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 $O(m^{3/2-ε})$ 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 $\tilde{O}(m^{(4ω-1)/(2ω+1)}) = O(m^{1.478})$ time algorithm for counting simple 3-paths between vertex pairs.

For any fixed measure $H$ that maps graphs to real numbers, the MinH problem is defined as follows: given a graph $G$, an integer $k$, and a target $τ$, is there a set $S$ of $k$ vertices that can be deleted, so that $H(G - S)$ is at most $τ$? In this paper, we consider the MinH problem on trees. We call $H$ "balanced on trees" if, whenever $G$ is a tree, there is an optimal choice of $S$ such that the components of $G-S$ have sizes bounded by a polynomial in $n/k$. We show that MinH on trees is FPT for parameter $n/k$, and furthermore, can be solved in subexponential time, and polynomial space, if $H$ is additive, balanced on trees, and computable in polynomial time. A measure of interest is the Inverse Geodesic Length (IGL), which is used to gauge the connectedness of a graph. It is defined as the sum of inverse distances between every two vertices: $IGL(G)=\sum_{\{u,v\} \subseteq V} \frac{1}{d_G(u,v)}$. While MinIGL is W[1]-hard for parameter treewidth, and cannot be solved in $2^{o(k+n+m)}$ time, even on bipartite graphs with $n$ vertices and $m$ edges, the complexity status of the problem remains open on trees. We show that IGL is balanced on trees, to give a $2^{O((n\log n)^{5/6})}$ time, polynomial space algorithm. The distance distribution of $G$ is the sequence $\{a_i\}$ describing the number of vertex pairs distance $i$ apart in $G$: $a_i=|\{\{u, v\}: d_G(u, v)=i\}|$. We show that the distance distribution of a tree can be computed in $O(n\log^2 n)$ time by reduction to polynomial multiplication. We extend our result to graphs with small treewidth by showing that the first $p$ values of the distance distribution can be computed in $2^{O(tw(G))} n^{1+\varepsilon} \sqrt{p}$ time, and the entire distance distribution can be computed in $2^{O(tw(G))} n^{1+\varepsilon}$ time, when the diameter of $G$ is $O(n^{\varepsilon'})$ for every $\varepsilon'>0$.