Algorithms and Hardness for Multidimensional Range Updates and Queries

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 arbitrarily large integer arrays in one or more dimensions. These problems allow range updates (such as filling all points 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 point 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 O?(n) time overall. Other than a few specific cases, until now, higher dimensional variants have been largely unexplored. Despite their tightly-knit complexity in one dimension, we show that variants induced by subsets of these operations exhibit polynomial separation in two dimensions. In particular, 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 general approaches to designing such algorithms that can be generalised to online and higher dimensional settings. First, we give almost-tight O?(n^{3/2}) time algorithms for single-update variants where the update and query operations meet a set of natural conditions. 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-?}) 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 O?(m^{(4?-1)/(2?+1)}) = O(m^1.478) time algorithm for counting simple 3-paths between vertex pairs

Fair Allocation of Two Types of Chores

We consider the problem of fair allocation of indivisible chores under additive valuations. We assume that the chores are divided into two types and under this scenario, we present several results. Our first result is a new characterization of Pareto optimal allocations in our setting, and a polynomial-time algorithm to compute an envy-free up to one item (EF1) and Pareto optimal allocation. We then turn to the question of whether we can achieve a stronger fairness concept called envy-free up any item (EFX). We present a polynomial-time algorithm that returns an EFX allocation. Finally, we show that for our setting, it can be checked in polynomial time whether an envy-free allocation exists or not