维度上针对整数数组引入了数据结构问题的网格范围类。这些问题允许范围更新（例如用常数填充范围内的所有单元格）和查询（例如查找范围内值的总和或最大值）。在这项工作中，我们将考虑这些操作以及将当前范围的最小值，最大值或总和以及一个常数替换为一个范围内的每个单元格的更新。在一个维度上，已知可以利用段树来促进这些操作中任何$ n $的时间，总共$ \ tilde {O}（n）$。到目前为止，除了一些特殊情况，尺寸较大的变体尚未开发。我们证明，在不伪造一些流行的猜想的情况下，没有真正的二次时间算法可以同时支持这些更新的某些对。从积极的方面来看，我们表明对于由其他子集引起的变体，可以获得真正的二次方程式算法。我们提供了两种设计此类算法的方法，这些方法可以推广到在线和更高维度的设置。首先，我们为单更新变体提供几乎紧密的$ \ tilde {O}（n ^ {3/2}）$时间算法，其中更新操作分布在查询操作上。其次，对于其他变体，我们提供了用于简化为具有特殊几何形状的实例的通用框架。使用这个 我们表明，用于计算稀疏图中顶点对之间长度为2和3的路径和步长的$ O（m ^ {3 / 2- \ epsilon}）$时间算法意味着某些变量确实是次二次数据结构；为此，我们给出了一个\\ tilde {O}（m ^ {（4 \ omega-1）/（2 \ omega + 1）}）= O（m ^ {1.478}）$用于计算简单3的时间算法-顶点对之间的路径。