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We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O (( k +1)log 2 n ) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x -monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O (log 2 n ) time; for a query point x 0 ∈ ℝ, it can report, in O (log n ) time, the point on the lower envelope with x -coordinate x 0 ; and for a query point q ∈ ℝ 2 , it can return all k pseudo-lines lying below q in time O (log n + k log 2 n ). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D , all input arcs intersecting D can be reported in O ( n 1/2+ɛ + k ) time, where k is the output size and ɛ > 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O (log 2 n ) time. 

We model the societal task of redistricting political districts as a partitioning problem: Given a set of n points in the plane, each belonging to one of two parties, and a parameter k, our goal is to compute a partition P of the plane into regions so that each region contains roughly s = n/k points. P should satisfy a notion of "local" fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in P if it belongs to the minority party. A group D of roughly s contiguous points is called a deviating group with respect to P if majority of points in D are unhappy in P. The partition P is locally fair if there is no deviating group with respect to P.This paper focuses on a restricted case when points lie in 1D. The problem is non-trivial even in this case. We consider both adversarial and "beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are "runs" of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of s, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists. 

Let T be a set of n planar semi-algebraic regions in R^3 of constant complexity (e.g., triangles, disks), which we call _plates_. We wish to preprocess T into a data structure so that for a query object gamma, which is also a plate, we can quickly answer various intersection queries, such as detecting whether gamma intersects any plate of T, reporting all the plates intersected by gamma, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in R^3 (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in R^3. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if T is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^(4/3)) storage (where the O^*(...) notation hides subpolynomial factors) and answers an intersection query in O^*(n^(2/3)) time. Alternatively, by increasing the storage to O^*(n^(3/2)), the query time can be decreased to O^*(n^(rho)), where rho = (2t-3)/(3(t-1)) < 2/3 and t≤3 is the number of parameters needed to represent the query arcs.